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Open Channel Flow Ansys Cfd Project | Design & Implementation

I

CFD Modelling of Turbulent Flow in Open-Channel Expansions

Building, Civil, And Environmental Engineering

Presented in Partial Fulfillment of the Requirements for the Degree of Master of Applied Science in Civil Engineering at

Abstract

CFD Modelling of Turbulent Flow in Open-Channel Expansions Sahar Najmeddin, M. A. Sc. Channel expansions provide a transition from a narrow to a relatively wide channel- section, which is necessary in many hydraulic structures. In the transition, flow tends to separate from its diverging sidewalls and create turbulent eddies, if the angle of divergence exceeds a threshold value. This phenomenon can cause undesirable flow energy losses and erosion to the sidewalls locally and even further downstream. Previously, researchers have tried to optimise the transition’s horizontal shape in order to reduce flow separation; the results are inconclusive. The purpose of this study is to extend earlier investigations about fitting a hump in the vertical to eliminate flow separation. This study uses the CFD modelling approach. This approach permits an efficient and systematic exploration of the effects of different angles of divergence, crest height of the hump and the Froude number of subcritical flow. The model results are validated using existent analytical solutions under simplified conditions and available experimental data for a limited number of cases. Flow quantities presented in this thesis include details of the velocity, vorticity, eddy structure, and cross-sectional area of flow reversal; these quantities are distributed at selected vertical and horizontal planes, and are available for the cases of with and without a hump. It is shown that the use of a hump effectively reduces flow separation and eddy motion in the transition. This is because the flow is forced to accelerate over the hump, and as a result, the otherwise adverse pressure gradient, which is known to be responsible for flow separation, diminishes. A hump in the vertical can easily be incorporated into the bed of existent channel expansions, and would be less expensive to construct than to modify the horizontal shape (or the sidewalls) of existent expansions. The results presented in this thesis are of practical values for the optimal design of humps.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

Iv

First of all, I would like to thank my supervisor, Dr Samuel Li, for his guidance and supervision Li for the past two years and I take this opportunity to express my gratitude for his support. Also, I would like to thank the examiners Dr. M. Paraschivoiu and Dr. L. Wang as well as the thesis committee chair Dr. A. Zsaki, for their comments and suggestions that helped prepare the version of this thesis.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

Last but not least, I would like to thank my dear parents and my dear sister for their endless love and support through the whole duration of my studies. Also, these achievements would not be possible without my beloved husband’s unconditional love and support.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

List of Figures..............................................................................................................................viii List of Tables.................................................................................................................................xii List of Symbols.............................................................................................................................xvi CHAPTER ONE INTRODUCTION........................................................................................1 1.1 Background…………....................................................................................................1 1.2 Specific objective...........................................................................................................2 1.3 Scope of the work..........................................................................................................3 CHAPTER TWO LITERATURE REVIEW..........................................................................5 2.1 Experimental studies of subcritical flow in expansions..............................................5 2.2 Channel expansions fitted with a local hump................................................................5 2.3 Flow patterns in a channel expansion............................................................................6 2.4 The effect of solid surface friction……………………...............................................12 2.5 Geometric shape of expansions...................................................................................12 2.6 Energy loss analysis for expansions............................................................................13 2.8 Hydraulic behaviour of channel expansions................................................................15 CHAPTER THREE MODELLING METHODOLOGIES..................................................17 3.1 Model domain..............................................................................................................17 3.2 Reynolds-averaged continuity and momentum equations...........................................19

Vi

3.3 Turbulence models.......................................................................................................21 3.4 The volume of fluid method...................................................................................... 26 3.5 Boundary conditions....................................................................................................26 3.6 Initial conditions..........................................................................................................28 3.7 Finite volume meshes..................................................................................................29 CHAPTER FOUR THE ENERGY PRINCIPLE .................................................................35 4.1 Energy balance for flow in expansions........................................................................35 4.2 The concept of specific energy....................................................................................37 4.3 E-y curve......................................................................................................................37 4.4 Critical flow and the concept of the Froude number...................................................38 4.5 Flow over a step in the vertical (hump).......................................................................40 4.6 Flow in a combination of a horizontal expansion and a vertical step..........................42 4.7 Limitations on the use of the energy principle.............................................................42 CHAPTER FIVE RESULTS...........................................................................................……44 5.1 The model channel.......................................................................................................44 5.2 Considerations of mesh resolution and downstream channel extension ….................49

R

5.6 Velocity field at different Froude numbers..................................................................65 5.7 Vorticity field...............................................................................................................71

Vii

5.8 u-velocity contours at different cross-sections….….…………………………..……80 5.9 Comparison of model results with theoretical and experimental data …………...….92 CHAPTER SIX DISCUSSIONS AND CONCLUSION...........................................................94 6.1 Discussions..................................................................................................................94 6.2 Conclusion...................................................................................................................96 6.3 Suggestion for future research.....................................................................................98 REFERENCES.............................................................................................................................99

Figure 2.1

Streamlines computed for flow in a 1:3 symmetric sudden expansion. The Reynolds number Re is: (a) 40, (b) 50, (c) 80, and (d) 140. The flow patterns are

In (B-D) (From

Manica and de Bortoli, 2003)...................................................................................8 4).

The

centerline..................................................................................................................13

Figure 3.1

Geometry of the model channel, showing dimensions of various channel sections and bottom configurations. Uniform water flow enters (arrow) the model channel from the relatively narrow channel section............................................................18

Figure 3.2

A sample finite volume mesh system used for flow computations, showing the inflation of meshes near all solid walls. The solid surface on the top is set to be slippery...................................................................................................................30

Figure 4.1

Plan view of a channel expansion..........................................................................35

Figure 4.2

The specific energy curve and its application in the expansion problem…………….....38

Figure 4.3

Water surface profile for flow over a vertical step (hump) fitted on the bottom of a uniform channel. The depth of flow decreases over the hump on the basis of the energy Principle. From cross section CS 2 to cross section CS 3, the bed level rises and the water pressure decreases; from cross section CS 3 toward downstream, the water pressure increases while the bed level drops.................................................41

Figure 4.4

Specific energy diagram for a channel expansion.................................................41

Figure 5.1

Figure 5.1 (a) Velocity vectors at an elevation of 230 mm above the channel bed for Run FB4 for the Froude number of 0.5. The maximum velocity is 0.86 m/s. (b) A close up of velocity vectors in (a). Regions of eddies are observed near both sidewalls...................................................................................................................52

Figure 5.2

Flow streamlines at an elevation of 230 mm above the channel bed in the expansion for Run FB4. The Froude number is 0.5..................................................................53

Figure 5.3

Velocity vectors in the expansion at an elevation of 200 mm above the channel bed for Run FB4. The Froude number is 0.5. The maximum velocity is 0.89m/s.........56

Figure 5.4

A comparison of velocity vector around the exit of the expansion between a flat bottom channel (a) and a channel with a 1/2" hump (b). The elevation is 200 mm above the channel bed............................................................................................58

Figure 5.5

Velocity streamlines at an elevation of 200 mm above the channel bed for Runs FB4 and HH1. The Froude number is 0.5................................................................59

Figure 5.6

Velocity streamlines at an elevation of 230 mm above the channel bed for Run FB4. The Froude number is 0.7................................................................................68

Figure 5.7

Velocity streamlines at an elevation of 200 mm above the channel bed for Runs FB4 (panel a) and HH1 (panel b).The Froude number is 0.7...................................70 Figure 5.8 Velocity vectors and associated vorticity in the expansion. The area occupied by eddies is shown as high-vorticity region near the sidewall, in red color. The maximum velocity is 0.86 m/s. The results are for Run FB4……………………72 Figure 5.9 Vorticity contours at an elevation of 200 mm above the channel bed for a flat- bottom expansion, (FB4).......................................................................................73

X

Figure 5.10 Vorticity contours at an elevation of 200 mm above the channel bed, for the expansion with a 1/2" hump (HH1).......................................................................73 Figure 5.11 Vorticity contours at an elevation of 200 mm above the channel bed for a flat bottom expansion (FB4) .The Froude number is 0.3.............................................76 Figure 5.12 Vorticity contours for flow at an elevation of 200 mm above the channel bed for a flat bottom expansion. The Froude number is 0.3 (HH1)......................................78 Figure 5.13 Vorticity contours at an elevation of 200 mm above the channel bed for a flat bottom expansion. The Froude number is 0.7 (FB4)............................................................79 Figure 5.14 Vorticity contours at an elevation of 200 mm above the channel bed for a flat bottom expansion. The Froude number is 0.7(HH1).............................................80 Figure 5.15 Percentage in area of backward flow in a flat-bottom expansion (FB4), and expansions with a hump (HQ1and HH1)...............................................................82 Figure 5.16 u-velocity contours for Run FB4 at Cross-section 7(before the exit of the

Expansion) ……………… ………………………………………………………83

Figure 5.17 u-velocity contours for Run FB4 at Cross-section 7(before the exit of the

Expansion) ……………… ………………………………………………………83

Figure 5.18 u-velocity contours for Run FB4 at Cross-section 4(at the middle of the

Expansion)…………………………………………………………………….…..84

Figure 5.19 u-velocity contours for Run HH1 at Cross-section 4 (at the middle of the

Expansion)……………………………………………………………..…….........84

Figure 5.20 u-velocity contours for Run FB5 at Cross-section 3 (before the middle of the

Xi

Figure 5.21 u-velocity contours for Run HH2 at Cross-section 3 (before the middle of the

Expansion)……………………………………………………………….……….. 86

Figure 5.22 u-velocity contours for Run FB6 at Cross-section 7 (before the exit of the

Expansion)………………………………………………………………….…......87

Figure 5.23 Percentage in area of backward flow at different cross sections for a flat-bottom expansion (FB4), at different Froude numbers......................................................88 Figure 5.24 Percentage in area of backward flow at different cross-sections for a flat-bottom expansion (FB4), and expansions with a hump (HQ1 and HH1)..........................90 Figure 5.25 u-velocity contours for Run FB4 at Cross-section 8 ( the exit of the expansion).The Froude number is 0.7....................................................................91 Figure 5.26 u-velocity contours for Run HQ1 at Cross-section 8 (the exit of the expansion).The Froude number is 0.7...................................................................91

St Of Tables

Table 3.1 Mesh spacing (Δy) for various target values of the dimensionless wall distance (y+).........................................................................................................................32 Table 5.1 Geometric properties of channel expansions used in 13 model runs, for which the Froude number is equal to 0.5.These parameters, include upstream channel width (B1), downstream channel width (B2), upstream channel length (L1), expansion length (L2), downstream channel length (L3), expansion angle(α), downstream extension length (L4), hump crest height (H), and mesh resolution(∆x)............... 45 Table 5.2 The width ratio for different divergence angles......................................................46

Table 5.3

Parameters and boundary specifications for nine model runs with the Froude number of 0.5.Solid walls are taken as non-slippery .Turbulence model used is k-ω..........................................................................................................................48

Table 5.4

Average velocities at different elevations above the channel bed, before and after the entrance to the expansion for Run FB4............................................................54

Table 5.5

Minimum velocities at different elevations above the channel bed, before and after the entrance to the expansion for FB4...........................................................54

Table 5.6

Percentage in area of eddies at different elevations above the channel bed, in the expansion and downstream of the expansion for Run FB4...................................55

Table 5.7

Percentage in area of eddies near the right and left sidewalls in the expansion for Run FB4.................................................................................................................55

Table 5.8

Average and minimum velocities before and after the entrance to the expansion for a flat-bottom expansion, and expansions with a hump for the elevation of 200 mm above channel bed. .........................................................................................56 Table 5.9 Percentage in area of eddies in the expansion and between the downstream end of the expansion and downstream of the model channel for a flat-bottom expansion, and expansions with a hump…………………………………………………......57 Table 5.10 Average velocities at different elevations above the channel bed, before and after the entrance to the expansion for Run FB5............................................................60 Table 5.11 Minimum velocities at different elevations above the channel bed, before and after the entrance to the expansion for Run FB5...................................................61 Table 5.12 A comparison of velocity drops between Runs FB4 and FB5...............................61 Table 5.13 Percentage in area of eddies at different elevations above the channel bed, in the expansion and downstream for Run FB5.............................................62 Table 5.14 Percentage in area of eddies near the right and left sidewalls in the expansion for Run FB5.................................................................................................................62

Table 5.15

Average and minimum velocities before and after the entrance to the expansion for a flat-bottom channel (FB5) and channels with a hump for the elevation of 230 mm above the channel bed.....................................................................................63 Table 5.16 Percentage in area of eddies in the expansion and between the downstream end of the expansion and downstream end of the model channel for flat-bottom channel (FB5) and channels with a hump for elevation of 230 mm above the channel bed..........................................................................................................................64

Xiv

Table 5.17 Average and minimum velocities before and after the entrance to the expansion for a flat-bottom expansion (FB6), and expansions with a hump (HQ3 and HH3)for elevation of 200 mm above the channel bed...........................................65 Table 5.18 Percentage in area of eddies in the expansion and between the downstream end of the expansion and downstream of the model channel for a flat-bottom expansion (FB6), and expansions with a hump (HQ3and HH3)for the elevation of 200 mm above the channel bed...........................................................................................65

Table 5.19

Percentage in area of eddies at different elevations above the channel bed, in the expansion and downstream for Run FB4 at the Froude number

Table 5.20

Percentage in area of eddies in the expansion and between the downstream end of the expansion and downstream of the model channel for a flat-bottom expansion, and expansions with a hump, for the Froude number of 0.3 for the elevation of 200 mm above the channel bed..............................................................................67 Table 5.21 Percentage in area of eddies at different elevations above the channel bed, in the expansion and downstream for Run FB4 for the Froude number of 0.7...............68 Table 5.22 Percentage in area of eddies in the expansion and between the downstream end of the expansion and downstream of the model channel for a flat-bottom expansion, and expansions with a hump, for the Froude number of 0.7 for elevation of 200 mm above the channel bed.....................................................................................69 Table 5.23 Percentage in area of high vorticity area, at different elevations above the channel bed for Run FB4.....................................................................................................74 Table 5.24 Percentage in area of high vorticity at different elevations above the channel bed for Run FB5..................................................................................................................74

Xv

Table 5.25 Percentage in area of high vorticity at different elevations above the channel bed for Run FB6...........................................................................................................75 Table 5.26 A comparison of percentages in area of high-vorticity between a flat-bottom expansion and expansion with a hump, at different angles of divergence at an elevation of 200 mm above the channel bed. ........................................................75 Table 5.27 Percentage in area of high vorticity at different elevations above the channel bed for Run FB4and the Froude number of 0.3............................................................77 Table 5.28 Percentage in area of the high-vorticity for flat-bottom expansion and expansions with a hump. The Froude number is 0.3.The elevation is 200 mm above the channel bed ………….…......................................................................................78 Table 5.29 A comparison of percentage in area of high vorticity at different elevations above the channel bed for Runs FB4, HQ1and HH1 for the Froude number of 0.7........81 Table 5.30 Percentage in area of backward flow at different cross-sections for a flat-bottom expansion (FB5) and expansions with a hump (HQ2 and HH2)...........................85 Table 5.31 Comparison of energy loss coefficient kE between numerical model and

Experiments …….………………………………………………………………...92

Table 5.32 A comparison of the energy loss coefficient between this modelling study and the theoretical analysis by Henderson (1966)………………………………………..93

B = Upstream Channel Width At Cs1 (M)

2b = channel width at the entrance to an expansion, CS 2 (m) 3b = channel width at the exit of an expansion, CS 3(m)

D

/ = percentage in area occupied by eddies between the downstream end of an expansion and

Ex

E = percentage in area occupied by eddies in an expansion (%) lE = percentage in area occupied by eddies to the left sidewall (to an observer facing

R

E = percentage in area occupied by eddies to the right sidewall (to an observer facing

Rf = Froude Number ( - )

h = vertical distance between the water surface and the point of interest (m) eh = energy loss due to flow separation and eddy motion (m)

T

U =velocity tangent to the wall at a distance of y

A

V = average velocity after the entrance to an expansion (

B

V = average velocity before the entrance to an expansion (

A

Vmin, = minimum velocity after the entrance to an expansion (

B

Vmin, = minimum velocity before the entrance to an expansion (

Or

= percentage in area of high-vorticity region at the entrance to an expansion (%)

Or

=percentage in area of high-vorticity region to the left of an expansion (to an observer

Or

= percentage in area of high-vorticity region to the right of an expansion (to an observer

Y = Depth Of Flow (M)

1y = depth of flow upstream of an expansion or at CS 1 (m) 2y = depth of flow at the entrance to an expansion or at CS 2 (m)

Background

In general, channel transitions are defined as changes in cross-sectional area in the direction of open channel flow. Transitions can also include changes in bed level. A channel expansion is a transition that connects a relatively narrow upstream channel-section to a wide downstream channel-section. Such a transition is an important component of many hydraulic structures. Due to an increase in cross-sectional area, channel expansions cause flowing water to decelerate.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

Under steady flow conditions, flow deceleration will lead to an increase in water pressure that in turn triggers flow separation and creates turbulent eddy motions. These turbulent eddy motions can exist over a long distance downstream of the transition. They cause undesirable energy losses and sidewall erosion.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

In order to control energy losses and erosion, we need to improve our understanding of the problem of flow in a channel expansion. This problem has not been thoroughly investigated in the past. A lack of thorough knowledge about the problem has motivated this study. From the energy conservation perspective, it is particularly important to be able to reduce or even eliminate areas of turbulent eddy motions in the expansion.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

The problem of turbulent flow in an open channel expansion is very complicated, with turbulent eddy motions, flow separation and so on. It is difficult to use the analytical approach to obtain solutions to the problem even under highly simplified conditions. The experimental approach may be taken to tackle the problem, but experiments are very expensive to carry out.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

Therefore, it may not be feasible to experimentally investigate the effects of many factors that potentially control the behaviour of flow in a channel expansion. In this study, the CFD modelling approach is taken. This approach permits an efficient and systematic exploration of the effects of such factors as the angle of divergence, the crest height of a hump fitted at the channel bed and the Froude number on the flow in an expansion. The idea of using a hump in the vertical to reduce eddy motions and flow separation in expansions is interesting, because it is an easier and more economic solution compared to optimising the horizontal shape (or the sidewalls) of existing expansions.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

This study focuses on subcritical flows (with the Froude number less than unity) as they prevail in open channels. A number of important questions need to be answered. How does the flow field, in particular the distribution of eddy motions, vary with the angle of divergence? How does the velocity field change with elevation above the channel bed? In what way we can evaluate flow reversal in the expansion? Qualitatively, the use of a hump fitted at the channel bed is known to help reduce flow reversal and eddy motions, but quantitatively, how efficient is the use of it? Will answers to the above questions be different at different Froude numbers?

The Objectives Of This Study Are

 to numerically simulate subcritical flow in channel expansions with different angles of divergence. We will consider an angle of divergence equal to 10.34º, 7.54º and 5.04º in order to match experimental conditions. This will allow data comparison.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

 to quantify areas of eddy motions and flow reversal in expansions (at different cross- sections and longitudinal planes).  to investigate the effects of the Froude number.

Figure: System Model & Architecture for Open Channel Flow Ansys Cfd

 to determine the effectiveness of fitting a hump at the channel bed to suppress eddy motions and flow separation. We will consider two simple humps: one with a crest

). These Selections Are Consistent With

experimental setup used in previous studies.

Scope Of The Work

To achieve the above-mentioned objectives, the rest of this thesis is organized as follows. Chapter 2 gives a summary of previous studies on the topic of flow in expansions, including experimental and analytical studies on flow separation and the formation of turbulent eddies.

Previous works on other established facts concerning the design of hydraulically efficient channel expansions are reviewed in this chapter. This chapter also summarises previous studies about the effects of a hump in the vertical on the reduction of eddy motions and flow reversal.

Chapter 3 describes the modelling methodologies used in this study. This chapter provides the theoretical background and fundamental concepts of CFD modelling of free surface flow. The k-ω turbulence model is explained in details, and reasons for choosing this model are discussed. Details of the boundary conditions, including conditions at inlet, outlet, bottom and sidewalls of the model channel are discussed. Also, the choice of meshes, including volume mesh and inflations (near bottom and sidewalls) for accuracy improvement is discussed in this chapter.

In Chapter 4, the energy principle and the concept of specific energy for flow in channel expansions are discussed, together with the concept of E-y curve. The expected behaviour of flow in channel expansions and over a hump is also explained.

Chapter 5 presents the numerical results of the flow field in a flat-bottom expansion and

Z

) hump. The results include velocity, flow streamlines, vorticity contours, and along-channel velocity contours at different cross-sections along the length of expansions. In longitudinal planes at different elevations above the channel bed, areas of eddy motions are determined. In cross-sections, eddy motions are evaluated as regions where flow reversal occurs.

Percentages in area of low-velocities and high-vorticity areas are evaluated. A comparison of the percentages between a flat-bottom expansion and expansions with a 1/4″ (or 0.00635 m, with

), Are Made. Percentages In Area Of

backward flow (flow reversal) at different cross-sections are evaluated and compared. This chapter also discusses the effects of the Froude number, ranging from 0.3 to 0.7. In chapter 6, we draw conclusions and make suggestions for future work.

Iterature Review

2.1 Experimental studies of subcritical flow in expansions Previously, a number of researchers have experimentally studied the problem of subcritical flow in expansions. Alauddin and Basak (2006) took the experimental approach to study flow separation in an expanding transition and further downstream. The purpose was to design an expanding transition with minimum flow separation and hence small energy head losses. The authors made measurements of velocity profiles at the inlet, in the expansion, and at the outlet of a sudden expansion, and determined a transition profile closely matching to the shape of separating streamlines in the expansion. Presumably, using the transition profile to build an expanding transition diminishes the energy-dissipating effects of eddies associated with flow separation.

Alauddin and Basak (2006) reported that such an expanding transition gave an overall efficiency of 80.3%, representing an improvement from previous work. The overall efficiency is defined as the ratio of the potential-energy gain to the kinetic-energy loss as water flows through the transition. A higher efficiency means less energy head losses. Alauddin and Basak (2006) suggested that all other existent transitions had a lower efficiency because of their abrupt ending at the downstream end. They concluded that the provision of a smooth outlet by their transition profile was the key to virtually eliminate flow separation and eddy formation.

Hannel Expansions Fitted With A Local Hump

Ramamurthy and Basak (1970) conducted an experimental study of flow separation in an expanding transition and the suppression of flow separation by fitting a simple hump at the channel bottom. Flow through expansions would encounter an adverse pressure gradient, decelerate, and therefore separate from the sidewalls. As a result, turbulent eddies form along the sidewalls and cause flow energy dissipations. Ramamurthy and Basak (1970) showed that both the angle of divergence and the length of transition relative to the inlet dimension have effects on flow separation. They also showed that inserting a hump in the bottom profile could suppress flow separation. As they explained, the specific energy head remains the same before and after a horizontal transition, if the energy losses due to friction are negligible. In the presence of a hump, there is a loss of velocity head up to the crest of the hump, and a gain of velocity head after passing the crest.

The results of Ramamurthy and Basak (1970) were based on measurements of depth and velocity at a number of selected sections including the entrance and exit sections of the transition. Without a hump, the flow became more turbulent and asymmetric at a larger Reynolds number, and even with a hump, the same asymmetry appeared at large Reynolds numbers, as revealed by velocity contours. Flow separation was reduced or eliminated after inserting a hump.

In conclusion, expanding transitions fitted with a hump perform well in terms of flow energy conservation. The experimental investigation of Ramamurthy and Basak (1970) was limited to a couple of humps with specific crest heights.

Flow Patterns In A Channel Expansion

2.3.1 Asymmetric behaviour of flow in symmetric expansion Some previous researchers of fluid flow in expansions have dealt with the case of sudden expansions (see e.g. Mehta 1979, 1981; Graber 1982; Nashta and Garde 1988; Manica and Bortoli 2003). The results reported in their studies highlighted the effects of different expansion ratios on mean flow velocities, fluid pressure distributions and turbulence characteristics.

Specially, the asymmetric behaviour of the flow field in perfectly symmetric expansions is very important. First, Abbott and Kline (1962) observed asymmetric flow patterns in their experimental investigations. They also found that the Reynolds numbers and turbulence intensities have no effect on flow pattern. Filetti and Kays (1967) found that the flow in rectangular channel with expansion ratios of 2.125 and 3.1 is asymmetric.

Also, Mehta (1979) conducted an experiment with a two-dimensional rectangular channel and found that the flow is symmetric at an expansion ratio of 1.25 and asymmetric at expansion ratios of 2.0, 2.5 and 3.0. He also found that the expansion ratio has an important effect on asymmetrical behaviour of channel expansions. According to Graber (1982), flows are symmetric in symmetric, two-dimensional rectangular channels with the expansion ratio of less than 1.5. He presented the cause of the asymmetric behaviour of the flow is a static instability of the flow system.

The stability of the system depends on the forces acting on the system and their variation as the system undergoes a small deflection. If the change in stabilizing momentum reaction exceeds the change in the destabilizing pressure force, the system is stable. If not, the asymmetric behaviour of the flow would be expected. The results of the stability analysis of Graber (1982) show that channels have the limitation that the maximum Froude number is less than 0.2. The result also predicts instability for expansion ratios greater than 1.5 that is in good agreement with experimental observations.

Manica and Bortoli (2003) considered laminar flow with low Reynolds number in a symmetric sudden expansion (Figure 2.1). Their numerical results show that below a certain critical Reynolds number (about 50), the flow pattern is symmetric about the channel central line; the two vortices in the expansion corners are more or less of the same size. The symmetric flow pattern becomes unstable at Reynolds number above the critical value. Pair of steady asymmetrical vortices appears as one recirculation region grows at the expense of the other.

Figure 2.1 Streamlines computed for flow in a 1:3 symmetric sudden expansion. The Reynolds number Re is: (a) 40, (b) 50, (c) 80, and (d) 140. The flow patterns are symmetric when Re = 40

Re 

in (b-d) (from Manica and de Bortoli, 2003). Nashta and Garde (1988) presented the results of analytical and experimental investigations in channels with a sudden expansion with expansion ratios of 1.5 to 3 for subcritical flow. The theoretical analysis assumed that the optimum shape for the transition is the shape in which the total energy loss, defined as the sum of the friction loss at the bed and loss due to expansion, is minimum. The functional relationship for the total loss is the objective function to be minimized for obtaining the optimum transition profile. To solve this problem, the boundary condition at one end is relaxed and the solution is forced to satisfy the relaxed condition.

For

B ratios equal to or greater than 1.5, the flow is asymmetric (where

Of The Main Channel And

B is the width of expanded channel). In such a case, the centerline of the jet does not coincide with the axis of expansion and the angle between the two lines, called

The Deflection Angle, Dependents On

B . Nashta and Garde (1988) observed that velocity

). The Reason For This

uniformity is the diffusion of turbulence generated at high shear zone, which has influence on velocity distribution. Nashta and Garde (1988) derived the mean velocity profile and separating streamline in the expansion and found that discharge and hence the Reynolds number have no effect on mean velocity and streamline.

Nashta and Garde (1988) concluded that the transition Lebedev’s equation with

N

around 0.6 is preferable for determining the transition shapes because of the smaller separation

Bn

is the highest degree in the normalized distance. The lengths of longer and shorter eddy can be predicted by Abbott and Kline’s curve (Abbott and Kline, 1962). An expression for energy loss in a sudden expansion was obtained from the continuity, momentum and energy equations. The length scales for velocity and shear

X And

B , where x is the distance from the entrance of the transition. Foumeny et al. (1996) carried out experiments and showed that the asymmetric behaviour of the flow is related to the Reynolds number. Also, Mehta (1981) showed that the flow patterns become more asymmetric and unsteady with increasing expansion ratios. Smith and Yu (1966) observed in a rapid expansion that the fluid flowing from upstream follow one sidewall and large turbulent eddies exist between the flowing jet and the other sidewall.

Flow Characteristics In Expansions

Mehta (1979, 1981) showed that flow separation can take place on both sides of the expansion with the maximum velocity line deviating from the centerline of the expansion. Mehta (1979) investigated flow separation in two-dimensional, sudden rectangular channel with width ratios ranging from 1.25 to 3.0, using numerical model and experiments.

The experimental investigations covered both symmetric and asymmetric flow patterns. Mehta (1979) reported that flows in two-dimensional sudden expansions are asymmetric and unstable, with three-dimensional character when the expansion ratio is larger than 1.25;

Has No Influence On The Asymmetric Behaviour

of the flow. Also, unequal pressure occurs in eddy pockets on both sides of the axis of symmetry. All the important parameters of flow are influenced by the expansion ratio. Mehta (1981) conducted an experimental study of the behaviour of the mean flow pattern and the turbulent characteristics for two-dimensional flow through large, sudden expansions. The flow patterns become more asymmetric and unsteady with increasing expansion ratios, whereas the degree of turbulence does not change except the peak values develop earlier and decay faster compared to cases of low expansion ratio.

Seetharamiah and Ramamurthy (1968) presented the idea of using a triangular sill in a channel expansion to decrease separation and eddy formation downstream of the expansion. The sill increases the transition length, and therefore reduces the sharpness of deceleration. They assumed that the amount of energy losses over the sill is negligible and showed two stages of deceleration of subcritical flow in the specific energy diagram. Seetharamiah and Ramamurthy (1968) also pointed out that the geometry of the sill can be chosen such that the theoretical retardation is nearly uniform along the sill, although it can be chosen to accelerate the flow up to the crest of the sill in canals where to reducing silting resulting from deceleration in the transition is very important.

Swamee and Basak (1991) presented a design method for subcritical expansions for rectangular channels, in order to achieve a minimum head loss. By analysing a large number of profiles, they obtained an equation for the design of rectangular expansion, producing the optimal bed-width profile.

Swamee and Basak (1992) discussed an analytical method for the design of expansions that connect a rectangular channel section with a trapezoidal channel section for subcritical flow. They suggested that flow separation in the expansion and the associated energy losses were considerably reduced through the optimal design of bed-width as well as side-slope profiles, on the basis of the momentum and energy equations. They claimed that the optimal profiles represent an improvement from the design of Vittal and Chiranjeevi (1983) in terms of reducing flow head losses. Swamee and Basak (1993) used the optimal control theory for the design of rectangular-to-trapezoidal expansions for gradually varied subcritical flow. They obtained equations for bed-width, side-slope and bed profiles based on the minimization of the transition

Head Losses

Escudier et al. (2002) conducted an experimental study of turbulent flow in a sudden expansion with an expansion ratio of 4 and an aspect ratio of 5.33. A laser Doppler anemometer was used to measure mean flow velocity fluctuations and the Reynolds shear stress. They reported that the flow downstream of the expansion is asymmetric. They concluded that the effect of inlet of expansion is the reason for asymmetrical behaviour of flow.

The Effect Of Solid Surface Friction

Babarrutsi et al. (1989) experimentally investigated re-circulating flows in a sudden open- channel expansion, considering the frictional effects of the channel bed. Their measurements showed that the re-circulating flow rate and length decrease due to friction. They recommended the use of materials with insignificant roughness height when studying flow separation in expansions.

Geometric Shape Of Expansions

As shown in Figure 2.2, channel expansions can be classified into five categories, namely, straight line expansions, square end expansions, cylindrical quadrant expansions, warped expansions, and wedge expansions.

Hinds (1927) considered different empirical design methods for various geometries of flumes and siphons. The design criterion was to minimize the transition length and energy losses. To achieve this objective, Hinds (1927) assumed a water-surface profile as two reversed parabolas with equal length, merging tangentially with the upstream and downstream water surface. The energy principle was used to expresses the expanding width as a function of the distance from the inlet of the expansion, with an assumed energy loss coefficient. Hinds (1927) concluded that an S-curved warped wall expansion is the most suitable one.

Smith and Yu (1966) examined the results of Hinds (1927) and found that the S-curved warped wall expansion is inefficient results, and causes flow separation. Smith and Yu (1966) recommended a straight walled diverging expansion (Figure 2.2, “straight line” type).

All the expansions have geometry symmetric about the centerline.

Energy Loss Analyse For Expansions

Using a rational method based on the concept of specific energy, Vittal and Chiranjeevi (1983) attempted to determine the boundary shape of expansions and flow conditions, with minimum head losses. Through experiments, they derived functions for the geometric features of expansions such as the bed width, bed elevation and sidewall slope. Their method was for designing a trapezoidal expansion.

When subcritical flow passes through an expansion, there is a decrease in velocity and an increase in pressure. Without any change in bed level, the water surface will rise by a vertical distance that is equal to the amount of drop of velocity head between the entrance and exit of the expansion. This means a conversion from kinetic to potential energy. However, this conversion is accompanied by energy losses. Mathematic analyses of energy losses in expansions of arbitrary geometry are difficult to perform. Therefore, it is a good alternative to find the energy losses for specific expansions and then extend the results by careful interpolations. Kalinske (1944) found that the rate of loss of energy in a 30° expansion is more than that in a sudden expansion. He found a major portion of the energy in the expansions is lost by direct conversion into heat at the high shear region in the fluid, and the total loss of energy is much higher than the turbulence energy.

Using the energy principle, Skogerboe et al. (1971) expressed the head loss in an expansion as a function of the velocity head at the entrance of the expansion. However, there was a small head loss correction which will varied for each expansion. The head loss coefficient was expressed as a function of the inlet Froude number as well as the expansion ratio, instead of a Skogerboe et al. (1971) argued that since a unique relationship exists between the head loss coefficient and specific energy ratio for any particular geometry of open channel expansion, and since a unique relationship between the specific energy ratio and the inlet Froude number, a unique relationship exists between the inlet Froude number and head loss coefficient for any particular expansion geometry. This means that the Froude number is a factor to consider.

As stated in Morris and Wiggert (1972, p. 185), the efficiency of energy conversion requires the flow profile to be continuous and as smooth as possible. Also, the profile should be tangent to the water-surface curves in the upstream and downstream sections of the expansion.

In summary, for the design of expansions, the water-surface profile is computed using the energy principle. The accuracy of computations relies on more or less guessed energy losses.

Hydraulic Behaviour Of Channel Expansions

An expansion with a large amount of change of the differential kinetic energy to potential energy is considered to be hydraulically efficient. In fact, the rise of the water surface or the recovery of energy head is less than the theoretical vertical distance (Hinds 1927). Smith and Yu (1966) considered expansions as gradual if the total central angle between sidewalls, is smaller than 28°10'. Separation can occur when  reaches 19°, except at the expansion ratio less than 2. Note that  = 28°10' corresponds to a 1:4 ratio of flare; this is a rapid expansion. Except when the expansion ratio is between 1 and 2, separation cannot be avoided. However, reducing  to avoid flow separation in an expansion is not practical because the length of the expansion will increase and the cost to build such an expansion is high.

According to Smith and Yu (1966), in a rapid expansion, flow from contracted section leans toward one of the sidewalls and a large turbulent eddy forms between the jet and the other sidewall. A straight wall flare is better than curved wall flare of equal length (Smith and Yu, 1966), because when the curved wall flare is used, the central angle between wall tangents continuously increases, and the flow may separate on one side of the expansion when the central angle becomes too large. They concluded that the same benefit could be obtained at less cost by using a shorter gradual expansion so that there is no justification for using the rapid expansion.

Kalinske (1944) found that the rate of loss of energy in a 30° expansion is more than the sudden expansion. The author indicated that a major portion of the energy in the expansion is lost by a direct conversion into heat at the high shear region in the fluid, and the total loss of energy is much higher than the turbulence energy.

Hapter Three Modelling Methodologies

This study aims to model subcritical turbulent flow in the expansion section of an open channel. CFX software (ANSYS 2010a, 2010b) was used to create an appropriate model channel of different geometric configurations and to predict the three orthogonal components of water velocity along with the depth of flow in the model channel. This chapter begins with a description of the model domain. Then, the hydrodynamics equations and turbulence closure schemes are presented. This is followed by specifications of boundary and initial conditions.

Next, strategies for the generation of finite volume meshes for flow computation are discussed.

Odel Domain

The model domain for flow computations consists of an upstream channel section, an expansion and a downstream channel section, with or without a hump fitted at the channel bottom of the expansion section (Figure 3.1). In some cases, the model domain allows for an additional extension channel section at downstream. The length of the expansion section matches that of an the physical model are available for comparison.

The velocity and pressure fields of steady state are computed for given conditions of inflow at the upstream end and outflow at the downstream end of the model channel. Inclusion of an upstream channel section of efficient length allows the development of realistic flow profiles or vertically distributed flow velocities that approach the expansion, whereas inclusion of a downstream channel section is helpful for removing possible end effects, which are artificial, on the computed flow field in the expansion.

(B) Plan View

Figure 3.1 Geometry of the model channel, showing dimensions of various channel sections and bottom configurations. Uniform water flow enters (arrow) the model channel from the relatively narrow channel section.

3.2 Reynolds-averaged continuity and momentum equations

Ontinuity Equation

Let (u, v, w) denote the three orthogonal components of the instantaneous velocity field in the Cartesian coordinates (x, y, z). The positive direction of the z-axis points upward. For an incompressible fluid, the equation of continuity is given by

(3.1)

Open channel flows are always fluctuating or turbulent; practically it is extremely difficult to resolve the instantaneous flow field. One way to deal with turbulent flow is to split the instantaneous velocity into mean components (U, V, W) and fluctuations (u, v, w) through the so-called Reynolds decomposition, expressed as

(3.4)

Substitutions of Eqs. (3.2)-(3.4) into Eq. (3.1) give:

(3.5)

The operation of the Reynolds time average gives rise to

(3.6)

Since the averages of velocity fluctuations are zero, resultant equation become the Reynolds-

(3.7)

The three Reynolds-averaged velocity components (U, V, W) are unknowns.

Omentum Equations

Let ρ denote the density of water, t denote the time,  denote the dynamic viscosity of water, and p denote the instantaneous pressure field. The momentum equations for open channel flow can

( 3.10)

On the left hand side of the above equations, the first term is a transient term that describes the right side of the equations, there are a pressure gradient term, and three molecular diffusion constant with respect to time and space.

Because of the difficulty in dealing with the instantaneous velocity field, the Reynolds decomposition is applied to Eqs. (3.8)-(3.10). Similar to the instantaneous velocity components decomposed in Eqs. (3.2)- (3.4), the instantaneous pressure p is split into a mean value P and a fluctuation p, i.e.

(3.11)

Substituting Eqs. (3.2)-(3.4) and (3.11) into (3.8)-(3.10) and taking the Reynolds average yield

(3.14)

In these equations, there are four unknowns: the Reynolds-averaged pressure field P and the three unknown Reynolds-averaged velocity components (U, V, W). In addition, the Reynolds average operation produces six extra unknown quantities involving velocity component fluctuations. These quantities are the so-called specific Reynolds



, some of which are identical. These unknown shear stresses must be modelled, giving rise to a turbulence closure problem.

The Concept Of Turbulent Eddy Viscosity

In this study, turbulence closure makes use of the concept of turbulent eddy viscosity t. The specific Reynolds shear stresses are related to the mean flow strain rates using the Boussinesq

(3.20A,B,C)

These terms can readily be evaluated once the Reynolds-averaged velocity components. The eddy viscosity is obtained from the k-ω turbulence model.

The Standard K-Ω Model

For open channel flow applications, a variety of turbulence models have been developed, each with certain advantages and disadvantages. The choice of a specific turbulence model depends on the type and nature of the flow field to be simulated and the desired accuracy of results. The k-ω model is one of the most commonly used turbulence models.

This model is the first two-equation model of turbulence proposed by Kolmogorov (1942). It is a two equation model; i.e. it includes two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. Kolmogorov (1942) chose the kinetic energy of turbulence, k, as one of his turbulence parameters, and modeled the partial differential equation that governs the behaviour of k. His second parameter was the dissipation per unit turbulence kinetic energy i.e.

K

. This is the so-called ω. In his k-ω model, ω satisfies a partial differential equation similar to the equation for k. Kolmogorov’s justifications for introducing the

 Therefore, K

t has the dimension of time.

S

1 . Given that the most common processes in fluid motion are unsteadiness, convection, diffusion, dissipation, dispersion and production, Kolmogorov (1942) combined the physical processes with dimensional arguments and proposed a transport equation for ω as below

(3.21)

where β and ζ are two new closure coefficients. This equation is not written in terms of

Fact,

is the mean square vorticity of the “energy containing” eddies and k is the kinetic energy of the motion induced by this vorticity. Therefore, it is better to write the equation in terms of equations, closure coefficients and relationships are as follows:

Lim 

It means that the eddy viscosity depends on ~ rather than ω and this makes the eddy viscosity a function of k and ω and effectively, the ratio of the turbulence-energy production to the turbulence-energy dissipation.

(3.24)

where Pr is the turbulence production term, given by

(3.25)

Closure coefficients and auxiliary relations are as below

(3.29)

The two tensors in Equation (3.28) are the mean rotation (vorticity) and mean-strain-rate tensors.

, Is Added To The Equation To

remove the original model’s sensitivity to the free stream value of ω, and to remove the sensitivity to the imposed boundary condition. This is good for applications to wall-bounded flows. The reciprocal of ω is the time scale on which dissipation of turbulence energy occurs.

While the actual process of dissipation takes place in the smallest eddies, the rate of dissipation is the transfer rate of turbulence kinetic energy to the smallest eddies. Therefore, the dissipation rate is set by the properties of the large eddies (scales with k and l). Therefore, ω is indirectly associated with the dissipative process.

The Volume Of Fluid Method

The interface between the gas and liquid, where the difference in density between these two phases is quite large, is considered as a free surface. The inertia of the gas could usually be neglected due to a low density. Therefore, the only influence of the gas is the pressure acted on the interface and it is not necessary to model details of the gas phase. Hence, the free surface is simply modelled as a boundary with constant pressure.

The volume of fluid method is used to determine the shape and location of free surface based on the concept of a fractional volume of fluid. A unity value of the volume fraction corresponds to a full element occupied by the fluid (or liquid), and a zero value indicates an empty element containing no fluid (or gas). A value of volume fraction between zero and one means that the corresponding element is the surface (or partial) element. The equation of the volume of fluid method for determining the shape of the free surface is given by

(3.30)

where F is the volume fraction.

Boundary Conditions

The appropriate use of boundary conditions is required to fully define the fluid flow problem. The external boundaries of the model domain are the inlet, outlet, sidewalls and channel-bed.

Nlet Condition

Inlets are used mostly for regions where inflow is expected. At the inlet where the fluid flows into the domain (Fig. 3.1), the imposed mass and momentum conditions is the normal speed vn. The magnitude of the inlet velocity is specified and the direction is taken to be normal to the boundary. The normal speed is steady and uniform. For instance, vn equals 0.78 m/s for some simulations. Also, the relative pressure at the inlet is specified

(3.31)

where the subscripts w and a indicate water and the air, respectively, g is gravity, η1 is the elevation (above the channel bed) of the free surface at the inlet, and z changes from zero at the channel bed to η1 at the free surface. A value for η1 is given (e.g. η1 = 0.25 m for some simulations).

At the inlet, the turbulence intensity and turbulence length scale are specified. The turbulence intensity is given in terms of a fractional intensity (5%), and the turbulence length scale is taken to be equal to η1.

Outlet Condition

At the outlet where the fluid leaves the model domain (Fig. 3.1), the appropriate condition to

(3.32)

where η2 is the elevation (above the channel bed) of the free surface at the outlet, and z changes from zero at the channel bed to η2 at the free surface. Usually, η2 must be known as part of the problem definition. For the case of flows in expansions, the elevation is not known in advance.

However, it is sufficient to provide an estimate of η2 using the energy principle (Henderson 1966) for the purpose of determining the distribution of the relative static pressure with depth below the free surface.

Solid Surface Condition

Channel sidewalls and the channel bed (Fig. 3.1) are solid surfaces where conditions to be imposed can be a no-slip wall, a free slip wall or specified shear. In this study, the no-slip wall condition is applied. The flow near to the no-slip wall is modelled using wall function approach.

Based on the wall function approach , the near wall tangential velocity in the log-law region is

W

, by means of a logarithmic relation. The logarithmic relation



, (3.34)

T

U is the known velocity tangent to the wall at a distance of

 Is The

shear stress of the wall,is the von Karman constant, and C is the log-layer constant that depends on the wall roughness.

W

h denote the initial depth of water. Initially, the volume of fraction for air is given by a step



, where z is the vertical coordinates pointing upward with z = 0 at the

1

. Initially, the relative pressure field, P, in water is



and is uniformly zero in the air (the pressure in the air is set to zero). Thus, the initial conditions for the pressure field and the volume fraction are consistent.

Finite Volume Meshes

It is desirable to use fine meshes for flow computations in order to capture detailed flow features such as eddies and velocity shears. Meshes were generated on the basis of a number of criteria: First, meshes are fine enough in order to resolve rapid spatial variations in the velocity fields, especially near wall boundaries. This requirement is satisfied by performing inflation on meshes adjacent to solid walls where eddies and velocity shear are expected to appear (Fig. 3.2).

Second, the structure of meshes used for flow computations must not affect the computational results. In other words, the model results produced should be independent of the configurations of the meshes used. The strategies used to satisfy this requirement were to several mesh systems of progressive fine sizes (e.g. 10*10-3 m, 7*10-3 m, 5*10-3 m and 4*10-3 m) and to carry out model runs using the different meshes under identical flow conditions. The independence of the computed flow field for these runs was verified through comparisons of the results among these runs.

Third, the total number mesh points must not be excessively larger, resulting in prohibitively high computation cost. An excessively large number of mesh points will also create difficulties in the post processing of model output.

Figure 3.2 A sample finite volume mesh system used for flow computations, showing the inflation of meshes near all solid walls. The solid surface on the top is set to be slippery. 3.7.1 Determination of the near-wall mesh spacing As described in ANSYS (2010b), for fluid flow of characteristic velocity Uo over a flat surface of characteristic length L, the Reynolds number can be defined as:

(3.36)

An empirical correlation between the wall shear stress coefficient cf and the Reynolds number is

(3.37)

where x is the distance along the plate from the leading edge, and the Reynolds number is based on x, i.e.

(3.38)

For near-wall spacing estimate, the definition of the dimensionless wall distance y+ is

(3.39)

where u is the friction velocity, which is unknown, y is the mesh spacing between the wall and the first mesh point away from the wall. If we can eliminate the unknown friction velocity, for target values of y+, we will be able to determine y. Using the definition

(3.40)

we can eliminate the friction velocity in Eq. (3.39) to yield

(3.41)

The wall shear stress coefficient cf in equation (3.41) can be eliminated using the empirical correlation given in Eq. (3.37), to give

(3.42)

Using the definition in Eq. (3.36), we may rewrite Eq. (3.42) as

Re 

, where C is some fraction. If C is assumed to be 0.5 with C1/14  0.952,

(3.44)

Except for very small Rex (very close to the leading edge of the flat surface), Eq. (3.44) is suitable for estimates of the near wall mesh spacing. For target values of y+, the mesh spacing y can be determined. Some sample calculations are shown in Table 3.1. Values for the characteristic quantities used in these calculations are: L = 0.30 m (the approximate length of the expansion in question) and Uo = 0.70 m/s (the approximate normal speed of water flow at the inlet). The viscosity of water is taken as ν = 10-6 m2/s. The Reynolds number [Eq. (3.36)] is ReL = 2.1×105.

Table 3.1 Mesh spacing (Δy) for various target values of the dimensionless wall distance (y+). The calculations shown in Table 3.1 provide some guidelines for generating appropriate meshes. It is well established that the wall function of the form

(3.45)

is valid for the wall distance y+ in the range of 30 < y+ < 400. In the equation, U denotes the tangential velocity parallel to the solid wall; κ (= 5.5) is the von Karman constant; the wall



, where y measures the normal distance of a point in question from the solid wall. From Table 3.1, we make several observations: First, if the wall function [Eq. (3.45)] is to be used as the condition at the solid wall, it would be acceptable to place the first mesh point at a normal distance of about 1*10-3 m from the solid wall. This is because the corresponding y+ value is between 30 and 40, and the first mesh point is within the logarithmic layer where Eq. (3.45) is valid. Note that if the first mesh point is placed at a normal distance of

Y (Mm)

less than 0.3*10-3 m, the corresponding y+ value is less than 11, meaning that the first mesh point is in the viscous sublayer. Second, if meshes are created using 10*10-3 m as the size, there is a great uncertainty whether or not the first mesh point away from the wall is inside the logarithmic layer, since the corresponding y+ value is larger than 340. The first mesh point is at best in the outer edge of the logarithmic layer, and there will not be enough mesh points inside the logarithmic layer for the k-ε model to work accurately. Meshes created using 7*10-3 m as the size give a slight improvement from 10*10-3 m meshes; the corresponding y+ value of the first mesh point is still too high, being about 250.

Third, if meshes are created using 5 or 4*10-3 m or smaller as the size and if subsequent inflation is performed on the meshes adjacent to the solid wall, the locally refined meshes for flow computations should contain several mesh points inside the logarithmic layer due to the presence of the wall. Also, the first mesh point from the solid wall of the refined meshes is about 1*10-3 m away from the wall, and the corresponding y+ value is between 30 and 50. This makes it ideal to apply the wall function [Eq. (3.45)] as the boundary condition at the solid wall. The independence of model results on the meshes (4*10-3 m, 5*10-3 m, 7*10-3 m and 10*10-3 m) will be verified.

Estimates Of Boundary Layer Thickness

Although we do not intend to resolve the boundary layer due to the presence of a solid wall, it would be constructive to obtain estimates of the boundary layer thickness δ. On the basis of the

(3.46A,B)

As in the calculations of near-wall mesh spacing [Eq. (3.43)], the Reynolds number Rex is some fraction of ReL. If we assume that the fraction is 50%, Eq. (3.46b) will be simplified to

(3.47)

which gives an estimate of δ = 4*10-3 m, for L = 30*10-2 m (the approximate length of the expansion), Uo = 0.70 m/s (the appropriate normal speed at the inlet), and  =10-6 m2/s (the viscosity of water).

If Δy is set to 4*10-3 m in Eq. (3.44), the corresponding y+ value is between 140 and 150. Eq. (3.47) appears to give conservative estimates or underestimates of δ based on



. Importantly, meshes created using 4*10-3 m as the size with subsequent inflation are expected to satisfy the requirement that the several mesh points are inside the logarithmic layer; this not only justifies the use of the wall function as the condition at the wall but also provides the condition for the turbulence model to work properly.

Energy Balance For Flow In Expansions

In this study the expansion connects a cross section of rectangular shape with a smaller width to a cross section of rectangular shape with a larger width. Figure 4.1 shows the plan view of the channel expansion. The width of the expansion changes from

B

at its downstream end (CS3) .With the assumption of hydrostatic pressure, an energy equation could be written between sections CS2 and CS3, as below:

H Is The Energy Loss In The

expansion; the subscripts 2 and 3 refer to CS2 and CS3 , respectively. Figure 4.1 Plan view of a channel expansion.

(4.2)

Equation (4.2) shows that the flow at the two cross sections has the same specific energy. The

Term

h in equation (4.1) is the energy loss in the expansion, including the energy loss

Fh Due To

friction at the channel bed and on the sidewalls, and the energy loss

(4.4)

For the special case where the channel is rectangular, the discharge per unit width of channel; q, is related to the depth of flow, y, and flow velocity, v, through the equation of

V 

, (4.5)

(4.7)

This equation is used for obtaining energy loss in the expansion due to flow separation and eddy motions. An energy loss coefficient can be defined as

The Concept Of Specific Energy

The concept of specific energy is very important in the study of open-channel flows. Using the channel bed as datum, the specific energy, E, is defined as

(4.9)

For the special case where the channel is rectangular, the equation of continuity is given by

(4.10)

where Q is the total discharge ; b is the width of the channel. The specific energy equation (4.9)

E-Y Curve

Consider water flow at two cross sections (CS2 and CS3) in an expansion, the corresponding per-unit-width discharges are q2= Q/b2 and q3= Q/b3. Since b3 > b2, we have q3 < q2. The state of flow at cross sections 2 and 3 is represented by the two E-y curves (marked by q2 and q3, respectively).

The so-called E-y curve (Figure 4.2) shows how E will vary with y for a given value of q in a horizontal channel. More interestingly, as water flows through a channel expansion, the state of flow will change; this is equivalent to moving from one specific energy curve to another.

If there is no energy loss between cross sections 2 and 3, i.e. when the specific energy at cross sections 2 and 3 is the same (represented by the vertical line). The depth of flow, y, (dashed, horizontal lines) will increase from upstream (the E-y curve marked by q2) to downstream (the E-y curve marked by q3).

Figure 4.2 The specific energy curve and its application in the expansion problem. Critical flow and the concept of the Froude number The concept of critical flow is graphically illustrated in the E-y curve (Figure 4.2). For a given per-unit-width discharge, the flow is critical when the specific energy of flow is at a minimum level. In a channel expansion when flow at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in the flow surface. The rapidly flowing flow is abruptly slowed and increases in height, converting some of the flow's initial kinetic energy into an increase in potential energy (Henderson, 1966).

The specific energy equation (4.9) is valid only for small slopes (< 10%), where the flow has negligible acceleration in the vertical and hence the pressure distribution is hydrostatic. The velocity coefficient is usually quite high, between 0.95 and 0.99 for the rivers. Since the effects of the velocity variations across the flow section are neglected, the velocity coefficient,, is assumed to be equal 1.0 in this study. This assumption implies that surface waves with high amplitudes are not generated and propagate during the expansion. This could be ensured when the Froude number is less than one at upstream, however, for large Froude numbers, the presence of such surface waves is inevitable.

R

F , is defined as the ratio of actual water velocity, v, to surface wave

Celerity,

gy .The Froude number at cross section CS1(Figure 4.1) is defined as

(4.13)

The Froude number is only defined for channel sections that have a free surface. When Fr < 1, the flow is said to be subcritical; when Fr > 1, the flow is said to be supercritical; when Fr = 1, the flow is said to be critical.

On the specific energy diagram (Figure 4.2), the parts corresponding to subcritical and

(The Crest Point C), Below Which We Have

supercritical flow (the lower limb), whereas above which we have subcritical flow (the upper limb). For critical flow denoted by the subscript c, the depth of flow, flow velocity and the

Flow Over A Step In The Vertical (Hump)

Consider an open channel of constant width but with a change in the bed level such as an upward step and divergence angle shown in Figure 4.1. If  is zero, the per-unit-width discharge will not change (Figure 4.3) .In Figure 4.3, CS2 and CS3 are cross sections, upstream of the vertical step and at the vertical step, respectively.

The behaviour of flow over a step in the vertical can be analyzed using the energy principle, written between cross sections CS2 and CS3

Z And

3z are the bottom elevations at the two cross sections, respectively. The maximum

Min

for the given per-unit-width discharge, q. Consider subcritical flow, represented by point A on the upper limb of the specific energy curve (Figure 4.4). Subcritical flow approaches the vertical step (hump).

Figure 4.3 Water surface profile for flow over a vertical step (hump) fitted on the bottom of a uniform channel. The depth of flow decreases over the hump on the basis of the energy Principle. From cross section CS 2 to cross section CS 3, the bed level rises and the water pressure decreases; from cross section CS 3 toward downstream, the water pressure increases while the bed level drops.

Figure 4.4 Specific energy diagram for a channel expansion.

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.