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공학석사 학위논문

Numerical Study of Backward Facing Step Flow

over Various Step Angles and Expansion Ratios

계단각과 확장율 변화에 따른

후방 계단 흐름에 대한 수치적 연구

2016년 8월

서울대학교 대학원

건설환경공학부

최 회 현

1

공학석사 학위논문

Numerical Study of Backward Facing Step Flow

over Various Step Angles and Expansion Ratios

2016년 7월

서울대학교 대학원

건설환경공학부

최 회 현

최회현의 석사학위논문을 인준함

2016년 7월

위 원 장 서 일 원 (인)

부 위 원 장 서 경 덕 (인)

위 원 Van Thinh Nguyen (인)

2

Abstract

To study the flow over hydraulic structures such as weirs, dams, sluice

gates, etc., a single-sided backward-facing step is a fundamental configuration

in order to investigate the flow pattern including turbulence structures,

separation, and reattachment length. The separation zone plays an important

role on the energy dissipation and scouring after the steps.

In this study, two-dimensional numerical simulations of flow over the

backward-facing step with various step angles and expansion ratios have been

carried out. An open source program OpenFOAM was used for the computation.

RANS approach was applied for turbulence calculation, in order to determine

a better approach for a specific application.

Numerical results have been validated against Ruck et al and Nakagawa

et al’s experiments. The step angles varied from 10 to 45 degree. The results

showed a good agreement with experiments on the major flow patterns,

however at the low Reynolds number flow, RANS model showed a worse

agreement. For the real application, Sejong weir was chosen as a case study. In

this study, the reattachment length was increased as the step angle increases.

Keywords: CFD, Backward step facing, Step angle, RANS, OpenFoam.

Student Number: 2014-22714

3

Contents

Abstract ............................................................................................................ 2

List of Figures .................................................................................................. 5

Chapter 1. Introduction .................................................................................... 9

1.1 General introduction ............................................................................... 9

1.2 Previous research ................................................................................. 10

1.3 Objective of the study .......................................................................... 12

Chapter 2. Methodology ................................................................................. 14

2.1 OpenFOAM .......................................................................................... 14

2.2 Reynolds averaged Navier-Stokes equations (RANS) ......................... 15

2.2.1. Governing equations .................................................................... 15

2.2.2 Turbulence model .......................................................................... 15

2.2.3 Boundary conditions ..................................................................... 17

2.3 Reattachment length ............................................................................. 18

2.4 Flow distribution .................................................................................. 19

2.5 Turbulent intensity ............................................................................... 19

Chapter 3. Result and Discussion ................................................................... 21

3.1 Validation with experimental data ....................................................... 21

3.1.1 Ruck’s experiment (1993) ............................................................. 21

3.1.2 Nakagawa and Nezu’s experiment ................................................ 32

3.2 Application ........................................................................................... 34

3.2.1 Nakagawa’s channel ...................................................................... 35

4

3.2.2 Sejong weir .................................................................................... 37

Chapter 4. Conclusion .................................................................................... 40

References ...................................................................................................... 42

Appendix 1 Ruck and Makiola(1993)’s experiment and simulation......... 45

Appendix 2 Simulation result of Nakagawa and Nezu(2010)’s experiment

.................................................................................................................... 76

Appendix 3 Simulation result of Sejong weir condition ............................ 79

5

List of Figures

Fig 1.1 Backward facing step flow 11

Fig 3.1.1 Reattachment length, ER=1.48, Re=15000 22






Fig 3.1.7Reattachment length, ER=3.27, Re=11000 26


Fig 3.1.9 Empirical equation and simulation, ER=1.48, Re=15000 28






Fig 3.1.15 Reattachment length and angle, ER=2.00 32

Fig 3.1.16 Nakagawa’s experiment and k-omega simulation 34

Fig 3.2.1 Reattachment length and step angles (Nakagawa’s condition) 36

Fig 3.2.2 Reattachment length and step angles (Sejong weir condition) 38

Fig A1.1 Comparison between exp and sim, ER=1.48, Re=11000, α=10° 45





6
















Fig 3.21 Comparison between exp and sim, ER=1.48, Re=64000, α=10° 55

Fig 3.22 Comparison between exp and sim, ER=1.48, Re=64000, α=15° 55









7


























8







Fig A2.1 Nakagawa’s conditions, Step angle =10° 76






Fig A3.1 Flow distribution with condition of Sejong weir, Step angle =10° 79




Fig A3.5Flow distribution with condition of Sejong weir, Step angle =30° 80


9

Chapter 1. Introduction

1.1 General introduction

The study on the flow over a backward-facing step is very important in

fluid mechanics. Because, backward-facing step is not only simplest geometry

on which the separation flow is caught in laboratory but also is commonly

observed in real hydraulic structure and HVAC systems. For example, the

protection of scouring which is constructed behind the weir formed back-ward

facing step.

The most important characteristics of backward facing step flow is the

internal flow in which there are separation and reattachment. Flow separation

and reattachment are caused by sudden changes in cross-section geometries.

Reynolds number, expansion ratio and step angle are main factors of the

internal flow. The structure of this flow such as reattachment length is changed

by these factors. So, researches with various conditions of the factors are

needed to understand this flow.

Many studies applied experimental and numerical approach to this flow.

Experiments can catch the separation, reattachment, turbulent structure in the

lab. On the other hand, numerical analysis has advantage to change geometry

in several angles and expansion ratios because of its flexibility.

On the other hand, 16 weirs have been built on the rivers of South Korea

during 4 major river projects (Han, Nakdong, Gum and Youngsan River).

Passing through several flood seasons, the bottom structures after the weir was

10

damaged due to increasing load when the gates were opened. The protection of

scouring was constructed but it is not sufficient during flood seasons. To design

the weir, calculating the maximum load and deciding proper form of the

protection zone have to be concerned. And the shape of protection looks like

backward facing step flow.

Moreover, designing weir needs manuals that have been validated by

many experiments [1, 2] based on various step angles and flow conditions [3,

4, 5]. However, it is very difficult, expensive, and sometimes impossible. Using

computational fluid dynamics has an advantage to optimize the design.

Eventually, Sejong weir was chosen as an application case for this study. The

actual flow conditions were applied for different step angles and the result has

been analyzed.

1.2 Previous research

The separation of flows caused by geometry is important to study on

hydraulic structures. Abbot et al, Seban and Goldstein emphasized the

importance of such flow [6, 7, 8]. Following these study, Denham et al and

Etheridge et al paid attention to the 2-D backward-facing step [9, 10]. To study

about boundary layer flow, backward-facing step was simplest configuration to

carry out. Amarly et al conducted experiment with laminar, transient, turbulent

flow, and simulated with k-epsilon model. The length of the separation region

was dependent on the Reynolds number [11]. Srba Jovic et al conducted

backward-facing step flow for low Reynolds number to calculate velocity and

11

shear stress [12]. Lee et al measure wall pressure fluctuation in backward-

facing step and investigated statistical properties [13]. Le et al simulated DNS

over a backward-facing step, and the log-law indicates that the turbulent

boundary layer is not fully recovered at 20 step heights behind the step. Hall et

al simulated k-omega sst model for backward facing step and compare with Le

at el’s DNS experiment [14]. Park [15] conducted simulation on water channel

and studied relative turbulent intensity effect to scouring.

In review of backward facing step studies, many experiment and

simulations have been done with orthogonal angle step. But In the real situation,

geometrical condition is formed with different step angles. Ruck et al conducted

experiment over various step angles and expansion ratios [16] and observed

that change of step angle impacts reattachment length and turbulent intensity.

To design a protection of scouring or air duct considering with step

angle, the relationship between flow structure and step angle is needed.

Fig 1.1 Backward facing step flow

In this study, the relationship between flow structure and step angle is

analyzed by numerical approach. Several fluid characteristics (air, water),

12

Reynolds number (inlet velocity, inlet height), expansion ratios were applied to

the flow and compared. This research can be applied to weir design, air duct

design which have inclined steps.

1.3 Objective of the study

The study proceeds in following order to examine the effect of step angle

g change in the backward facing step flow.

1. Model validation using the results of Ruck

2. Model validation using the experimental results of Nakagawa

3. Application changing the step angle with Nakagawa condition

4. Application changing the step angle with Sejong weir condition

The main objective of the study is to simulate the backward-facing step

flow over various step angles and expansion ratios. First, numerical

investigation of the air flow were validated based on observation data obtained

from Ruck at el. Three Reynolds numbers (Re=5000, 47000, 64000), seven

step angles ( =10°, 15°, 20°, 25°, 30°, 45°, 90°) and three expansion ratios

(ER=1.48, 2.00, 3.27) are applied. Secondly Nakagawa’s experiment [19] was

used to validate water flow over a step angle of 90°. The results were analyzed

with same conditions of Nakagawa’s channel and with different step angles (

=10°, 15°, 20°, 25°, 30°, 45°, 90°).Eventually, Sejong Weir was applied as an

application case study. Reynolds number was about 84000, expansion ratio is

2.25. and step angles ( =10°, 15°, 20°, 25°, 30°, 45°)

13

An open source CFD package named “OpenFOAM” is selected for

numerical analysis. In this study, RANS with k-omega model and LES

approach will be applied. Since it has been documented that k-epsilon cannot

well performed the separation zone and reattachment length, therefore applying

other model (k-omega) and different approach such as LES is needed to

improve the turbulence simulation of the flow over the steps.

14

Chapter 2. Methodology

2.1 OpenFOAM

For numerical simulations in backward-facing step flow, OpenFOAM was

used as numerical tool. It is a toolbox for the development of customized

numerical solvers, and pre/post-processing utilities for the solution of

continuum mechanics problems, using finite volume method. The following

explanation is from OpenFoam User Guide (http://cfd.direct/openfoam/user-

guide/introduction/#x3-20001)

OpenFOAM is first and foremost a C++ library, used primarily to create

executables, known as applications. The applications fall into two categories:

solvers, that are each designed to solve a specific problem in continuum

mechanics; and utilities, that are designed to perform tasks that involve data

manipulation. The OpenFOAM distribution contains numerous solvers and

utilities covering a wide range of problems.

One of the strengths of OpenFOAM is that new solvers and utilities can

be created by its users with some pre-requisite knowledge of the underlying

method, physics and programming techniques involved. OpenFOAM is

supplied with pre- and post-processing environments. The interface to the pre-

and post-processing are themselves OpenFOAM utilities, thereby ensuring

consistent data handling across all environments.

15

2.2 Reynolds averaged Navier-Stokes equations (RANS)

2.2.1. Governing equations

Reynolds decomposition is applied to the variables in three-dimensional

Navier-Stokes equations each of them is decomposed in a time-averaged part

and a fluctuating part. It yield the two following equations.

Momentum equation for incompressible fluid

1 ji i ij i i j

j i j j i j

uu u upu g u u

t x x x x x x

Continuity equation

0j

j

u

x

2.2.2 Turbulence model

To solve RANS equation i ju u makes 6 additional unknown variables. We

called it closure model. These are modeled according to the Boussinesq’s

approximation.

2

3

jii j t ij

j i

uuu u k

x x

where t is the eddy-viscosity; k is the turbulent kinetic energy; ij is

Kronecker’s delta.

16

The eddy-viscosity in momentum and advection-diffusion equations is

calculated by turbulent model. Most applications of hydraulics have used the k-

epsilon model based on Boussinesq hypothesis and empirically calibrated

equations for turbulent kinetic energy, k and energy dissipation, epsilon. But

this model have weaknesses in predicting separating flow. Therefore, many

other two-equation model have been postulated. The most prominent

alternative is the k-omega model proposed by Wilcox (1988).

In this study, for turbulent model k-omega model was used. The momentum,

k, omega equations are

*

2

1 1

1

2

3

ii j i i j i

i j

kT

k

iTij ij ij

j

u pu u u u u f

t x x

Pkku k k

t

uu S S k

t x

where

1

1

*

22

3

2

3

2.0

2.0

0.553

0.075

0.09

T

i

k ij ij ij

j

i j

i j T ij

j i

k

k

uP S S k

x

u uu u k

x x

17

2.2.3 Boundary conditions

The boundary condition at the inlet was applied experiment data. The

turbulent parameters of k-omega turbulence model are following

23

2k UI

where U is maximum velocity at the inlet (-2H, H means step height); I is the

turbulent intensity.

1

4k

Cl

where C is the turbulent model constant which usually has a value of 0.09. l

is the turbulent length scale.

Turbulent intensity, I was assumed as 5% in fast flow. At the high

Reynolds number flow and open-channel case, this can be used. l was

assumed as 7% of inlet height. . The wall function for specific dissipation rate,

is

2 2

logp vis

where p is centroid value in cells adjacent to solid wall. vis and

log are

the values for viscous and log-layer, respectively. It is known as

18

2

1

*

log 1/4

6vis

y

u

C y

where *u is the shear velocity.

The eddy-viscosity, t at the walls is replaced as

,

1ln

t wall

wall

y

Ey

At the outlet boundary condition is dealt with Dirichlet boundary (zero

gradient). And the condition for pressure was assumed zero gradient in all

boundary.

2.3 Reattachment length

The main purpose of studying backward-facing step flow is information for

boundary-layer flow behavior. In the past, the research works carried out

knowledge of backward-facing step flow and well understanding of internal

flow with separation. Recirculation and separation region occurs behind the

step and flow energy dissipated here.

The size of separation region is determined by reattachment length. To

calculate the reattachment length at a given simulation, scanned near wall flow

velocity at bottom and upper wall. To determine reattachment length of primary

19

separation behind the step, the position of the zero-mean-velocity line was

calculated. The point of reattachment were taken as the extrapolated zero-

velocity line down the wall.

2.4 Flow distribution

Flow rate was compared using a value normalized to a maximum velocity.

The maximum velocity at the inlet was used. The range of the normalized value

is from -1 to 1.

0

0 max,

xn

inlet

uu

u

u u

2.5 Turbulent intensity

Turbulent intensity was compared with that observed in direct experiments

of Rucks and Makiola.

2

0

0 max,inlet

uI

u

u u u

u u

The k value was converted to turbulent intensity as follows.

20

2

2

0

2 2 2 2

0 max,

3

2

1 3

2 2x y z

inlet

k

Iu

k u u u u

u u

21

Chapter 3. Result and Discussion

3.1 Validation with experimental data

3.1.1 Ruck’s experiment (1993)

The experiment of backward-step facing flow is conducted by Ruck &

Makiola. The geometry of it is single-side step with variation of the angles an

expansion ratios. Velocity data were measured by laser Doppler anemometry.

To validate model, the work performed for a Reynolds-number range between

5000 and 64000, which covered the transitional and turbulent regime of the 2D

backward-facing step flow. The definition of the Reynolds number is given by

0Reu D

Where 0u is maximum inlet velocity, D is the hydraulic diameter of the inlet

channel and equivalent to its height inletD h , and is the kinetic viscosity.

The results of Reynolds number 47000, expansion ratio 2.00 and various step

angles (10, 15, 20, 25, 30 ,45 and 90) were presented at KSCE2015 [20]. The

other results are presented at HIC2016 [21]. In addition, the results of flow

distribution and turbulence intensity are attached in Appendix 1.

All simulations were run on different grids. The results of the paper were the

most similar to the experimental data.

22

3.1.1.1 Reattachment length (ER=1.48)

Fig 3.1.1 Reattachment length, ER=1.48, Re=15000


0

1

2

3

4

5

6

7

8

0 20 40 60 80 100

Xr/

Hs

Step angle(α)

ER=1.48, Re=15000

Ruck(1993)

RANS(k-omega)

0

1

2

3

4

5

6

7

8

0 20 40 60 80 100

Xr/

Hs

Step angle(α)

ER=1.48, Re=47000

Ruck(1993)

RANS(k-omega)

23


When the expansion ratio is 1.48, reattachment length of the experiment and

simulation showed good agreements in most cases. But the poor results were

came out in Reynolds number are 5000 ~ 11000. Velocity profile, turbulence

intensity and reattachment length did not fit well. The k-omega model is

affected to the Reynolds number range. In this Reynolds number, the properties

of the fluid can be considered as transient regime. Amaly’s experiment showed

similar results. When Reynolds number is 15000 and step angles are when 10

~ 15 °, the reattachment length of simulation comes out higher than the

experiment. At the same condition, the velocity profile and turbulence intensity

fits well. If using a different turbulence model or other schemes, the more

accurate result is expected. The results from 47000 to 64000 showed very good

agreement. If the expansion ratio is 1.48 and Reynolds number is larger than

15000, the simulations are expected to be able to get accurate results using the

k-omega model.

0

2

4

6

8

10

0 20 40 60 80 100

Xr/

Hs

Step angle(α)

ER=1.48, Re=64000

Ruck(1993)

RANS(k-omega)

24




0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100

Xr/

Hs

Step angle(α)

ER=2.00, Re=15000

Ruck(1993)

RANS(k-omega)

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100

Xr/

Hs

Step angle(α)

ER=2.00, Re=47000

Ruck(1993)

RANS(k-omega)

25


When the expansion ratio is 2.00 and Reynolds numbers are 15,000 ~ 64000,

the results are compared to the experiment data. There was no experiment data

for the Reynolds number less than 11000. The reattachment length of the

simulations are almost similar to the experimental data. Reattachment length

of the simulation comes out lower than the experimental result at the step angle

90° regardless of Reynolds number. In this condition, max turbulence intensity

showed lager value than experiment data. Lower value at the 90° and the

maximum turbulent intensity seemed an association. The lager value of

maximum turbulence intensity can be interpreted energy diffused well then the

size of the reattachment decreased. If the expansion ratio is 2.00 and Reynolds

number is 15000, the simulation can be expected to get accurate result.

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100

Xr/

Hs

Step angle(α)

ER=2.00, Re=64000

Ruck(1993)

RANS(k-omega)

26


Fig 3.1.7Reattachment length, ER=3.27, Re=11000


0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100

Xr/

Hs

Step angle(α)

ER=3.27, Re=11000

Ruck(1993)

RANS(k-omega)

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100

Xr/

Hs

Step angle(α)

ER=3.27, Re=15000

Ruck(1993)

RANS(k-omega)

27

When the expansion ration is 3.27, the results of the simulation were

unstable. Lager expansion ratio, the bigger difference between the inlet and

outlet ( /outlet inletER h h ). When making the structural mesh, the grids are

always same from inlet to outlet in the z-direction. The difference between the

inlet and the outlet causes the difference of grid size. In this study, the flow rate

( Q U HW ) is constant. It means Reynolds number( Re /U H ) is

constant but grid size is increased. Simulation can be affected with Reynolds

number and grid size. Large Reynolds number needs fine grid. So, big

expansion ratio could cause mesh problem.

3.1.1.4 Empirical equation and simulation

Ruck and Makiola has proposed the following equation based on

experimental data from the same study

.

3

2.521

max 1

0

0 max,

/

n

n

x

xER

n r

inlet

out

in

ue

u

x x x

u u

hER

h

The simulation results and the equation were compared for the model

validation

28

Fig 3.1.9 Empirical equation and simulation, ER=1.48, Re=15000


0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Um

ax/U

0

X/Xr

x_n

90

45

30

25

20

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Um

ax/U

0

X/Xr

x_n

90

45

30

25

20

29



0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Um

ax/U

0

X/Xr

x_n

90

45

30

25

20

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Um

ax/U

0

X/Xr

x_n

90

45

30

25

20

30



0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Um

ax/U

0

X/Xr

x_n

90

45

25

20

0.15

0.35

0.55

0.75

0.95

1.15

0 0.5 1 1.5 2 2.5 3

축제목

X/Xr

x_n

90

25

20

31

When nx is less than 1.5, the simulation results showed good agreement to

the equation. In separation zone, nx is between 0~1. So, the numerical model

in this study is expected to get good results on separation zone.

3.1.1.5 Discussion

The relationship between step angle and reattachment length was analyzed

in expansion ratio of 2.00 which showed most good agreement with

experimental data. The reattachment length was zero when step angle is 10°,

which means that there is no separation at step angle of 10°.

The reattachment length showed the appearance of a steady incensement

from 15 ° to 30 °, the convergence value (about 7.5) after 30 °. This is consistent

with the results of Ruck. Looking at the test of Ruck, reattachment length in

many cases is increased gradually starting from 0 and showed convergence after

25 ~ 30 °.

In the model validation with ruck’s experiment, the comprehensive results

are as follows. When Reynolds number is bigger than 15000, expansion ratio

is same or smaller than 2.00 and in the separation zone, the model is expected

to get good agreement with experiment data.

32

Fig 3.1.15 Reattachment length and angle, ER=2.00

3.1.2 Nakagawa and Nezu’s experiment

Looking at previous research, the study about the effects of step angle to the

backward facing step flow was rare. In this study, it is expected to get 솟

relationship in water channel. The model validation of the previous chapter

used air condition, model validation of water was required. So, the experimental

data of Nakagawa and Nezu(2010) was used.

The experiment with water channel flow was conducted by Nakagawa &

Nezu. The geometry was single-sided backward facing step with orthogonal

step angle. Velocity data were measured by laser Doppler anemometry. The

33

case of which Reynolds number was about 23000 was chosen for the validation

study. The definition of the Reynolds number is given by

0Re inu H

Where 0u is maximum inlet velocity,

inH is water level of the inlet channel,

and is the kinetic viscosity (1.001x10^-6m2/s).

3.1.2.1 Boundary conditions

Boundary conditions are as follows.

1. 2D by width : flow direction(x), orthogonal to the bottom(y)

2. Steady state, 0du

dt , simulated until the flow is steady state

3. Boundary of top was slip (water-air intersection) and bottom was no

slip condition. 0bottomu

4. Outlet – zero gradient , 0du

ds , Neumann boundary condition for

outlet

5. Inlet- non-uniform and steady condition, and fully developed

value(specific velocity by experiment) with RANS,

6. 0.292 /inU m s

86inH cm , 106outH cm , 2stepH cm

Re 23400

: 90°

Kinematic viscosity : 1.00e-06m2/s (water)

34

3.1.2.2 Flow distribution

Fig 3.1.16 Nakagawa’s experiment and k-omega simulation

Velocity profile of simulation showed good agreement with Nakagawa’s

study. The reattachment length of simulation (5.212) is exactly same with

experiment (5.2). This model showed also good agreement with water channel

case.

3.2 Application

The virtual scenarios of two conditions were applied to examine the flow rate

characteristics in the actual channel. In the first scenario, while using the

boundary conditions of the Nakagawa's channel, flows were analyzed with

various angle steps. In the first scenario, while using the boundary conditions

of the real weir, flows were analyzed with various angle steps.

35

3.2.1 Nakagawa’s channel

Relationship between step angle and reattachment length is analyzed using

Nakagawa’s open channel condition. The expected result was gradually

incensement of reattachment length at small step angle and convergence after

25~30°

3.2.1.1. Boundary conditions








outlet

5. Inlet- non-uniform and steady condition, and fully developed

value(specific velocity by experiment) with RANS,

6. 0.292 /inU m s

86inH cm , 106outH cm , 2stepH cm

Re 23400

: 10°, 15°, 20°, 25°, 30°, 45°


36

3.2.1.4 Result and Discussion

Fig 3.2.1 Reattachment length and step angles (Nakagawa’s condition)

The result of the first application is as follows. Reattachment length was 0

when the step angle is 10°, increased gradually to 25° then after this step angle

was seem to be constant

This result indicates the following facts. In the air and water, effect of step

angle to reattachment length is same. There is no separation zone at the small

step angle. Then the separation zone is gradually increased and to be constant

after specific step angle.

Nakagawa, 5.2

0

1

2

3

4

5

6

0 20 40 60 80 100

Xr/

Hs

Step angle

Sim, RANS(k-omega)

Exp, Nakagawa(2010)

37

3.2.2 Sejong weir

The application was performed with real river condition based on previous

model validation and application with the open channel. The reasonswhy

choose Sejong weir were similarities of boundary conditions and geometrical

condition. The velocity used in the application is from Ministry of environment

and the inlet height was control inlet from k-water. The actual protection of

bottom of river was not 90° of step angle then step angle was changed from 10°

to 45°. The expected result was gradually incensement of reattachment length

at small step angle and convergence after 25~30°. There was no depth

directional velocity profile, the model of Sejong weir condition was not

validated. The roughness of the bottom was same with previous application

(Nakagwa’s condition)

3.2.2.1 Boundary conditions








outlet

5. Inlet- non-uniform and steady condition, and fully developed value

(specific velocity by experiment) with RANS.

38

0.08 /inU m s

1.15inH m , 3.65outH m , 2.6stepH m

Re 83916

: 10°, 15°, 20°, 25°, 30°, 45°


3.2.2.3 Result and Discussion

Fig 3.2.2 Reattachment length and step angles (Sejong weir condition)

The simulation result based on Sejong wier was similar to the results of

previous chapter in this study. The reattachment length was 0 at the step angle

of 10~15° and gradually increased. Reattachment length was 6.286 at 30° and

6.734 at 45°. The reattachment of 45° is 7% larger than 30°.

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50

Xr

Step angle

Sim(k-omega)

39

The second application result indicated that the step angle effect to the

reattachment length is similar at the big scale simulation. To get a accurate

results, 3D geometrical data and numerical analysis is needed. In addition,

depth direction velocity data would be necessary to get model validation at real

weir and river .

40

Chapter 4. Conclusion

This study was proceeded to analyze the relationship between the geometry

of the fluid flow and the backward step facing flow. The selected parameters

were the step angle and the rate of expansion.

The numerical methods were used to achieve the object of the study. RANS

approach has been mainly used and k-omega model was used as a turbulent

model. The vertical direction 2D simulation was used to examine the separation

phenomenon of the bottom part intensively. .

Using the Ruck’s experimental data (1993) of which various step angle,

expansion ratios and Reynolds numbers, the simulation was validated and

compared. Through the numerical analysis, it was confirmed that the

reattachment length was increased depending on the angle increases. In

addition, the reattachment length was found to be converged after 30 o.

After comparing the experiment value with using the air, the fluid property

was converted to water. Using the Nakagawa’s experimental data (2010) of

which step angle is 90 o, the simulation was validated. And the applications

were conducted changing angles. At the water condition, the result was also

similar with air. Reattachment length was increased depending on step angle

increase. And the length was showed convergence after 25 o.

As an application of actual river case, the simulations were performed using

the geometrical and flow data of Sejong weir to analyze the effect of step angles.

The same 2D simulation was conducted. The boundary conditions came from

41

K-water and Ministry of Environment. (This simulations are on the cluster). At

the actual weir condition, the result was also similar with other channels.

Reattachment length was increased depending on step angle increase. And the

length was showed convergence after 25 o.

In previous studies, it has been conducted mainly for the case of orthogonal

step angle. To analyze or design the real product (weir, HVAC system etc.), the

study which changes step angles is needed. Especially for the water channel,

most of studies covered orthogonal step angles. In this study, angle effects on

reattachment length and turbulent intensity was analyzed. It was confirmed that

the reattachment length was increased depending on the angle increases.

In future research, the numerical data could be compared with measured data

which is conducted on a laboratory scale (such as Nakagawa’s study) varying

the step angles. The comparison could clarify the effect of step angles. In

addition, measuring the velocity components along the vertical direction of

river and validating with simulation data would be needed for getting accurate

simulation results. Then 3D simulation with geometrical data around weir

would be helpful for designing protection of bottom to prevent scouring.

42

References

[1] 한국수자원학회, 2005, 하천설계기준

[2] 한국수자원학회, 1999, 하천공사표준시방서

[3] 한국수자원공사, 2009, 낙동강 살리기 18 공구사업 수리모형실험보고서

[4] 한국수자원공사, 2009, 낙동강 살리기 23 공구사업 수리모형실험고고서

[5]한국수자원공사, 2009, 낙동강 살리기 24 공구사업 수리모형실험보고서

[6] ABBOTT, D. E. & KLINE, S. J. 1962 Experimental investigations of

subsonic turbulent flow over single and double backward-facing steps. Trans.

A.S.M.E. D: J. Basic Engng 84, 317.

[7] SEBAN, R. A. 1964 Heat transfer to the turbulent separated flows of air

downstream of a step in the surface of a plate. Trans. A.S.M.E. C: J. Heat

Transfer 86, 259.

[8] GOLDSTEIN, R. J., ERIKSEN, V. L., OLSON, R. M. & ECKERT, E. R.

G. 1970 Laminar separation reattachment, and transition of flow over a

downstream-facing step. Trans. A.S.M.E. D: J. Basic Engng 92, 732.

[9] DENHAM, M. K. & PATRICK, M. A. 1974 Laminar flow over a

downstream-facing step in a two-dimensional flow channel. Trans. Inst. Chem.

Engrs 52, 361. DURST, F. & TROPEA,

[10] ETHERIDQE, D. W. & KEMP, P. H. 1978 Measurements of turbulent

flow downstream of a backward-facing step. J. Fluid Mech. 86, 545.

43

[11] Armaly, B. F., Durst, F., Pereira, J. C. F. & Sch ̈ onung, B.1983

Experimental and theoreticalinvestigation of backward-facing step. J. Fluid

Mech.127, 473-496

[12] Backward-Facing Step measurements at Low Reynolds Number,

Reh=5000, 1994, NASA

[13] Le H., MoinP., Kim J. (1997) Direct numerical simulation of turbulentflow

over a backward- facingstep.Journal of Fluid Mechanics. 330: pg. 349-374.

[14] Hall, Stephen D., and Tracie J. Barber. "A Detailed CFD and Experimental

Investigation of a Benchmark Turbulent Backward Facing Step Flow."

[15] Park, Sungwon. “Experimental Study of Local Scouring at the

Downstream of River Bed Protection”, Ph.d Thesis, Seoul National University,

2016

[16] Ruck, B., and B. Makiola. "Flow separation over the inclined step."

Physics of Separated Flows—Numerical, Experimental, and Theoretical

Aspects. Vieweg+ Teubner Verlag, 1993. 47-55.

[17] Pope, Stephen B. Turbulent flows. Cambridge university press, 2000.

[18] Baek, Dong Hae, and Van Thinh Nguyen. "사행수로에서 3 차원 이송-

확산 방정식의 수치모의.” 제 40 회 대한토목학회 정기 학술대회, 2014.10,

885-886 (2 pages)

[19] Nakagawa, Hiroji, and Iehisa Nezu. "Experimental investigation on

turbulent structure of backward-facing step flow in an open channel." Journal

of Hydraulic Research 25, no. 1 (1987): 67-88.

44

[20] Choi, Hoi Hyun, Van Thinh Nguyen. "Numerical Analysis of Backward

Facing Step Flow with Varying Step Angles." 대한토목학회

학술대회,(2015.10): 195-196.

[21] Hoi-Hyun Choi, Van Thinh Nguyen, John Nguyen. “NUMERICAL

INVESTIGATION OF BACKWARD FACING STEP FLOWS OVER

VARIOUS STEP ANGLES.”. HIC2016. Paper NO. F20160406-1063

45

Appendix 1 Ruck and Makiola(1993)’s experiment and

simulation

Fig A1.1 Comparison between exp and sim, ER=1.48, Re=11000, α=10°


46



47



48



49



50



51



52



53



54



55

Fig 3.21 Comparison between exp and sim, ER=1.48, Re=64000, α=10°

Fig 3.22 Comparison between exp and sim, ER=1.48, Re=64000, α=15°

56



57



58



59



60



61



62



63



64



65



66



67



68



69



70



71



72



73



74



75


76

Appendix 2 Simulation result of Nakagawa and

Nezu(2010)’s experiment

Fig A2.1 Nakagawa’s conditions, Step angle =10°


77



78



79

Appendix 3 Simulation result of Sejong weir condition

Fig A3.1 Flow distribution with condition of Sejong weir, Step angle =10°



80


Fig A3.5Flow distribution with condition of Sejong weir, Step angle =30°


81

초 록

단일방향 후방계단흐름은 하천 하상 보호공, 위어, 댐 등과 같은

수리구조물 주위의 유체흐름을 연구하는 데 있어 분리구역, 재부착 길이,

난류강도와 같은 유체의 특성을 조사하기 위한 가장 기초적인 고려대상이다.

특히 계단 이후에 나타나는 분리구역은 에너지 상쇄와 쇄굴에 중요한 역할을

한다.

본 연구에서는 계단각과 확장률에 따른 후방계단흐름을 2 차원

수치해석으로 분석하였다. 오픈소스 프로그램인 OpenFoam이 계산을 위해

사용되었으며 Reynolds averaged Navier Stokes equation(RANS) 접근을

이용해 난류모델을 해석하였다.

선행 연구들은 직각 계단인 흐름에 대해 대부분 진행되었고 본

연구에서는 계단간의 영향에 대해 분석하기 위해 계단각을 10~45도로 변화

시켰다. 수치해석의 결과는 Ruck(1993), Nakagawa(2010)의 실험을 이용해

검증하였으며 대부분 실험결과 일치하는 것으로 판단되었다.

실제상황에서의 결과를 살펴보기 위해 세종보의 조건들을 이용하여 응용된

시뮬레이션이 실행되었다.

연구결과 계단각이 커짐에 따라 재부착 길이가 지속적으로 증가하는

것을 알 수 있었다. 또한 계단각이 25~30 도보다 커질 경우에는 일정한

값으로 수렴하는 형태를 보였다.

주요어: 전산유체역학, 후방계단흐름, 계단각, RANS, OpenFoam

학 번: 2014-22714

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