Scour Caused by a Propeller Jet

Scour Caused by a Propeller Jet

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Jian-Hao Hong 1; Yee-Meng Chiew, M.ASCE 2; and Nian-Sheng Cheng 3

Abstract: The scour hole induced by a propeller wash is experimentally investigated with noncohesive sediments. The associated scour profile at the asymptotic condition, which can be expressed by a combination of three polynomials, comprises (1) a small scour hole beneath the propeller, (2) a primary scour hole downstream of the small scour hole, and (3) a deposition mound farther downstream of the primary scour hole. The time-dependent maximum scour depth induced by propeller jets is closely related to the densimetric Froude number, a reference time scale, offset height relative to the propeller diameter, and sediment size. A semiempirical formula, which shows good agreement with the experimental data, is proposed to describe evolution of the maximum scour depth. An equation for the determination of the time to initiate scour is also proposed. Moreover, empirical equations for the maximum asymptotic scour depth induced by a propeller and threedimensional submerged offset jets agree well with the experimental data. The critical condition for the initiation of scour caused by propeller and offset jets is also presented. DOI: 10.1061/(ASCE)HY.1943-7900.0000746. © 2013 American Society of Civil Engineers. CE Database subject headings: Scour; Sediment transport. Author keywords: Propeller scour; Propeller wash; Scour; Sediment transport.

Introduction In the continual proliferation of global maritime trade, vessels with more powerful engines are being used more extensively. Driven by large-diameter propellers or side thrusters, the wash of these high velocity jets can seriously impact the seabed. Hamill et al. (1998) found that the strong jet flow can last for a distance of several propeller diameters from the propeller. Near such an intense jet flow, seabed material can easily be entrained, and severe erosion could occur on the bed or bank of navigation channels and around harbor structures. The impingement of propeller or thruster jets is more serious where large ships navigate in shallow water with a minimum keel clearance, for example, the berthing of the Ro-Ro ferries and unassisted maneuvering of ships with bow and stern thrusters. Scour induced by propeller or thruster wash has become one of the most important issues for the design and maintenance of navigation channels and harbor structures. Evidence reported by Bergh and Cederwall (1981) shows severe damages induced by propeller jets in 25 quay structures in Swedish harbors. Chait (1987) documented scour damage of many ports in South Africa. To minimize propeller-jet-induced scour and better protect port facilities, an improved understanding of scour due to propeller wash is imperative. 1

Research Fellow, DHI-NTU Centre, Nanyang Technological Univ., N1.2-B1-02, 50 Nanyang Ave., Singapore 639798; Associate Research Fellow, Taiwan Typhoon and Flood Research Institute, Taichung 40763, Taiwan. E-mail: [email protected] 2 Professor, School of Civil and Environmental Engineering, Nanyang Technological Univ., N1-01a-27, 50 Nanyang Ave., Singapore 639798 (corresponding author). E-mail: [email protected] 3 Associate Professor, School of Civil and Environmental Engineering, Nanyang Technological Univ., N1-01a-27, 50 Nanyang Ave., Singapore 639798. E-mail: [email protected] Note. This manuscript was submitted on August 15, 2012; approved on February 11, 2013; published online on February 13, 2013. Discussion period open until February 1, 2014; separate discussions must be submitted for individual papers. This paper is part of the Journal of Hydraulic Engineering, Vol. 139, No. 9, September 1, 2013. © ASCE, ISSN 0733-9429/2013/ 9-1003-1012/$25.00.

To this end, Blaauw and van de Kaa (1978), Bergh and Magnusson (1987), Verhey (1983), Hamill (1987, 1988), and Hamill et al. (1999) have experimentally studied propeller wash. Blaauw and van de Kaa (1978), Verhey (1983), Hamill (1988), and Aberle and Soehngen (2008) focused on the prediction of the maximum scour depth in the absence of a berthing structure. Hamill et al. (1999) extended their earlier studies by including the effect of propeller wash on quay walls. Through dimensional analysis, they showed that the densimetric Froude number, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fo ¼ U o = ðg 0 d50 Þ, plays the most important role in affecting scour depth, where U o = efflux velocity; d50 = median grain size; g 0 ¼ ½ðρs − ρÞ=ρg = relative gravitational acceleration; g = gravitational acceleration; ρ = density of water; and ρs = density of sediment particles. Hamill et al. (1999) not only investigated the effect of propeller wash on the quay walls but also time variation of the maximum scour depth. However, they did not propose any suitable reference time scale for the temporal development of the scour hole induced by a propeller wash. The aim of this study is to investigate the development of a scour hole with noncohesive sediments due to the jet induced by a rotating propeller. Measured experimental data are analyzed to show the similarity of the temporal variations of scour profiles. Influences of various parameters on the time-dependent maximum scour depth are also studied. Moreover, the study also examines the similarity of the maximum scour depth at the asymptotic condition, ds;me , induced by a propeller and that by a three-dimensional submerged offset jet.

Experimental Setup The experiments were carried out in a tank 1.8 m wide, 4 m long, and 1 m deep in the Hydraulics Laboratory, Nanyang Technological University, Singapore. A sediment recess 0.25 m deep, 1.8 m long, and 1.2 m wide was constructed inside the tank. Fig. 1 shows the layout of the experimental setup. Two propellers with diameter Dp ¼ 10 and 21 cm and two uniformly distributed sediment particles with a median grain size of d50 ¼ 0.24 and 0.34 mm were used in the experiments. The geometric standard deviations,

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Tank

ht


Dp

C

yo

hc

ds

False Floor

False Floor

Bed Features Zone C Deposit Mound

Zone B Primary Scour Hole

Zone A Small Scour Hole

Fig. 1. Schematic layout of experimental setup

σg ¼ ðd84 =d16 Þ0.5 , in which d16 and d84 are the particle size for which 16 and 84% are finer by weight, respectively, of the two sediments were 1.48 and 1.40. These values signify that the particle size distributions of both sediments were fairly uniform. Three offset height ratios, yo =Dp , where yo = offset height, and six different speeds of propeller rotation in terms of revolutions per minute (rpm) were tested. Table 1 summarizes the flow and geometric characteristics of the tests conducted in this study. For the purpose of comparison, 3 data sets of Karki et al. (2007) and 11 data sets of Hamill (1987) were also included. The tail water depths used in this study were

0.6 m and 0.45 m for Runs 1–3 and Runs 4–11, respectively. Before commencement of each experiment, the sediment was first leveled using a sand leveler. The tank was then slowly filled with water to avoid disturbing the leveled sand bed. Once the predetermined water depth was reached, the propeller was turned on. To investigate the time-dependent scour profiles induced by the rotating propeller, the latter was temporarily turned off at a specific time, at which juncture a point gauge was used to measure the geometry of the scour hole. The scour profile was measured at time = 0.5, 1, 2, 4, 8, 16, 32, and 64 h after commencement of each test. Since the experiments were performed in a confined tank without flow

Table 1. Summary of Test Conditions Experiment number R-1 R-2 R-3 R-4 R-5 R-6 R-7 R-8 R-9 R-10 R-11 H-1 H-2 H-3 H-4 H-5 H-6 H-7 H-8 H-9 H-10 H-11 K-1 K-2 K-3

Dp (cm)

d50 (mm)

yo (cm)

Uo (m=s)

Fo

Rj

ht (cm)

Type of scour

10 10 10 21 21 21 21 21 21 21 21 6.1 6.1 6.1 6.1 6.1 15.4 15.4 15.4 15.4 15.4 15.4 2.66 2.66 2.66

0.34 0.34 0.34 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.76 0.76 1.46 1.46 1.46 0.76 0.76 0.76 1.46 1.46 1.46 0.71 0.71 0.71

15 15 10 10.5 10.5 10.5 10.5 21 21 21 21 12.5 17.5 7.5 12.5 17.5 17.5 17.5 17.5 17.5 17.5 17.5 2.66 3.99 5.32

0.682 0.484 0.484 0.379 0.482 0.557 0.666 0.379 0.482 0.557 0.666 1.219 1.219 1.689 1.689 1.689 0.857 1.285 1.715 0.853 1.279 1.706 1.080 1.080 1.080

9.19 6.53 6.53 6.08 7.73 8.94 10.69 6.08 7.73 8.94 10.69 10.99 10.99 10.99 10.99 10.99 7.729 11.59 15.46 5.548 8.318 11.095 10.07 10.07 10.07

68,176 48,443 48,443 79,580 101,177 117,014 139,920 79,580 101,177 117,014 139,920 74,355 74,355 103,058 103,058 103,058 132,016 197,965 264,067 131,344 196,921 262,664 28,716 28,716 28,716

60 60 60 45 45 45 45 45 45 45 45 — — — — — — — — — — — 16 16 16

Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Propeller wash Squared offset jet Squared offset jet Squared offset jet

Note: In Column 1, R denotes the present study, K the study by Karki et al. (2007), and H the study by Hamill (1987); Dp = propeller or orifice diameter; d50 = median diameter pffiffiffiffiffiffiffiffiffiffiffiffiffiffiof sediment particle; yo = offset height (from seabed to center of propeller axis); Uo = efflux velocity; Fo ¼ densimetric Froude number ¼ U o = ðg 0 d50 Þ; Rj ¼ Reynolds numbers of the jets ¼ U o Dp =ν. 1004 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2013

J. Hydraul. Eng. 2013.139:1003-1012.

(a) Scouring processes along 6

Scouring Process The scouring process due to a propeller wash is complex, owing to the interaction between the jet affected by the rotating propeller and the resistance of the bed sediment. The evolution of a typical scouring profile along the longitudinal direction, which is illustrated in Fig. 2, may be divided into four stages: 1. Initial stage: During the initial stage, no obvious scour hole is formed directly beneath the propeller, but a main or primary scour hole is formed downstream of the propeller (Fig. 2). A large dune or deposition mound overlaid by smaller ripples are observed to form downstream of the scour hole. 2. Developing stage: During the developing stage, the size of the primary scour hole increases with time. However, since the rotating propeller not only induces the jet longitudinally but also entrains fluid from behind the propeller, a small scour hole forms directly beneath the propeller. At this stage of development, the bed

(b) Pictures for scouring processes

(c) Schematic diagram for

the centerline of propeller

scouring processes

location of propeller

Initial stage

4

0.5 h

Scour hole

0.5h

2

ds (cm)

ripples/dunes 0 -2 Flow

-4

Deposit crest

-6 Dp = 10 cm, yo = 10 cm rpm = 450, d50 = 0.34 mm

-8 -10

Location of propeller

0.5 h


Developing stage

4

ds (cm)

Plain view

2.0h

2.0 h main scour hole

2 0 -2

small scour hole

-4 -6 0.5 h 2.0 h

-8 -10


Stabilization stage

8.0 h

8.0h

Asymptotic stage

64.0 h

64h

4

ds (cm)

2 0 -2 -4 -6

0.5 h 2.0 h 8.0 h

-8 -10


4 2

ds (cm)


and sediment supply, one could safely assume that the turning off and on of the propeller would not affect the overall development of the scour profile, as was suggested by Hamill (1987). The variables used in the study (Table 1) were chosen on the basis of results from previous studies on local scour due to both propeller and three-dimensional submerged offset jets, e.g., Blaauw and van de Kaa (1978), Verhey (1983), and Chiew and Lim (1996). These studies reveal that the densimetric Froude number, Fo ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi U o = ðg 0 d50 Þ and offset height ratios, yo =Dp , are important parameters in affecting the resulting scour hole. To this end, these variables were systematically varied in this study. It must be stated that in pure jet studies, the velocity is simply the average velocity (¼ Q=A) of the flow where the cross-sectional area, A, is that of the jet source. For example, if the jet is from a circular pipe, then A is just the area of a circle. With a propeller jet, the velocity or efflux velocity, U o , is defined as the maximum velocity measured just downstream of the propeller. In this study, it was the velocity measured at the location 0.5Dp downstream of the propeller, where Dp = diameter of the propeller.

0 -2 -4 0.5 h 2.0 h 8.0 h 64 h

-6 -8 -10 -20

0

20

40

60

80 100 120 140 160

x (cm)

Fig. 2. Schematic diagram of scouring process induced by a propeller under a typical clearance condition JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2013 / 1005

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y Low rpm with high clearance High rpm with low clearance

Deposit Crest Dp y0

(l s , me – xs , me )

hc,me

ε

x

ds,me xsf xs,me ls,me lc,me

le,me

Fig. 3. Schematic diagram of scour profile under asymptotic condition induced by a propeller jet (definition of symbols is applicable to the scour hole formed by the propeller with low revolutions per minutes with high clearance—dashed line)

20 Temporal scour hole profiles Dp = 0.21 m, d50 = 0.24 mm y0/Dp = 0.5, 250rpm

15 10 Scour depth, ds (mm)

level between these two scour holes remains intact (no erosion). 3. Stabilization stage: During the stage of stabilization, the size of both the primary and small scour holes increases with time. This causes the length of the plane bed region between the two scour holes to decrease as they approach each other. 4. Asymptotic stage: After 64 h of testing, both the small and primary scour holes reach the asymptotic stage, at which point they are merged together. Beyond this time, the dimensions of the scour hole essentially remain unchanged. The longitudinal dimension of the main scour hole during the asymptotic stage is about 10 times that of the diameter of the propeller. Two typical scour profiles at the asymptotic condition induced by a propeller wash are shown in Fig. 3. The dashed and solid lines in the figure represent the conditions for a low propeller speed with a high offset height and a high propeller speed with a low offset height, respectively.

5 0 -5 -10

0.5 h 1.0 h 2.0 h 4.0 h

-15 -20

8.0 h 16.0 h 32.0 h 64.0 h

-25 0

(a)

50 100 150 200 250 300 Longitudinal distance from propeller plane, x (cm)

350


0.8 0.6 0.4

Scour Profiles

d s / d s,me


C

Temporal Variations of Scour Profiles The measured temporal developments of the scour profile along the centerline of the propeller, with Dp ¼ 0.21 m, d50 ¼ 0.24 mm, and rpm ¼ 250, are presented in Figs. 4 and 5 for low and high offset heights (yo =Dp ¼ 0.5 and 1), respectively. Figs. 4(a) and 5(a) show the dimensional temporal variations of the scour profiles. The maximum scour depth at the asymptotic condition, ds;me , and length of the primary scour hole (ls;me − X sf ) in the streamwise direction are used as the normalizing variables for the vertical and horizontal dimensions of the scour hole, respectively (Fig. 3). The dimensionless coordinates (^x, y^ ) of the scour hole, where x^ ¼ ðx − xsf Þ=ðls;me − xsf Þ, x = horizontal distance from the

0.2 0 -0.2 -0.4

0.5 h 1.0 h 2.0 h 4.0 h

-0.6 -0.8

8.0 h 16.0 h 32.0 h 64.0 h

-1 -1

(b)

-0.5

0

0.5

1 1.5 2 (x-xsf ) / (ls,me-xsf )

2.5

3

3.5

4

Fig. 4. Typical scour profiles for low offset height (yo =Dp ¼ 0.5) at different times: (a) dimensional scour profiles; (b) nondimensional scour profiles

1006 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2013

J. Hydraul. Eng. 2013.139:1003-1012.

10

5 Scour depth, d s (mm)

Scour Profiles under Asymptotic Condition

Temporal scour hole profiles Dp = 0.21 m, d50 = 0.24 mm y0/Dp = 1.0, 250rpm

0

-5 0.5 h 1.0 h 2.0 h 4.0 h

8.0 h 16.0 h 32.0 h 64.0 h

-15 0

(a)

50 100 150 200 250 300 350 Longitudinal distance from propeller plane, x (cm)

400


0.8 0.6 0.4 d s / ds,me

0.2 0 -0.2 -0.4

0.5 h 1.0 h 2.0 h 4.0 h

-0.6 -0.8

8.0 h 16.0 h 32.0 h 64.0 h

y^ ð0 ≤ x^ ≤ 1Þ ¼ a0 þ a1 x^ þ a2 x^ 2 þ a3 x^ 3

for main scour hole ð1Þ

-1 0

1 2 (x-xsf ) / (ls,me-xsf )

3

4

30

Fig. 5. Typical scour profiles for high offset height (yo =Dp ¼ 1.0) at different times: (a) dimensional scour profiles; (b) nondimensional scour profiles

propeller, xsf = horizontal distance from the propeller to the beginning of the primary scour hole, ls;me = length of scour hole from the propeller to the end of the primary scour hole, y^ ¼ y=ds;me , y = vertical distance from the undisturbed bed level, and ds;me are computed accordingly. The nondimensional scour profiles at different time intervals are shown in Figs. 4(b) and 5(b) for shallow and high offset heights, respectively. The main distinguishing feature of the shape of the scour profiles between the low and high offset height conditions is the scour depth at the transition of the small and primary scour hole, ε ¼ 0 for high offset height and ε > 0 for low offset height, as shown in Fig. 3. The data in the figures reveal that the nondimensional scour profiles, including the small scour hole, primary scour hole, and the deposition mound, generally collapse for both the low and high offset height conditions. This is particularly true for the primary scour hole. Moreover, the collapse is more apparent for tests conducted with the low offset height. At the initial and early developing stage of the scouring processes (time ≤ 1 h), the data, as shown in Figs. 4(b) and 5(b) [see full circles (•) and open circles (○)], exhibit slightly greater scatter. It may be inferred that the scouring rate is more uncertain at the early stage of the scouring processes. With time, when the development stage enters the later developing stage and beyond (time > 1 h), the variations in the scouring rate decrease and the data collapse more readily, indicating that the asymptotic scour profiles under different propeller diameter, speed, and offset height exhibit a similar behavior. The scour profile in the asymptotic condition is discussed in greater detail in the following section.

Equlibrium scour hole profile Dp = 0.21 m, d50 = 0.24 mm


-1

(b)

10 0 -10 -20

rpm y0/Dp

-30

200 250 300

0.5 0.5 0.5

-40 0

50

(a)

100 150 200 Longitudinal distance, x (cm)

250

300

30 Equlibrium scour hole profile Dp = 0.21 m, d 50 = 0.24 mm



-10

Fig. 6 shows the asymptotic scour profiles for tests conducted with different offset height ratios. The results show that the dimension of both the small and primary scour holes associated with a low offset height ratio is larger than that with a high offset height ratio. The same applies to the dune that forms downstream of the scour hole. Moreover, for a given offset height ratio, the dimension of scour hole increases with an increase in the propeller speed (rpm). For the case with a low offset height ratio (yo =Dp ¼ 0.5), the shape of the primary scour hole is more rounded, with a shorter scour length (ls;me − X sf ). and deeper scour depth as compared to that with a larger offset height ratio (yo =Dp ¼ 1.0). Fig. 6 shows that the scour-depth-to-scour-length ratio of the primary scour hole with yo =Dp ¼ 0.5 is approximately twice that with yo =Dp ¼ 1.0. Published results, e.g., those by Rajaratnam (1981), Chatterjee et al. (1994), Ali and Lim (1986), Chin et al. (1996), and Dey and Sarkar (2006), have shown the existence of a similarity between the centerline profile of scour holes formed downstream of a pure jet. It is surmised that a similarity also exists for the scour profile, including the small scour hole, primary scour hole, and dune, formed by a propeller jet. To verify this hypothesis, the measured dimensionless scour profiles are compared with three independently deduced polynomials as follows:

10 0 -10 -20

rpm y0/Dp

-30

200 250 300

1.0 1.0 1.0

-40 0

(b)

50

100 150 200 Longitudinal distance, x (cm)

250

300

Fig. 6. Scour profiles: (a) low offset height ratio, yo =Dp ¼ 0.5; (b) high offset height ratio, yo =Dp ¼ 1.0


J. Hydraul. Eng. 2013.139:1003-1012.

y^ ð^x > 1Þ ¼ b0 þ b1 x^ þ b2 x^ 2 þ b3 x^ 3

1

for deposition mound ð2Þ

1 Dp = 0.21m d50= 0.24 mm y0 = 0.105m

ds / ds,me

0.5

0

ds / ds,me

-0.5

Data d50 (mm) H-1 1.46 H-2 0.76

-1

-1.5 -1

-0.5

0

0.5 (x-xsf ) / (ls,me-xsf )

1

1.5

2

Fig. 8. Comparison of Hamill’s (1987) data and proposed model

remarkably well with the experimental data. The data for both the low and high offset height ratios generally support the existence of similarity of the scour hole profile and that the proposed polynomials may be used to describe the scour profile. To ascertain the general applicability of the proposed model, the experimental data of Hamill (1987) is used for comparison, as shown in Fig. 8. It must be noted that the experimental setup of Hamill is slightly different from that of the present study, as is shown in the inset in Fig. 8. Since his propeller was placed at the edge of the sediment recess, i.e., the interface of the solid and loose boundary, the small scour hole directly beneath the propeller did not form; only the primary scour hole and deposition mound were observed. Despite this difference, Hamill’s data compare very well with the model proposed here, except for the height of the deposition mound.

The scour depth at any time t, ds;t , is assumed to be a function of 11 independent variables as follows:

rpm y0 /Dp 200 250 300

-1

ds;t ¼ f1 ðU 0 ; Dp ; d50 ; y0 ; ρ; g; ρs ; μ; σg ; t; ht Þ

0.5 0.5 0.5

-1.5 -1

-0.5

0


1

1.5

2

1 Dp = 0.21m d50= 0.24 mm y0 = 0.21m

0.5

0

-0.5

200 250 300

gDp gd50 gyo gds;t ρ gt ; ¼ f ; ; ; 2 U 2o U 2o U 2o U 2o ðρs − ρÞ U o

1.0 1.0 1.0

-1.5 -1

-0.5

0


1

1.5

ð4Þ

where μ = fluid viscosity; and ht = tail water depth. Rajaratnam (1981) had earlier conducted a study of erosion by a plain wall jet and found that when the jet Reynolds number, Rj ¼ U o Dp =μ (where μ = fluid viscosity) exceeds 10,000, the effect of fluid viscosity could be neglected. Since the values of Rj (Table 1) range from 2.8 × 104 to 2.64 × 105 , the viscosity term is ignored in subsequent analyses. The nonuniformity of the bed sediment likely will affect the scouring processes induced by the propeller jet. However, in this study the effect of the nonuniformity of sediment particle is not considered. Moreover, the tail water depths used in this study are reasonably high, and its effect is not examined in the study. Using the Buckingham π theorem and choosing g, ρ, and U o as the fundamental variables, one gets

rpm y0 /Dp -1

(b)

0

Temporal Variation of Maximum Scour Depth

-0.5

(a)

0.5

for small scour hole ð3Þ

in which a0–3 , b0–3 , and c0–3 are coefficients. Since all the polynomials are cubic functions, four boundary conditions are needed to determine the four coefficients related to each of the equations. The four boundary conditions associated with Eq. (1), which describes the primary scour hole profile, are as follows: (1) at x ¼ 0, y ¼ ε, where ε is a local degradation (Fig. 3); (2) x ¼ xs;me , y ¼ −ds;me ; (3) x ¼ xs;me , dy=dx ¼ 0; (4) x ¼ ls;me , y ¼ 0, where xs;me = horizontal distance of maximum scour depth from propeller. Similarly, the coefficients b0 , b1 , b2 , and b3 , related to Eq. (2), which depicts the deposition mound, are determined with the following boundary conditions: (1) x ¼ ls;me , dyðy < 0Þ=dx ¼ dyðy > 0Þ=dx; (2) x ¼ ls;me , y ¼ 0; (3) x ¼ lc;me , y ¼ hc;me ; and (4) x ¼ le;me , y ¼ 0, where lc;me = horizontal distance of crest of deposition mound from propeller, hc;me = height of deposition mound, and le;me = distance of downstream end of dune from propeller. For the small scour hole, one may use a similar concept as that used for Eq. (1) to determine the coefficients in Eq. (3). Figs. 7(a and b) show the measured nondimensional scour profiles compared with that computed using Eqs. (1)–(3) with yo =Dp ¼ 0.5 and 1.0, respectively. For the case of the low offset height ratio with yo =Dp ¼ 0.5, not only do all the experimental data collapse into one curve but the proposed model also agrees

ds / ds,me


y^ ð^x < 0Þ ¼ c0 þ c1 x^ þ c2 x^ 2 þ c3 x^ 3

Hamill (1987) Dp = 0.154m y0 /Dp = 1.14 rpm = 400

ð5Þ

2

Fig. 7. Dimensionless scour profiles: (a) offset ratio, yo =Dp ¼ 0.5; (b) offset ratio, yo =Dp ¼ 1.0

Rearrangement of Eq. (5) leads to ds;t y y t ¼ f 2 Fo ; o ; o ; Dp Dp d50 ðDp =U o Þ


J. Hydraul. Eng. 2013.139:1003-1012.

ð6Þ

In the preceding equation, Fo ¼ U o =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðρsρ−ρÞgd50 and Dp =U o ,

3 Data of Hamill (1988) Propeller jet

which is denoted by tR, is the proposed time scale. Using the measured data, a nonlinear regression analysis yields the following dimensionless equation for the estimation of time-dependent scour depth induced by a propeller wash: k 3 ds;t U0 t ¼ k1 log10 ð7Þ − k2 Dp Dp

d50 = 1.46 mm Dp = 6.1 mm rpm = 1800 Simulation

d s,t /Dp

2

y0/Dp 1.23 2.05 2.87

Run H-3 H-4 H-5

1

0

k1 ¼ 0.014F1.120 o

yo Dp

k2 ¼

−1.740

yo Dp

1.882F−0.009 o

k3 ¼ 2.477Fo−0.073

2.302

yo Dp

0.53

yo d50 yo d50

yo d50

2

−0.170

ð8Þ

−0.441

3

4 5 log 10(Uot/Dp)

6

7

Fig. 10. Time-dependent maximum scour depth due to propeller wash: comparison between data of Hamill (1987) and present approach for different offset heights (yo =Dp )

ð9Þ

−0.045

ð10Þ

Fig. 9 shows the overall comparison between the proposed model and the time-dependent scour depth. The R-squared value is 0.983. In general, the proposed model agrees well with the experimental data. It must be stated that Eq. (7) is only valid for 0.5 < yo =Dp < 2.87 and 5.55 < Fo < 11.1. For the purpose of checking the performance of the proposed model in detail, Figs. 10–12 show a comparison of the computed results with the published experimental data of Hamill (1987) and Karki et al. (2007). The data were selected to include a wide range of densimetric Froude number, Fo , offset height ratio, yo =Dp , and relative submergence, yo =d50 . The calculated normalized time variations of scour depths (ds;t =Dp ) with log10 ðU o t=Dp Þ are plotted together with the corresponding measured normalized scour depths for comparison. Figs. 10 and 11 show the comparison between the data of Hamill (1987) and those computed using the proposed model for a different offset height, (yo =Dp ), and propeller speed, respectively. Physically, for a condition with given propeller diameter, propeller speed, and sediment size, the scour depth decreases with an increase in offset height. The comparison presented in Fig. 10 shows that the agreement between the measured data and computed results is excellent.

On the other hand, for a condition with given propeller diameter, sediment size, and offset height, the scour depth increases with an increase in propeller speed. Fig. 11 shows the results of the comparison, indicating that the proposed equation can predict the experimental data well. In addition, an attempt is also made to predict the timedependent scour depth due to a square offset jet (Karki et al. 2007). Here, the propeller diameter in Eq. (7) is replaced by the width of the square orifice. Fig. 12 shows a comparison between 2 Data of Hamill (1988) H_7-9, propeller jet

d s,t /Dp

d50 = 0.76 mm Dp = 15.4 mm y0 = 17.5 cm Simulation

1

Run rpm H-6 400 H-7 600 H-8 800

0 1

2

3 4 log 10(Uot/Dp)

5

6

Fig. 11. Time-dependent maximum scour depth due to propeller wash: comparison between data of Hamill (1987) and present approach for different propeller speeds (revolutions per minute)

3 R2 = 0.983 4 Data of Karki et al. (2007) Square offset jet

2 3

ds,t /d0

Predicted ds,t /Dp


where

d50 = 0.71 mm d0 = 2.66 mm F0 = 10.07

2

1 Run K-1 K-2 K-3

1

y0/d0 1.0 1.5 2.0

0

0

2

0

1

2

4

6

8

log10(Uot/d0)

3

Observed d s,t/Dp

Fig. 9. Comparison between observed and computed time-dependent scour depth using the proposed method

Fig. 12. Time-dependent maximum scour depth due to squared offset jet: comparison between data of Karki et al. (2007) and present approach for different offset height ratios (yo =Dp )


J. Hydraul. Eng. 2013.139:1003-1012.

Table 2. Range of Data Collected on Scour Caused by Offset and Propeller Jets Remark

0.91–1.00 Propeller jet 0.47–12.76 Circular jet 2.03–2.87

Square jet

0.50–1.50

Propeller jet

Note: yo =do = offset ratio; Fo ¼ densimetric Froude number ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uo = ðg 0 d50 Þ; ds;me =do = dimensionless scour depth.

100

A close examination of Eq. (7) reveals that when ds;t ¼ 0, the equation reduces to log10 ðU o t=Dp Þ ¼ k2 . Using Eq. (9), one may calculate the time needed to initiate the scouring, ts :

Submerged jets Circular jet

Propeller jet

Square jet

y0 /d0

y0 /d0

y0 /d0

0.50 1.00 1.50

1.00 1.50 2.00

0.50 - 0.81 1.83 - 1.97 3.94

10

dsem/do

p

ds;me =do

Fo

Hamill (1987) 1.14 5.55–7.73 Chiew and Lim 0.50–15.75 13.14–60.74 (1996) Karki et al. 1.00–2.00 10.00 (2007) Present study 0.50–1.50 6.08–10.69

Initiation of Propeller Scour

−0.009 yo 2.302 yo −0.441 t s ¼ 10k2 ¼ 10½1.882Fo ðDp Þ ðd50 Þ D

yo =do

Investigators

ð11Þ

dsem d0

Uo

= 0.21Fo

1

This positive value of ts is due to the fact that when a propeller starts rotating, the propeller jet requires a certain time before scouring of the seabed commences. In Eq. (11), the exponent of the densimetric Froude number is negative, signifying that ts increase with a reduction Fo . On the other hand, the exponent of yo is 1.861 (positive), which means that ts increases with an increase in offset height yo . In addition, the exponents of Dp and d50 are −2.302 and 0.441 (or 0.446 if the d50 in Fo is also included), respectively. This implies that a larger propeller diameter and larger sediment particles require less and more time to initiate scouring, respectively. Using Eq. (11), one can estimate the time to start the scouring process, which ranges from 0.06 to 12.1 s for the experiments conducted in this study. The result shows that the seabed scour induced by propeller wash will occur within a very short time period after the propeller begins rotating. However, the characteristics of the propeller will produce different efflux velocities, which may result in different scour depths with the same propeller speed. More experimental data are required to improve the applied range of the proposed equation.

Comparison of Maximum Scour Depth between Propeller and Three-Dimensional Submerged Offset Jets Due to the resemblance, both in terms of their geometrical likeness and the cause of formation, of the scour hole formed by a propeller jet and three-dimensional submerged offset jets, this section attempts to explore their similarities and differences. Accordingly, the experimental data from the published literature and those from this study, as tabulated in Table 2, are used in the following analyses. Fig. 13 shows the effect of Fo on the dimensionless scour depth ds;me =do . In Fig. 13, do denotes the nozzle diameter of the circular jet, width of the square jet, or propeller diameter. The solid line and its corresponding equation are fitted using the data of a circular wall jet (Chiew and Lim 1996). By definition, a wall jet exists when the offset height ratio yo =do ¼ 0.5. The data in Fig. 13 are collected from circular and square jet and propeller jet experiments. In general, Chiew and Lim’s (1996) formula for circular wall jets (yo =do ¼ 0.5) forms the upper limit of

Chiew and Lim (1996) water wall jet formula

0.1 1

10 Fo

100

Fig. 13. Comparisons of ds;me =do and Fo between offset and propeller jets

the scour depth. The result shows that the data associated with Fo ≤ 10, which are mostly associated with propeller jet scour, tend to merge into the trend formed by the offset jet data at higher Fo. Moreover, the scour depths at the asymptotic condition caused by propeller and offset jets with the same given Fo and yo =do are essentially the same. The data in Fig. 13 also show that Fo is not the only parameter affecting the depth of scour; the offset height also plays an equally important role. Chiew and Lim (1996) and Karki et al. (2007) reported that the reduction in scour depth with offset distance can be attributed to jet diffusion. Fig. 14 shows the combined effect of yo =do and Fo . For a given value of Fo , Fig. 14 clearly shows that the dimensionless scour depth ds;me =do decreases with an increase of yo =do . 15

Type of jet Circular

10

ds,me/do


the data of Karki et al. (2007) and those computed using the proposed equation. In general, good agreement is achieved except for the slight deviations at the initial stages of scouring. Overall, the proposed model not only can simulate the time variation of maximum scour depth due to propeller wash but can also be applied to the prediction of scour depth induced by a three-dimensional submerged offset jet. The propeller diameter is chosen as the length scale, so it is convenient for practical use because the maximum equilibrium scour is an unknown and not involved in the calculation. Moreover, Eq. (7) does not take the time to equilibrium into account for the time variation of scour depth, which may be another advantage of this equation.

Fo 13.1 - 13.9 24.4 - 25.0 57.3 - 60.7

Square

10

Propeller

6.08 - 6.11 8.94 - 9.19

Simulation

Fo = 60

5 Fo = 10

Fo = 25

0 0.5

1

10 yo /do

100

Fig. 14. Comparison of experimental data and maximum scour depth equation


J. Hydraul. Eng. 2013.139:1003-1012.

The dashed line in Fig. 14, which shows the simulated results based on Eq. (12), describes the trend of the data well. It must be noted that the exponent of the first term on the right-hand side of the equation is positive, indicating that the dimensionless scour depth ds;me =do is directly proportional to Fo . In contrast, the exponent for the second term on the right-hand side of the equation is negative, signifying that ds;me =do is inversely proportional to yo =do . The empirical model clearly can express the physical meaning of the relationship of scour depth, densimetric Froude number, and offset height. It must be stated that Eq. (12) is valid for 0.5 < yo =Dp < 2.87 and 5.55 < Fo < 11.1.

Initiation of Propeller and Jet Scour Using the first term on the right-hand side of Eq. (12) for a given value of y=do , one can obtain the critical densimetric Froude number, Foc , for the initiation of jet scouring as follows: Foc ¼ 4.114

y0 ; do

y0 ≥ 0.5 do

ð13Þ

Note that Eq. (13) is only valid within the same limits as those for Eq. (12). Fig. 15 shows the functional relationship of Eq. (13) and the experimental data of Chiew and Lim (1996), Karki et al. (2007), Hamill (1987), and the present study. In this figure, the lefthand side of the line represents the no-scour condition, while the right-hand side of the line denotes the formation of a scour hole. All the experimental data, including tests conducted under both a pure or propeller jet, expectedly fall below the straight line, confirming

20 10

No scour depth

0.5

1

Laboratory experiments on seabed scour due to a propeller wash were conducted with two uniform sands and different offset heights and propeller speeds in a tank without current. The following conclusions are drawn from the study: 1. The asymptotic scour profile induced by propeller wash, which includes the small scour hole beneath the propeller, primary scour hole, and the deposition mound, follows a particular geometrical similarity that can be expressed by a combination of three polynomials with suitable boundary conditions. 2. Using the experimental data, a semiempirical formula [Eq. (7)] is proposed to estimate the time-dependent maximum scour depth. The simulated results compare well with the present experimental data and those from the published literature. By replacing the propeller diameter with the width of a square orifice, Eq. (7) may also be used to predict the time variation of maximum scour depth due to a square offset jet. 3. Using Eq. (7), the time to initiate propeller scour, ts , may also be computed; the result shows that ts decreases with decreasing densimetric Froude number and propeller diameter but increases with increasing sediment size and offset height. The experimental data also show that propeller-wash-induced scour commences within a very short duration after the propeller begins its rotation. 4. The densimetric Froude number Fo and offset ratio, yo =do , play the most important role in affecting the scour dimensions caused by both offset and propeller jets. In general, for the same Fo value, the wall jet induces the deepest scour depth. The wall jet equations proposed by Chiew and Lim (1996) formed the upper bound for scour depths induced by a submerged circular offset and a propeller jet. Eq. (12) can be used to account for the scour depth induced by an offset and a propeller jet and to deduce the critical condition for the initiation of scour. Eq. (13) may be used for scour design to compute the threshold condition for the initiation of scour caused by either an offset or propeller jet at sewage outfall, quay, and other related hydraulic, port, and coastal structures.

Notation

Scour

0.1

Conclusions

The authors would like to thank the Maritime and Port Authority of Singapore (MPA), Hamburg Port Authority (HPA), and DHI-NTU Centre for funding this research. In addition, Indra Susanto, Neo Hui Shan, and Ang Xiu Xuan are gratefully acknowledged for their assistance in performing the experiments.

Critical curve Fo = 4.114 yo/do, yo/do

1

that scouring indeed has taken place in all instances. Eq. (13) in the figure may be used for scour design to compute the threshold condition for the initiation of scour caused by either an offset or propeller jet at sewage outfall, quay, and other related hydraulic, port, and coastal structures.

Acknowledgments

Submerged offset jets Chiew and Lim (1996) Circular jet Square jet Karki et al. (2007) propeller jet Hamill (1987) propeller jet Present

yo /do


Fig. 14 also reveals that a critical offset height ratio exists beyond which scouring ceases altogether for a given Fo -value. On the other hand, for the same yo =do value, a similar critical densimetric Froude number also exists below which scouring does not occur. In both these instances, which are clearly illustrated in the figure, the intensity of the jet has become too weak to entrain the bed sediments. Given the foregoing reasoning, by considering the critical condition for the initiation of scouring, an empirical equation that relates ds;me =do with Fo and yo =do is fitted to the data as follows: 0.955 −0.022 d y y0 13 s;me ¼ 0.265 Fo − 4.114 0 ; do d0 d0 y ð12Þ × o ≥ 0.5 d0

10

100

Fo

Fig. 15. Critical condition for initiation of scour induced by threedimensional submerged offset and propeller jets

The following symbols are used in this paper: A = cross-sectional area of jet source; a0–3 = coefficients; b0–3 = coefficients; C = propeller tip to bed clearance; c0–3 = coefficients;


J. Hydraul. Eng. 2013.139:1003-1012.


Dp = diameter of propeller; do = diameter of nuzzle, width of squared orifice or propeller diameter; ds;me = maximum asymptotic scour depth; ds;t = maximum time-dependent scour depth; d16 = particle size is 16% finer by weight; d50 = median sediment size; d84 = particle size is 84% finer by weight; Fo = densimetric Froude number; g = acceleration due to gravity; hc;me = maximum height of deposit mound at asymptotic condition; ht = tailwater depth; k1 = adjustment factor; k2 = adjustment factor; k3 = adjustment factor; lc;mc = horizontal distance from propeller tip to deposit crest; le;me = horizontal distance between propeller tip and end of deposit crest; ls;me = length of scour hole determined from location of propeller to end of main scour hole; n = number of propeller revolutions per second; Q = flow rate of jet source; Rej = Reynolds number of jet; S = relative sediment density; T = time; ts = time to start scouring processes; U o = efflux velocity; W s;me = maximum scour width at asymptotic condition; X = horizontal distance; X 0 = length of zone of flow establishment; xsf = horizontal distance from propeller hub to starting location of main scour; xs;me = horizontal distance between propeller tip and position of maximum scour depth; x^ = ðx − xsf Þ=ðls;me − xsf Þ dimensionless horizontal distance from hub of propeller; y = vertical distance; yo = offset height; y^ = y=ds;me ; ε = factor; μ = dynamic viscosity of fluid; ρ = density of water; ρs = density of sediment particles; and σg = geometric standard deviation of sediment particles.

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