Post-trenching with a Trailing Suction Hopper Dredger

Transcription

1 Post-trenching with a Trailing Suction Hopper Dredger Master thesis Hydraulic Engineering Kevin Van de Leur This thesis may not be released before

2 Post-trenching with a Trailing Suction Hopper Dredger Master Thesis Kevin Van de Leur December 13, 2010 Dredging Development Department Royal Boskalis Westminster NV Papendrecht, The Netherlands Faculty Civil Engineering & Geosciences Delft University of Technology Delft, The Netherlands

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4 Preface This master thesis forms the culmination of the time I spent studying at the university in Delft. Taking the trip down memory lane I can t be anything less than grateful for the opportunities I had. One of these opportunities was my graduation at Boskalis. The combination of performing a literature study, execute DIY model tests and develop my own numerical model made my graduation a very diverse and interesting project. Hereby I would like to express my gratitude towards the members of my committee, their support and knowledge was very valuable. Special gratitude goes out to Mark Biesheuvel for his patience and guidance during the past nine months. I am convinced that the broad education and extracurricular activities in Delft formed me to be more than an engineer. Therefore I can only hope that the university will provide the same opportunities for future students. Kevin Van de Leur Rotterdam, the Netherlands December 13, 2010 Graduation committee: Prof.dr.ir. C. van Rhee ir. G.L.M. van der Schrieck ing. M. Biesheuvel Dr.ir. R.J. Labeur Delft University of Technology Delft University of Technology Royal Boskalis Westminster N.V. Delft University of Technology iii

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6 Abstract It is common practice to protect subsea pipelines by embedding them into the soil. Trenches can be made before or after the pipelines have been laid. In the latter case, the excavation process is called post-trenching. The essence of post-trenching, as handled in this thesis, is erosion of sand by a waterjet. The literature study focused on the processes of jets and erosion. A lot of research has been done in the field of water jets and usefull information is widely available. Nevertheless the available information on the subject of impinging jets is rather limited and the validity remains questionable. Water jets used for post-trenching create high flow velocities for which the traditional erosion equations are not valid. Therefore use is made of a special set of equations for high speed erosion. With the information provided by the literature study a description of jetting in sand was made. The known processes were arranged resulting in a set of equations. Following the rules for scaling the set of equations was converted into a properly scaled model. Preliminary model tests were conducted to observe the jet-process and narrow down the possible jet angles. These preliminary tests were followed by scale model tests to determine the erosion depth for different nozzle angles. A numerical model was developed to simulate the jet erosion. Since the known erosion equations could not model the erosion behaviour of a jet, a turbulence term was introduced. The results of the simulations were compared with the model tests. Though the numerical erosion model showed promising results, it could not be validated due to a lack of data. The most important conclusions are that soil can be eroded to the desired depth, a data-set has been created and much insight is gained with respect to the post-trenching process. Last but not least, a numerical model was made that can prove to be useful after better validation. v

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8 Contents 1 Introduction Background Post-trenching with a TSHD Problem definition Jets Introduction Circular turbulent jet Flow development region Developed flow region Impinging circular jet Oblique impinging circular jet Conclusion Erosion Introduction Basic principles Sedimentation Erosion Van Rijn formula Van Rhee stability parameter Approximation of the van Rhee formula Conclusion Hofland stability parameter Description of jetting in sand Introduction Jet regimes Penetrating jet regime for translating jet Deflective jet regime for translating jet Transitional jet regime for translating jet Processes Jet vii

9 CONTENTS Turbulence Erosion depth Centrifugal force Gravity current Settling Breaching Scaling of jet processes Introduction Scale factors Similarity Dimensionless indicators Froude Reynolds Scaling Jet pressure and flow velocity Turbulence Erosion velocity Erosion depth Centrifugal force Gravity current Settling Breaching Grain diameter Scale scenarios, indicators and effects Froude scenario Reynolds scenario Velocity scale n u = 1 scenario Conclusion Small scale model tests 47 7 Large scale testing 49 8 Erosion model Introduction Model overview Fluid dynamics Turbulence model Boundary conditions Erosion calculations Data gathering Flow driven erosion Turbulence driven erosion viii

10 8.4.4 Computation of the bed level for the next time step Results erosion model Introduction Stability Stationary jet Trailing jet Low flow velocity High flow velocity Conclusion Conclusions and recommendations Conclusions based on model tests Conclusions based on numeric erosion model Recommendations A Small scale tests 79 B Conversion of CPT to density 81 C Mass balance 83 D Test results 85 E Boundary conditions 87 F Stagnation point 91 F.1 Introduction F.2 Pressure gradient F.3 Turbulence F.3.1 Friction velocity F.3.2 LES F.3.3 Hofland parameter ix

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12 List of Figures 1.1 TSHD Prins der Nederlanden Pipe configurations with draghead and nozzle Circular jet Flow development Impinging circular jet Oblique impinging jet Equations approximating the Shields curve Shields curve Erosion forms Scour hole Translating penetrating jet Translating deflecting jet Translating jet in transitional regime Erosion front Settling in transitional regime Domain for erosion model Flow chart of erosion model Patches Samplepoints Calculation of slope angle Wall function Erosion in the EM Translation in the EM Bed level for a translating jet Area of transition between cilindrical jet inlet and rectangular domain Cross section jet inlet Velocity distribution stationary jet Pressure distribution stationary jet xi

13 LIST OF FIGURES 9.5 Turbulence distribution stationary jet Simulation results Cross section of domain after simulation This figure is not available Cross section of domain after simulation test E.1 File boundary conditions pressure E.2 File boundary conditions velocity E.3 File boundary conditions turbulence E.4 File boundary conditions turbulent dissipation F.1 Pressure in stagnation point F.2 Pressure gradient F.3 Penetrating jet F.4 Flow pattern for LES solver F.5 Erosion with LES solver F.6 Determination of Shields parameter xii

14 List of Tables 5.1 Results for Froude scaling Results for Reynolds scaling Results for velocity scale n L = Boundary conditions in the erosion model xiii

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16 Chapter 1 Introduction 1.1 Background The energy demand of modern society is constantly increasing due to factors as growth of the world population and increasing welfare. The majority of this energy demand is fulfilled by conventional sources such as oil and gas. Transport of these commodities by pipeline is common practice, both on land and offshore. Offshore pipelines are subject to forces of nature but also to man made hazards. A major problem is formed by subsea currents. First of all the currents can erode soil supporting a pipe. The pipe will start to deform and, when stresses become too large, it will fail. The same failure mechanism can occur when the currents are strong enough to transport a pipeline. Another problem is formed by shipping. In shallow water near busy shipping routes, pipelines must be embedded to minimize the chance of damage by shipping, debris or whatsoever. Finally there is the problem of buoyancy. Depending on the substance that is transported by the pipeline, it has to be embedded or covered with stones to prevent buoyancy of the pipeline. It is clear that protection is important since maintenance or reparations are time-consuming and expensive efforts. One way to protect pipelines is embedding them in the soil, this process is called trenching. When the pipeline has already been laid the soil underneath the pipeline must be excavated, this is called post-trenching. 1

17 Introduction 1.2 Post-trenching with a TSHD Although several methods exist to embed pipelines, in this thesis the focus goes out to the options involving Trailing Suction Hopper Dredgers (TSHD). An TSHD is a ship capable of sucking up sand which they collect in an onboard hopper. After the sand has been collected the TSHD sails to a dump site and discharges its load. Figure 1.1: TSHD Prins der Nederlanden The usual dredging process can be used to create a trench and lay a pipeline in this trench afterwards. The disadvantage of this approach is that the dredged sand has to be lifted to the water surface and dumped somewhere. The process can be made more efficient if the sand does not have to be handled. This is were (post-) trenching with a jet comes in. For post-trenching the suction pipe is connected to the outlet of the dredge pump. At the outlet the pump does not create suction but pressure. The suction pipe now becomes a pressure pipe thrusting out water. The draghead is replaced by a nozzle which concentrates the outflow of water, see figure

18 (a) Draghead (b) Nozzle Figure 1.2: Pipe configurations with draghead and nozzle 1.3 Problem definition The basic principle behind post-trenching is the excavation of sand under a pipeline, over a length that allows the pipe to sink under its own weight. Given the fact that most pipelines are relatively inflexible, the trench must be of considerable length. The exact length of the trench depends on the properties of the pipe and the desired depth. When jetting a trench, sand is brought into suspension. After a certain amount of time this suspended sand will settle again at the sea bottom and in the trench. In the ideal situation the sand does not settle in the trench, or only after the pipeline has been lowered. This means that the jet has two functions. First of all it must create a trench of sufficient depth. Secondly the sand must be jetted out of the trench to prevent it from settling in the trench again. These considerations lead to the main research question: Determination of the optimum combination of the jet-angle, propagation speed and jet-pressure resulting in the largest deepening of a pipeline when post-trenching with a TSHD. 3

19 Introduction In addition to this question a second objective is formulated: Modeling of flow and erosion caused by a water-jet. To arrive at an answer for these questions, the study is divided in three parts: a literature study, model tests and the simulations with a model. The literature study is described is chapters 2 and 3. Chapter 2 gives a description of submerged jets. The physics of a free flowing submerged jet are described after which a look is taken at impinging jets. As a result of a flow impinging on a sand bed, erosion occurs. This subject is treated extensively in chapter 3. The basic principles of erosion are explained and the equations describing erosion are presented. Subsequently the erosion equation for high flow velocities is presented. Chapter 3 is concluded with an extensive derivation of the high speed erosion equations as a prelude to the second part of this thesis, the model tests. Chapter 4 describes all the known relevant process for jetting in sand. These processes are converted to scale rules in chapter 5. These two chapters form the basis of the model tests and ensure correct scaling between model and prototype. The small scale model tests are described in chapter 6. The different influences of e.g. the flow velocity or positioning of the jet are assessed and analysed. This section of the thesis is concluded with chapter 7 describing the large scale tests. Productions and erosion depths are looked into and an answer to the main research question is formulated. In chapter 8 the developed erosion model is described and insight is given into the workings of this model. Following this chapter the results of the simulations are presented in chapter 9. 4

20 Chapter 2 Jets 2.1 Introduction In this chapter will be explained how jet induced flows develop. A lot of research in this field was done by Rajaratnam [11]. The conclusions of his research will be the main subject of this chapter. The circular free flowing turbulent jet will be described first after which impinging jets are handled. 2.2 Circular turbulent jet First of all a description will be made of a free flowing circular turbulent jet in water. In figure 2.1 a definition sketch of a circular turbulent jet can be seen. Along the axis of the jet, the flow can be divided into a flow development region and a region of fully developed flow. The boundary between both regimes lies at a distance of approximately 6.D 0 from the jet. Figure 2.1: Definition sketch of circular turbulent jet [11] 5

21 Jets Flow development region The flow development region is characterized by the following equation: With u m,s = u 0 (2.1) u m,s = Maximum core velocity [ ] m s u 0 = Velocity at jet nozzle [ ] m s Immediately after the nozzle the flow will start to diverge, see figure 2.2. The reason is that a shear stress is present between the flow and surrounding fluid due to a difference in velocity. The surrounding stagnant fluid will be accelerated, thereby increasing the total amount of flow. This is called entrainment. Figure 2.2: Definition sketch of flow development region [11] The amount of entrainment in the region of flow development is described according to Albertson et al. [4]: With Q s = 1 + 0, 083. s ( ) s 2 + 0, 013. (2.2) Q 0 D 0 D 0 Q s = Flow at distance s from nozzle ] Q 0 = Flow at jet nozzle [ m 3 s D 0 = Diameter of jet at nozzle [m] s = Position along flow trajectory [m] [ ] m 3 s 6

22 Due to entrainment the total amount of flow will increase. Under the assumption that there are no frictional losses and the impulse remains constant, it is found that the average velocity decreases along the jet axis. The some holds if frictional losses are taken into account. Note that the potential core velocity will remain constant, but the average velocity decreases Developed flow region In the region of developed flow the turbulence generated at the boundaries between the jet flow and stagnant fluid has penetrated to the axis of the jet. At this point the velocity in the potential core will start to diminish along the flow trajectory. From experimental results the following relation was derived [11]: u m,s = 6, 3 s or u m,s = 6, 3.u 0.D 0 u 0 D 0 s (2.3) With this equation, the velocity in the center of the jet is calculated. For different reasons it is interesting to know the velocity at a distance from the centerline. This can be described with the following function [12]: With u (s, r) = u m,s.e 108( r s) 2 (2.4) u (s, r) = Velocity at distance s from nozzle and distance r from centerline [ ] m s r = Distance to center [m] Now the velocity distribution in both x and y-direction has been described. The entrainment for the developed flow region can be described with a rather simple formula [11]: Q s Q 0 = 0, 32. s D 0 or Q s = 0, 32. s.q 0 D 0 (2.5) The amount of entrainment increases constantly along the jet-axis. 7

23 Jets The assumption on which these formulas are based is that the impulse is constant along the flow trajectory: I s = I 0 = ρ.u 0.Q 0 = ρ.u 0 2. π 4.D 0 2 = constant (2.6) With I s = Impulse at distance s from nozzle [ ] I 0 = Impulse at nozzle kg.m s 2 [ ] ρ = Density of jetted medium kg m 3 [ ] kg.m s Impinging circular jet The next step is to consider a non-free flowing or impinging jet. point of impact the flow spreads radially outwards, see figure 2.3. At the Figure 2.3: Impinging circular jet [2] The jet will initially develop similar to a circular turbulent jet. At a distance of 0, 86. x H from the nozzle the velocity will start to decrease as the pressure builds up to a maximum at the impingement point [11]. The velocity in this point is zero. When the circular jet deflects, the pressure will drop with increasing distance to the point of impingement. Pressure is converted into velocity. 8

24 The velocity along r can be calculated with the next relation [11]: u m 1, 03 = r or u m = 1, 03.u 0.D 0 u 0 D 0 r (2.7) This relation has the same form as equation (2.3). It must be noted that the stand-off distance is missing although it was varied in the data analysed to derive this formula (8 SOD 24). The same observation was done by Rajaratnam. To support the results Rajaratnam uses the Buckingham-pi theorem to show that the distance between the jet and bed vanishes from the equation. However, this theorem focuses on dimensional analysis and gives no information about the physical validity. This can be illustrated by ( the ) fact that one would expect the appearance of the stand-off distance H D0 in equation (2.7). The stand-off distance is dimensionless and will therefore play no role in dimensional analysis. Equation (2.7) is considered as not reliable and will therefore not be used. 2.4 Oblique impinging circular jet The next step is to formulate a relation for oblique impinging circular jets, see figure 2.4. Theoretical analysis done by Beltaos [2] led to the formulation of the next equations: And u m = h (φ b, θ b ) r or u m = h (φ b, θ b ).u 0.D 0 u 0 D 0 r (2.8) With h (φ b, θ b ) = 1, cos φ b. cos θ b ( ) sin φb 2 (2.9) cos 2 θ b + sin θb sin φ b φ b = Angle between jet and surface in x, z plane [degrees] θ b = Angle between jet and surface in x, y plane [degrees] If θ b = 0, equation (2.9) goes over in: Now equation (2.8) becomes: h (φ b, 0) = 1, 1 sin φb. 1 + cos (φ b) 1 u m = 1, cos (φ b). u 0.D 0 sin φb 1 r (2.10) (2.11) 9

25 Jets Figure 2.4: Oblique impinging circular jet [2] Again the conclusion is that the stand-off distance has no influence, although this parameter was varied between values of 15 and 47 during execution of the tests. The absence of this factor again raises questions with respect to the reliability. Another point to mention is that the tests were done using high velocities up to 90 m s but the medium used was air. 2.5 Conclusion Relations were found to describe both free-flowing and impinging jets. The relations describing the circular turbulent jet are widely accepted and reliable. The research done in the field of impinging circular jets has produced relations in which the stand-off distance is absent. The reliability of these relations is questionable and they will not be used. 10

26 Chapter 3 Erosion 3.1 Introduction The main principle of post-trenching is erosion of soil. This chapter will give an explanation of trafitional erosion equations and explain how erosion is defined. Subsequently the equations for high speed erosion are given after which an approximation is derived. 3.2 Basic principles Erosion velocity is defined as the difference between erosion and sedimentation: With v e = E S ρ s. (1 n 0 c b ) v e = Erosion velocity [ ] m [ ] s S = Settling flux kg m 2.s ] E = Pick up flux [ kg m 2.s ρ s = Density of the grains [ ] kg m 3 n 0 = Porosity of the settled original bed [ ] c b = Near bed concentration [ ] (3.1) For low near bed concentrations the variable c b can be neglected [14] and the equation is simplified. However, this will not be done at this point. More important, in order to solve the equation, parameters E and S have to be defined. 11

27 Erosion Sedimentation The settling flux is defined as: S = ρ s.w s.c b (3.2) With w s = Settling velocity [ m s ] The parameter w s takes into account the effect of hindered settlement. Hindered settlement means that the fall velocity of the grain is lower than in clear water. The main reason is that water has to flow upwards while sediment particles sink. The upward directed flow hinders the settling of particles, hence the term hindered settlement. With w s = (1 c b ) α.w 0 (3.3) α 4 for average sand of 120 < D < 300 µm w 0 = Settling velocity for a single particle [ ] m s A lot of expressions exist for the settling velocity of a particle. The grain size and fall velocity of a particle determine whether or not the flow around a particle is turbulent or laminar. The settling velocity can be defined as [13]: 4. (ρ s ρ w.).g.d 50.ψ w 0 = (3.4) 3.ρ w.c D The drag coefficient in this equation can be described with the next empirical formulas: C D = 24 Re p Re p 1 (3.5) C D = 24 Re p + 3 Rep + 0, 34 1 Re p 2000 (3.6) C D = 0, 4 Re p 2000 (3.7) 12

28 Substitution of (3.5) for the laminar case or (3.7) for the turbulent case into (3.4) and (3.3) respectively leads to: w s = (1 c b ) α. ψ..g.d ν for laminar flow (3.8) w s = (1 c b ) α.1, 8..g.D 50.ψ for turbulent flow (3.9) With ψ Shape factor (0, 7 for [ sand) ] ν = Kinematic viscosity m 2 [ s] ρ w = Density of water kg m 3 g = Gravitational acceleration [ ] m s 2 D 50 = Median grain diameter [m] C D = Drag coefficient = ρs ρw ρ w [ ] Substitution of the drag coefficient for transitional flow is not as straight forward as for laminar and turbulent flow. Therefore use will be made of an empirical formula in the transitional region. This is the formula of Budryck valid for 0,1 mm < D 50 < 1 mm: (ρ s ρ w ).D w 0 = 8, 925 (3.10) D 50 In this equation the settling velocity is in mm s, D 50 in mm and the densities in ton. m 3 The settling velocity for the transitional regime is now defined as well: (ρ w s = (1 c b ) α s ρ w ).D , 925 (3.11) D 50 13

29 Erosion Erosion Under the influence of flow, soil particles are subjected to shear stress. Depending on the rate of flow the shear stress will vary. Each particle in a bed will begin to move when a certain amount of shear stress is applied, this onset of motion is called the critical shear stress (τ crit ). The bed shear stress is often expressed by the dimensionless Shields parameter: With θ = 2 τ = u (3.12) (ρ s ρ w ).g.d 50 g..d 50 θ = Shields parameter [ ] τ = Bed shear stress [ ] N m 2 u = Shear velocity [ ] m s The onset of motion is defined if the critical shear stress is substituted in equation (3.12). To calculate the erosion velocity it is necessary to know the value of the pick-up flux (E). 3.3 Van Rijn formula The pick-up flux can be calculated with the Van Rijn pick-up function [15]. And With φ = E ρ s. = 0, D 0,3 g..d 50 ( ) θ 1,5 θcr (3.13) θ cr D = D g ν 2 (3.14) φ = Pick up function [ ] θ cr = Critical shields parameter [ ] D = Particle parameter [ ] [ ] ν = T = Kinematic viscosity m 2 s T = Temperature [Celsius] In this thesis the temperature is assumed to be 20 degrees Celsius although this number is fairly high for post-trenching conditions. 14

30 With the erosion and sedimentation quantified, the erosion velocity can be calculated. However, the function of Van Rijn is only valid for low velocities < 1 m s. For post-trenching the jet velocities are much higher and the function is not usable in this form. 3.4 Van Rhee stability parameter For higher flow velocities different processes govern the erosion process thereby influencing the erosion velocity. To solve this problem, Van Rhee [14] alters the Van Rijn formula by introducing an extended version of the Shields parameter. A favourable result of this approach is that the Van Rhee formula is still valid for low flow velocities. The modified Shields parameter is defined as follows: ( sin (ϕ β) θ cr = θ cr + v e. n ) l n 0 1. sin ϕ k l 1 n l. (1 n 0 ) With (3.15) θ cr = Modified Shields parameter [ ] ϕ = Angle of internal friction [degrees] β = Slope angle [degrees] k l = Permeability at loose state [ ] m s n l = Porosity at loose state [ ] The formula for the modified Shields parameter can be divided into three contributions that influence the erosion velocity. The first term between brackets accounts for a sloping surface. The second term accounts for dilatancy. The flow over the bed leads to a shear stress that is exerted onto the grains, at a certain point this leads to shearing of the soil particles. The shearing of soil particles goes hand in hand with an increase in pore volume. Because the pore volume must be increased an underpressure builds up causing a hydraulic gradient, water wants to flow into the soil to fill up the pores. The effect of the underpressure is that the soil particles are less likely to erode, increasing the value of the critical Shields parameter. The third contribution takes into account the effect of seepage flow and the effect on the stability of the grains [10]. 15

31 Erosion Combination of equations (3.13) and (3.15) gives a set of formulas to calculate erosion at high flow velocities. With φ = E ρ s. g..d = 0, D 0,3 ( ) θ θ 1,5 cr (3.16) θ cr φ = Modified pick up flux [ ] Combination of equations (3.1), (3.2) and (3.16) leads to: v e = 1 ( (1 n 0 c b ). φ. ) g..d c b.w s (3.17) This function cannot be solved analytically due to the occurrence of the erosion velocity at both sides of the equation. One solution is to solve the equation numerically. Another solution is to find an approximation for the function Approximation of the van Rhee formula Parts of this paragraph are not available. The next step is to show that the erosion velocity is mainly a function of the flow velocity and grain diameter. In order to achieve this, the next paragraphs will handle the following variables: k 0, D, θ and θ cr. The formulation for k 0 is given in the next paragraph. The relations concerning D, θ and θ cr in the subsequent paragraphs can be found in Van Rijn [16]. Permeability The permeability of a settled bed (k 0 ) is given in Verruijt [17]: With k 0 = c.ρ w.d 50 2.n 0 3 µ. (1 n 0 ) 2 (3.18) c = Coefficient for void and grain ] shape, approx. 0, 005 0, 010 [ ] µ = Dynamic viscosity [ kg m.s The relation with the grain diameter becomes: k 0 D 50 2 (3.19) 16

32 Particle parameter The second variable to re-write is the particle parameter: And thus: D = D 50. ( ) 1.g 3 ν 2 (3.20) D D 50 (3.21) Shields parameter The Shields parameter is more laborious to derive. First the mobility Shields parameter is looked into. This parameter is defined as follows: With θ = f 0 8. u 2 g..d 50 (3.22) f 0 = Darcy Weisbach friction coefficient [ ] u = Flow velocity [ ] m s The friction coefficient can be defined by: With f 0 = 8.g C 2 (3.23) C = Chézy coefficient [ ] m 0,5 The Chézy coefficient for hydraulically rough flow is defined as follows: ( ) 12.h C = 18. log (3.24) With k s h = Water depth [m] k s = Effective bed roughness [m] In equation (3.24) the water depth is used instead of the hydraulic radius because the domain is infinite. Equation (3.24) can be approximated with the Strickler formula (for range C = 40 to 70) : s ( ) 1 h 6 C = 25. k s (3.25) 17

33 Erosion Combination of equations (3.23) and (3.25) gives: f ( h k s ) 1 3 (3.26) The Shields parameter can now be defined as: θ = ( h ks ) u 2. (3.27) g..d 50 For large flow velocities the erosion behaviour of a bed is different, so-called sheet flow will occur. When flow velocities become very high, thin layers of sediment are eroded instead of single particles. This phenomenon complicates the determination of the effective bed roughness. Most researchers relate the effective bed roughness to the median particle diameter and the Shields parameter (k s D 50.θ n ). However, this relation is still subject of discussion. For example Wilson [18] claims that n = 1, but this is only valid for θ < 4, 5. While for higher values of the Shields parameter Matou sek [8] proposes n = 1, 7. For convenience the relationship as given by Madsen et al. [5] will be used: This is written as: Substitution of (3.29) into (3.27) leads to: k s = 15.D 50 (3.28) k s D 50 (3.29) ( ) 1 h 3 D 50 u 2 θ =. (3.30) 64 g..d 50 The water depth is assumed to be constant as are the relative density and gravitational acceleration, giving: θ u2 D (3.31) 18

34 Critical Shields parameter The critical Shields parameter can be derived from the Shields curve. It can also be calculated by a set of equations that approximate the Shields curve, see figure 3.1. Figure 3.1: Equations approximating the Shields curve [16] With the help of these equations the Shields parameter can be plotted as a function of the particle parameter, see figure 3.2. Figure 3.2: Shields parameter as function of particle parameter [16] The particles that will be used in the model tests have a median grain diameter of 170 µm or a particle parameter of 4,3. This value is a result of the assumption that the temperature is 20 degrees Celsius. A lower temperature will affect the kinematic viscosity and thereby the formula for the critical Shields parameter. According to figure 3.1 the equation to describe the Shields curve for D = 4, 3 is the following: θ cr = 0, 14 D 0,64 (3.32) 19

35 Erosion Or, substituting equation (3.20): θ cr = 0, 14 ( ( ) 1 ) 0,64 = D 50..g 3 ν 2 0, 14 ( ) 0,213 (3.33) D 0, g ν 2 The relation between the critical Shields parameter and grain diameter is: Conclusion θ cr 1 D 50 0,64 (3.34) The erosion velocity was approximated with the next equation: v e... (3.35) Substitution of the equations for permeability (3.19), particle parameter (3.21), Shields parameter (3.31) and the critical Shields parameter (3.33) into (3.35) gives: 3.5 Hofland stability parameter v e D 50 x.u x (3.36) In areas with high turbulence the turbulent fluctuations can lead to erosion of particles. The Shields parameter given in equation (3.12) depends solemnly on the flow velocity. The Hofland stability parameter describes the effect of flow and turbulence on the stability of particles. Hofland [6] alters the Shields parameter by adding a turbulence term (k) to obtain the following formula: θ H = max[ ū + α k Lm. Lm y ]2 g..d 50 (3.37) 20

36 With θ H = Hofland stability parameter [ ] ū = Average flow velocity [ ] m s α = Coefficient = 6 [ ] k = Turbulence [ m 2 s 2 ] L m = Length scale of turbulence [m]y = Vertical distance to bed [m] The length scale in this formula is still undefined. In the case of jets a suitable length scale is given by Celik et al [3]. This turbulence length scale depends on the jet diameter and is defined as follows: L m = c µ.0, 5.D 0 (3.38) c µ = Coefficient [ ] With the introduction of equations (3.37) and (3.38) the Shields parameter is made suitable for areas with high turbulence intensities e.g. the stagnation point of a jet. 21

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38 Chapter 4 Description of jetting in sand 4.1 Introduction The previous chapters described the generation of flow by a jet and the erosion of sand as a result. The next step is to investigate the development of the trench as a result of jet-induced erosion. This chapter describes the different relevant processes. 4.2 Jet regimes Some of the first authors to describe the behaviour of a jet impinging on an erodible bed were Kobus et al. [7]. Two forms of erosion were described, these forms are shown in figure 4.1. Figure 4.1: Erosion forms of impinging jets [7] 23

39 Description of jetting in sand Form I shows a regime in which the jet deflects on the soil, gradually creating a trench. This regime occurs when the flow velocities perpendicular to the bottom are relatively small. In this thesis the regime corresponding with Form I is called the deflective jet regime. Form II occurs when the flow velocity perpedicular to the bottom is high. This can be seen in figure 4.1, the flow at the point of impingement is highly turbulent. Instead of gradually creating a scour hole the jet penetrates the soil at the point of impingement. Erosion mainly takes place at the sides of this vertical scour hole. This jet regime will be called the penetrating jet regime Penetrating jet regime for translating jet The behaviour for translating jets, with respect to the two different jet regimes is very similar to the behaviour for stationary jets. The main difference is that the moving scour hole is not symmetric because one of the sides is open, see figure 4.2. Figure 4.2: Scour hole for penetrating translating jet [?] The penetrating jet regime is characterised by the fact that the majority of loosened sediment flows away through the created trench. This process is clearly visible in figure 4.3. The jet translates from left to right and the eroded sediment stays trapped in the trench. As can be seen the majority of the flow stays trapped and the transported sand will settle in the trench. This means that the soil is loosened after passage of the jet, but the bottom level will hardly decrease. 24

40 Figure 4.3: Topview of scour hole for translating penetrating jet Deflective jet regime for translating jet The deflective jet regime is characterised by the fact that the majority of loosened sediment flows away in any other direction than the trench. Subsequently little sediment will settle in the trench and, depending on the set-up, a significant increase in depth can be measured. An example of the deflective jet regime has been given in figure 4.4. Figure 4.4: Frontview of translating deflecting jet Transitional jet regime for translating jet Between the penetrating and deflecting jet regime there exists a zone in which both phenomena occur. This means that part of the sand can flow away out of the trench and the other part becomes trapped in a vortex over the length of the trench. After passage of the jet, there will still be loosened sand in the trench, although less in comparison with a full penetrating jet. Although the vortex is only clearly visible in reality or on film, figure 4.5 gives an overview of the transitional jet regime. 25

41 Description of jetting in sand Figure 4.5: Topview of translating jet in transitional regime 4.3 Processes The different regimes that can occur when a jet impinges on a erodible bed have been described. This paragraph describes the processes and other relevant parameters that play a role when jetting in sand. Some of the parameters will be treated extensively in preparation of chapter Jet The flow velocity at the nozzle of a jet depends on the pressure drop over the nozzle, assuming a velocity head of zero at the entrance of the nozzle. The relation for this pressure drop is: p = 1 2.ρ.u 0 2 (4.1) Turbulence Turbulence plays an important role in the erosion process and mixing/transport of particles. The amount of turbulent diffusion can be quantified by means of the Reynolds number. The formulation for the Reynolds number is: Re = u 0.D 0 υ (4.2) Erosion depth The relations concerning the erosion depth are derived from a two-dimensional graphic representation of reality, see figure 4.6. The striped arrows depict the flow trajectory of the jet. Perpendicular to this flow trajectory erosion 26

42 will occur at a certain rate (v e ). Along the trajectory the angle between the erosion velocity and trailing velocity will increase. The reason is that the erosion velocity decreases whilst the trailing velocity remains constant. This means that the increase in depth ( S y ) becomes gradually smaller. More important, the erosion depth seems to depend on the ratio between the erosion and trailing velocity. Figure 4.6: Angle α of erosion front With the help of these considerations the following relation is derived: With cos α = v e v trail (4.3) α = Angle between trail direction and erosion direction [ ] The variable α basically describes the shape of the trench and can be used to derive an expression for the erosion depth. The gain in depth per step ( S) is described as: ds y = sin (90 α).ds = cos α.ds = The total erosion depth becomes: s y = s 0 v e v trail ds (4.4) v e v trail.ds (4.5) 27

43 Description of jetting in sand With: s y = Erosion depth [m] Assuming that the development of the flow velocity is the same as for a free flowing jet, substitution of equations (3.36) and (2.3) into equation (4.5) gives: s y s 0 ( ) x x 6,3.u D 0.D 0 50 s.ds = D 50 x (6, 3.u 0.D 0 ) x s 1..ds (4.6) v trail v trail 0 sx The solution of this integral is: s y D 50 x.(6, 3.u 0.D 0 ) x. x v trail s x + c (4.7) This equation is only valid when the stand off distance is larger than 6,3.D 0. The constant c can be determined using the point of impingement with s y = 0 and s = 6, 3.D 0. c = x.d 50 x.(6, 3.u 0.D 0 ) x (6, 3.D 0 ) x = x.d 50 x.6, 3.u x 0.D 0 (4.8).v trail v trail Substituting c into equation (4.7) the following formula is obtained for the erosion depth: s y x.d 50 x (6, 3.u 0.D 0 ) x s x + x.d 50 x.6, 3.u x 0.D 0 (4.9).v trail v trail The limit of this function becomes: lim s y = c e. 6, 3x.D 50 x.u x 0.D 0 (4.10) s v trail The value of c e is determined by the difference between the exact and approximated value of the erosion velocity. It also accounts for the the stand-off distance Centrifugal force The curvature in the flow trajectory introduces centrifugal forces. The result is that particles are forced towards the bottom. This affects the density distribution over the height of the flow. The force that opposes this displacement towards the bed is the drag force. The drag force on a particle is determined by the particle properties and not the properties of the surrounding flow. This means that settling of a particle is mainly determined by the grain diameter. 28

44 The centrifugal force for a single particle is given by: With F c = m.u2 r = 1 6 π. (ρ s ρ w ).D 50 3.u 2 r (4.11) F c = Centrifugal force [N] m = mass [kg] The drag force acting on a particle is calculated with: With F D = C D. 1 4.π.D ρ.w c 2 (4.12) F D = Drag force [N] w c = Settling velocity due to centrifugal force [ ] m s Equaling equations (4.11) and (4.12) gives the velocity at which a particle moves to the bed due to centrifugal acceleration. The assumption is made that the radius of the flow trajectory is equal to the erosion depth, thus r = s y. w c = 4..D 50.u 2 3.s y.c D (4.13) Although this expression looks very much like the settling velocity of a particle due to gravity, it is not the same. In this case the driving force is the centrifugal acceleration. The total settling velocity of a particle is the sum of settling due to gravity and centrifugal acceleration. w s,c = w s + w c = 4..g.D50 4..D 50.u + 2 (4.14) 3.C D 3.s y.c D Or w s,c = 4..D50. g + u2 (4.15) 3.C D s y With w s,c = Total settling velocity [ ] m s From the second term on the right-hand side of this equation follows: u 2 s y > g (4.16) 29

45 Description of jetting in sand If the influence of the centrifugal force on the settling of a particle has more influence on the total settling velocity than the influence of gravity, the displacement due to centrifugal forces cannot be neglected Gravity current The particles suspended in the flow must be lifted out of the trench to create the largest possible erosion depth. What basically happens is that kinetic energy must be converted into potential energy, velocity is transformed into height. This tells whether or not eroded particles are able to settle outside, or remain trapped in the trench. In the case of a non buoyant waterjet in open air the energy equation becomes: or 1 2.m.u v 2 = m.g.h max (4.17) u v 2 = 2.g.h max (4.18) With: m = Mass of water [kg] u v = Vertical velocity component of waterjet at height zero [ ] m s h = Maximum height the waterjet reaches [ ] m s Rewriting the former equation leads to the formulation of the Froude number: With: F r = F r = Froude number [ ] u g.h (4.19) When post-trenching a sand water mixture with a higher density than the surrounding fluid must be lifted out of the trench. This density difference between the mixture and surrounding fluid acts as a driving force and can be expressed by the densimetric Froude number. 30

46 The densimetric Froude number is obtained if equation (4.19) is corrected for the differences in density: u F r d = g. ρm ρw ρ m.h (4.20) With: Settling F r d = Densimetric Froude number [kg] The tests, which were carried out in the scope of this thesis, used particles with D 50 = 170 µm. Therefore the formula for settling in the transitional regime should be used. The fall velocity for a particle in this regime was given in paragraph (ρ s ρ w ).D w 0 = 8, 925 (4.21) D Breaching When the jet has created a trench and the slope angle of the sidewalls is larger than the natural angle of repose, they will start to breach. This breaching process can be described with the so-called headwall velocity as explained by van der Schrieck [12]. This is the velocity at which the breach front moves away from the center of the trench. v wall =. k. cot ϕ (4.22) n 31

47

48 Chapter 5 Scaling of jet processes 5.1 Introduction In the ideal situation the post-trenching tests would be carried out on prototype scale. This means that a TSHD would be used to carry out the tests. Practical and financial considerations make this hardly possible. Therefore a model of the prototype must be made to do the tests on a smaller scale. The different scales of prototype and model imply that scaling effects can occur. Scale effects are deviations from the desired scaled behaviour. This chapter will give a short introduction on scaling according to van der Schrieck [12] after which scale rules are defined for the model tests to be executed. 5.2 Scale factors The scale factor of a quantity is the ratio between the value of that quantity in the prototype and model. n x = x p x m (5.1) For a constant the scale factor is always equal to one. If jetting in sand is considered, one of the parameters that must be scaled is the pressure generated by a jet. The equation for this pressure is: p = 1 2.ρ.u2 (5.2) In order to scale the pressure from prototype to model, the density and/or velocity can be scaled: n p = n ρ.n u 2 (5.3) It can be seen that scaling of the pressure depends on two different variables. Fortunately the scale of the product or quotient of two parameters is equal to the product or quotient of the scales of the two parameters. 33

49 Scaling of jet processes 5.3 Similarity In order to have a valid scale model there must be similarity between the prototype and model. Geometrical similarity means that the geometric dimensions are modeled at the same scale. Dynamic similarity refers to the similarity of the movements of objects. 5.4 Dimensionless indicators Indicators give the ratio between physical quantities of interest in model and prototype. In a valid scale model these quantities must have the same ratios in model and prototype. This leads to the demand that an indicator has scaling factor one Froude A well-known indicator giving dynamic similarity is the Froude indicator. The dimensionless Froude indicator gives the ratio between the forces of inertia and gravity. With F r 2 = u2 g.l (5.4) L = Length [m] Equality of the Froude number in model and prototype ensures correct scaling of gravity forces. Under the assumption that n g = 1 and the requirement that n F r = 1, the scale rule, representing inertia and gravity forces in the same ratio as in prototype, is as follows: Reynolds n u = n L (5.5) The dimensionless Reynolds indicator gives the ratio between inertial and viscous forces. With Re = u.l ν (5.6) Re = Reynolds number [ ] Correct scaling of viscous forces can be of importance when considering erosion. For example the shear stress over a bed is related to viscous forces. 34

50 The dimensionless particle parameter (D ) in equation (3.13) for the pickup flux depends on the kinematic viscosity showing the relevance of correct modeling viscous forces. Under the assumption that the viscosity is a constant (n υ = 1) the scale rule becomes: n u = 1 n L (5.7) 5.5 Scaling In chapters 3 and 4 (extensive) derivations were made to obtain proportionalities for the parameters describing the jet process in sand. Subsequently the theory for scaling was treated in the previous paragraphs. Now the theory will be applied to the proportionalities in order to obtain scale rules. After this a description will be given of the scaled down jet-process and the scale effects that occur when using different indicators Jet pressure and flow velocity The scale rule for pressure depends on the flow velocity and density. In both the prototype and model the jetted medium will be water, the density is constant. With n ρ = 1 the scale rule becomes: n p = n u 2 (5.8) Turbulence The Reynolds number was treated before. Using the same fluid in model and prototype the kinematic viscosity is constant and vanishes from the equation. n Re = n u.n D (5.9) Erosion velocity The proportionality for the erosion velocity with the grain diameter and flow velocity was described by equation (3.36). The scale rule is the following: n ve = n D50 x.n u x (5.10) 35

51 Scaling of jet processes Erosion depth The erosion depth was described by equation (4.10), the scale rule is: n sy = n D 50 x.n u0 x.n D0 n vtrail (5.11) Noticing that the flow and trail velocity both have the same scale factor n u, the scale rule for erosion depth is simplified. n sy = n D50 x.n u x.n D0 (5.12) Correctly scaling the erosion depth is the first step to geometrical similarity. Secondly the shape of the trench must be modeled correctly. Using equation (4.3) the scale rule for α is found. n cos(α) n v e n vtrail (5.13) Correct modelling of the trench shape requires n α =1. In other words, the erosion and trail velocity must be scaled by the same factor. This is also a requirement for dynamic similarity and thus: n ve = n vtrail (5.14) Now the scale rule for the trail velocity has been found as well Centrifugal force Independent of the flow regime the settling velocity due to centrifugal forces depends on the relation u2 s y. This leads to the next rule: Or n u 2 = n sy (5.15) n u = n sy (5.16) In this scale rule the Froude indicator can be recognised. Usage of another scaling indicator inevitably leads to scaling effects with respect to gravity currents and vertical transport Gravity current The effect of gravity current was accounted for by the densimetric Froude number. The important aspect of this indicator is the difference in density between the mixture and surrounding fluid. 36

52 The main concept (in this thesis) with respect to scaling is that the erosion velocity is scaled correctly. The same holds for the shape of the trench. With respect to the erosion of sand there exists dynamic and geometric similarity. This leads to the conclusion that the mixture density is the same in model and prototype. The scale rule for gravity currents with n ρm = n g = 1,becomes: n v = n h (5.17) Again it holds that, using n h = n L, the Froude indicator can be recognized Settling The scale rule for the settling of a particle in a transitional regime is derived from equation (3.10): n w0 = nd50 3 n D50 = n D50 (5.18) It can be seen that the settling of particles depends only on the grain diameter. Equation (3.10) from which this rule is derived uses the dimension mm for the grain diameter. This is not of any influence on the scale rule since the factor thousand, which converts the milimeters to meters is a constant. The validity of this scale rule is verified by plotting the settling velocity against the grain diameter, see figure 5.1. The relation takes on the form of a square-root function, for D 50 > 0, 3mm. Since equation (3.10) is valid for the range 0, 1 < D 50 < 1mm, this also holds for the scale rule. Figure 5.1: Settling in transitional regime 37

53 Scaling of jet processes Breaching The rule for breaching is derived from equation (4.22): n vwall = n k (5.19) Making use of the Kozeny-Carman formula (equation (3.18)) and assuming that the porosity in model and prototype is the same, the scale rule goes over in: n vwall = n k = n D50 2 (5.20) Grain diameter For the problem of jetting in sand it is important that the erosion velocity is scaled according to the used (dimensionless) indicator. If this condition is fulfilled, the shape and depth of the trench will be the same in model and prototype. This is the single most important concept for scaling in this thesis! Therefore the choice is made to alter the grain diameter in such a way that the erosion velocity will scale similar to the other flow velocities. Similar scaling of all velocities is also a requirement to attain dynamic similarity. This can best be explained by an example. Froude scaling means: n ve = n u = n L (5.21) If this is substituted into equation (5.10) it follows that: nl = (n D50 y ) x. ( n L 0,5 ) x (5.22) This is equation only valid if y = 1 x. The scale rule for the grain diameter is determined: n D50 = n 1 x L = 1 n L 1 x (5.23) It must be borne in mind that this result is only valid when the velocities in the model are scaled according to Froude (n u = n L ). Usage of another scale rule for the velocity will lead to another scaling factor for the grain diameter. Beforehand the scale factor cannot be determined since it depends on the scaling indicator, the factor for the grain diameter becomes: n D50 = n L y (5.24) 38

54 5.6 Scale scenarios, indicators and effects In paragraph 5.4 two dimensionless indicators were introduced. The first was the Froude indicator. When this indicator is kept the same in model and prototype correct scaling of gravity forces is ensured. However, this can lead to the wrong scaling of turbulence. It is not possible to apply a correct Froude scale and Reynolds scale at the same time. So one has to decide which forces are dominant in the processes under investigation. This means there is an offset between the desired and actual behaviour of the different parameters in the model. The offset in behaviour of a certain variable is called a scale effect. The second indicator was the Reynolds indicator. When this indicator is kept the same in model and prototype the viscous forces are scaled at the same scale as the inertia forces. These forces are of importance for e.g. the shear stress over a bed. This shear stress ultimately leads to erosion. Besides the scenarios of scaling according to the Froude and Reynolds indicators a third scenario can be used. This scenario assumes that the velocities in the model are equal to those in the prototype, only the geometric variables are scaled. The benefit is that the erosion velocity and shape of the trench are always modeled correct, although at the penalty of an overestimated density current. In the next paragraph the link between the scale rules and scaling scenarios is made. This can lead to the conclusion that a scale rule does not comply with the indicator used in the scenario. In this case a scale effect is present Froude scenario The flow velocity at the jet nozzle is taken as starting point. This velocity is scaled according to the Froude indicator (n u = n L ). The velocity at the jet nozzle depends on the pressure drop over the nozzle. Using the Froude scaling indicator it follows that the pressure scales with the length (n p = n u0 2 = n L ). The next process to consider is turbulence (n Re = n u0.n D0 = n 1,5 L ). The scale effects for turbulence are supposed to be negligible as long as a turbulent regime is maintained for the model. The erosion velocity depends directly on the flow velocity. It was stated before that, by altering the grain diameter, the erosion velocity will always be scaled according to the used indicator. Furthermore the trail velocity should 39

55 Scaling of jet processes be scaled similar to the flow velocity. So the flow-, trail- and erosion velocities are scaled similar and correct. The problem however is that the grain diameter is scaled using n D50 = n L 1 x. This scale rule implies that the grain diameter in the model must be larger than in the prototype. In itself this is no problem since the sand used in the model can be changed. However, the grain diameter has a direct influence on the settling and breaching velocities. Both the settling and breaching velocities will not comply with the indicator n u = n L. In case of the settling velocity (n w0 = n D50 = n L 1 x ) it means that particles will settle faster than in the prototype and the distance over which a particle settles is modeled too small. This is actually a favourable result since the model will give an underestimation of excavation capacities of the jet. The higher breaching velocity in the model (n vwall = n 2 D50 = n 2 x L ) mainly has an influence on the timescale for the breaching process to complete. There is a possibility that the different timescale affects the lay-out of the trench but this effect is assumed to be negligible. With the velocities covered a look is taken at the geometric variables. Again the jet is taken as a starting point. The diameter of the jet scales directly with the length (n D0 = n L ). It was derived that the erosion depth depends on the grain diameter and flow velocity. With the determination of the scale rule for the grain diameter, the rule for the erosion depth can be completed (n sy = n L (x. 1 x ). n L x.n L = n L ). The result is that the erosion depth scales directly with the length. No scale effects will occur with respect to the geometric variables. To complete the overview two more processes must be checked, centrifugal forces and gravity current. Derivation of the scale rule for both processes led to the formulation of the Froude scale rule. No scale effects will occur considering gravity currents and centrifugal forces. In table 5.1 an overview has been given of the scaling rules. The indicated scale indicator is the ratio between the desired scale and the actual scaling. 40

56 Velocity [ m ] s Variable Flow velocity* Scaling Desired Scale (Froude) scale effect nl nl 1 Geometry [m] Erosion velocity nl nl 1 Trail velocity* nl nl 1 1 Settling 1 nl 1 n L x 1 Breaching 2 nl 1 n L x Jet diameter* n L n L 1 Erosion depth n L n L Other Grain diameter* 1 n x 1 [µm] L n x 1 L n Gravity current [-] nl v = Centrifugal force nl n v = [-] Reynolds [-] n u. n L = 1 1 n L 1,5 Pressure* [bar] n L n L 1 Table 5.1: Results for Froude scaling The variables marked with an * can be controlled/chosen. The other variables depend on the settings of the controlable variables. The first and second column of table 5.1 show the different variables that must be scaled. The third column in this table gives the result of the scale rule as a function of the length scale. This is done for each variable as determined in paragraph 5.5. The fourth column gives the desired scale, again as a function of the length scale. For example, in this case it means that all velocities must be scaled with n L. If this is so, there is no scale effect and the scale effect has a value one. It is clear that the actual and desired scale are not the same for e.g. the breaching and settling velocity. For these two variables the scale effect can be determined. The scale effect is obtained by dividing the desired scale by the used scale. scale effect = desired scale actual scale = nl 1 2 n L x x = n L x (5.25) 41

57 Scaling of jet processes Assuming that the scaling factor between prototype and model is 10 or n L = 10 the scale effect for the breaching process can be quantified. scale effect = 10 x x = x (5.26) When the number of a scale effect is larger than one, the corresponding process is overestimated. For the breaching velocity this means that breaching in the model goes relatively faster than in the prototype. Vice versa, a number lower than one indicates underestimation of a process. The scale effect for the Reynolds number has a value of x. Apparently turbulence is largely underestimated in comparison with the prototype. It should be noted that when working with dimensionless variables like the Reynolds number, the desired scale is always equal to one Reynolds scenario An overview of scaling for the Reynolds scenario is given in table 5.2. Using Reynolds scaling it follows that the flow-, trail- and erosion velocities scale with the inverse of the length (n u = 1 n L ). Both the pressure and velocities are much higher in the model as in the prototype. Using n L = 10 the velocities will be a factor 10 larger. Again the grain diameter is scaled in order to accomodate correct scaling of the erosion velocity. This implies that scale effects occur for the settling and breaching velocities, both velocities are modeled too low. Apart from the timescale on which breaching takes place there are no serious problems. However, the fact that the settling velocity is modeled too low results in particles traveling further during model tests than they do in the prototype. This is dangerous since the model gives an overestimation of the prototype. For the nozzle diameter and erosion depth it follows that they scale with the length. With respect to the gravity current scale effects occur. The gravity current is largely overestimated. The result is that the gravity current will transport particles over a larger distance in model than prototype. This also means that particles are lifted out of the trench relatively too easy and the erosion depth is overestimated. The centrifugal forces are subject to scale effects as well. The forces acting on a particle, as a result of the curved flow trajectory are too high. This forces particles to the bed affecting the density distribution over the height of the flow. 42

58 Velocity [ m ] s Variable Scaling Desired Scale (Reynolds) scale effect Flow velocity* 1 n L 1 n L 1 Geometry [m] 1 1 Erosion velocity n L n L Trail velocity* n L n L Settling n L x n L 1 x 1 Breaching n L x n L 1 Jet diameter* n L n L 1 Erosion depth n L n L 1 Other Grain diameter* [µm] n x x L n x x L 1 1 n L nl = 1 Gravity current [-] n 1,5 L 1 31,6 1 n Centrifugal force nl L = 1 n 1,5 L 1 31,6 [-] Reynolds [-] n u.n L = Pressure* [bar] 1 n L 2 Table 5.2: Results for Reynolds scaling 1 n L 2 1 Besides the appearance of scale effects there is another problem with Reynolds scaling. Suppose the flow velocity in the prototype is 20 m s. Scaling at a length scale n L = 10 would mean that the flow velocity in the model should be 20 1 = 200 m s. The pressure needed, and subsequently the flow velocity, is 10 extremely high and other undesired effects such as cavitation could occur. It is not possible to produce these values with the available equipment Velocity scale n u = 1 scenario In this case the velocities in the model are the same as in the prototype. Nonen of the velocities is scale down. In order to arrive at the same erosion velocity in model and prototype the grain diameter cannot be scaled. Therefore the settling and breaching velocities are not influenced by scale effects. Logically the nozzle diameter and erosion depth scale with the length. After elaboration it follows that the Reynolds number scales with the length as well. Both the gravity current and centrifugal forces are overestimated. The ve- 43

59 Scaling of jet processes locities are the same as in the prototype which should lead to similarity, however the geometry has been scaled down leading to an overestimation of both variables. The results are given in table 5.3. Velocity [ ] m s Geometry [m] Variable Scaling Desired Scale n u = 1 scale effect Flow velocity* Erosion velocity Trail velocity* Settling Breaching Jet diameter* n L n L 1 Erosion depth n L n L 1 Other Grain diameter* [µm] Gravity current [-] 1 nl 1 3,16 Centrifugal force [-] 1 nl 1 3,16 Reynolds [-] n L 1 0,1 Pressure* [bar] Table 5.3: Results for velocity scale n L = Conclusion Three different methods for scaling have been looked into. Practical limitations lead to the conclusion that Reynolds scaling cannot be used. The variables subject to scale effects when using Froude are settling, breaching and turbulence. It was stated that the scale effect for breaching means that the timescale to complete the breach process is shorter. It will not affect the eventual lay-out of the trench. The scale effect for turbulence is neglected as long as both prototype and model display turbulent flow. This leaves settling as the only important variable showing scale effects. When using Froude the settling velocity is modeled too high which leads to an underestimation of the excavation capacity in the prototype. This is a beneficial result since the prototype will perform better than the model. 44

60 Using the velocity scale n u = 1, scale effects are present for the gravity current, centrifugal force and turbulence. Again the scale effect as a consequence of turbulence is neglected assuming the presence of turbulent flow. The gravity current cannot be neglected. This variable is overestimated meaning that the transport capacity of the current is larger than it should be. Particles are more likely to be transported out of the trench, possibly leading to an overestimation of the erosion depth. The centrifugal forces are overestimated as well. Particles are forced towards the bed leading to an distorted density distribution. The density near the bed will be relatively too high. The conclusion is that with Froude the scale effects are least intrusive. The Froude indicator is therefore the preferred scaling indicator. 45

61

62 Chapter 6 Small scale model tests This chapter is not available. 47

63

64 Chapter 7 Large scale testing This chapter is not available. 49

65

66 Chapter 8 Erosion model 8.1 Introduction The previous chapter described the results of the model tests. A disadvantage of these model tests is that the results can be scaled up, but only within a specific range. To overcome this problem a numerical model can be made. If the most important physical processes are captured in a numerical model, an unlimited number of settings can be tested. A numerical erosion model was developed in order to predict the erosion of a sand bed under the influence of a translating jet. In this chapter an explanation of the model will be given. 8.2 Model overview The erosion model (hereafter referred to as EM ) consists of two main parts. One part calculates the flow development over the bed and the second part calculates the erosion of the bed. Repetition of this process leads to an equilibrium state in which erosion does not take place any more. The first step is the creation of a domain in which the computations take place. The domain exists of a rectangular box with a cilinder on top. This cilinder acts as the jet inlet. The only physical boundaries are the sand bed (bottom of the domain) and the nozzle (walls of the cilinder). The other boundaries are virtual, the flow is not obstructed by these boundaries. Figure 8.1 gives an overview of the domain. 51

67 Erosion model Figure 8.1: Domain for erosion model When the domain has been constructed it is time to simulate the flow through the domain, the second step in the process. At the inlet a constant inflow of water is present. As a result a flow will develop throughout the domain. The cross flow generated by the translation of the jet is not modeled. This assumption is valid when the stand-off distance is relatively small and the jet flow is large enough in relation to the cross flow. The flow development is determined with the help of a steady state solver. A steady state solver searches for an equilibrium situation after a number of iterations. This implicitly leads to the rigid wall approach [1]. The flowfield has developed over the entire domain before the erosion is calculated. In reality the flow field will develop differently since erosion immediately plays a role, the changed bed affects the flow development. As a consequence the trench always develops gradually without steep walls, see the indentation in figure 8.1. Therefore the penetrative jet regime can not be modeled. If reference is made to figure 4.1, only the form on the left-hand side can be created. In addition to this, the used solver is a single-phase solver. This means that the fluid throughout the domain has a single and constant density. The effect of particles entering the flow and thereby increasing the density of the flow is not accounted for. This also means that there will be no sedimentation and the ridges at both sides of the trench, as seen in figure 4.1, will not appear. 52

68 After the flow distribution over the bed has been determined the results are exported from the CFD programme to calculate the erosion, step three. For a number of points the fluid motions are recorded at a specific distance to the bed. With the help of these fluid motions the erosion is calculated and the shape of the bottom of the domain is changed. Steps one to three are repeated until the sand-bed reaches an equilibrium state. Figure 8.2 shows the flow chart of the erosion model. Figure 8.2: Flow chart of erosion model A remark has to be made in addition to the flow chart. It can be seen that each loop the initial conditions of the velocity etc. are given as input for the computational fluid dynamics (CFD) software. This means that each time-step the flow through the entire domain must be calculated until the steady state solution is reached. To save time, a solution is to map the flow distribution of the previous time step onto the domain of the current timestep. In that case the flow distribution needs little time to adapt itself to the new lay-out of the sand bed. Although the used CFD software (OpenFOAM) can be used for this kind of mapping, it did not work. The reason why remains unknown. 53

69 Erosion model 8.3 Fluid dynamics During step two in the flow chart the flow distribution in the domain is determined. The computations in the domain are done using a program called OpenFOAM (referred to as OF ). This software is open-source and therefore it is possible to see which calculations the program performs before arriving at a solution. An assessment of these calculations and numerical methods lies outside the scope of this thesis, therefore OF will be considered a blackbox. Although this black-box concept is introduced, two subjects are considered in more detail for a better understanding of the model. These are the turbulence model and the boundary conditions Turbulence model In the case of post-trenching a highly turbulent flow is created by the jet. This poses a problem for the CFD simulation since turbulent flows are often unstable. To overcome this problem use is made of the Reynolds Averaged Navier Stokes (RANS) equations. These equations are time-averaged equations of motion for fluid flow. The RANS equations cannot be solved directly, they require an additional turbulence model. The turbulence model chosen is the SST k-ω turbulence model. The abbreviation SST stands for shear stress transport. The choice for the SST k-ω model is based on its good behaviour in adverse pressure gradients 1. These pressure gradients are encountered near e.g. inlets. Secondly the k-ω model proved to be more stable than the k-ε model during development of the erosion model. For some cases usage of the k-ε model led to severe convergence problems in and around the jet inlet Boundary conditions The boundary conditions are of great influence on the solution of a numerical calculation. In OF the boundary conditions are applied on a patch. A patch is a surface at the boundary of the computational domain. For example, the surface that represents the sand bed is a patch. In the erosion model three types of patches are used: inlet, outlet and wall. In table 8.1 an overview is given of the patch and the corresponding boundary conditions for several variables, see also appendix E. In figure 8.3 can be seen that the physical boundaries are walls and the virtual boundaries are outlets, water can flow away through these patches. The patch inlet generates an inflow of water in the domain. 1 k-omega model 54

70 Figure 8.3: Patches Inlet Outlet Wall Velocity (U) fixedvalue zerogradient fixedvalue Pressure (p) zerogradient fixedvalue zerogradient Turbulence (k) fixedvalue zerogradient kqrwallfunction Turbulent dissipation fixedvalue zerogradient omegawallfunction (ω) Table 8.1: Boundary conditions in the erosion model The fourth column of table 8.1 is considered to explain the different terms. The velocity at the wall is defined by the type fixedvalue. At the wall patch the value of the velocity is equal to zero, the wall does not move and neither does the fluid directly against the wall. The pressure at the wall is given the type zerogradient. This means that the normal gradient of the pressure is zero at the patch. Both the turbulence and the turbulent dissipation at the wall patch are defined with the help of a wallfunction. Generally the velocity profile close to the wall is divided in two layers, see figure 8.6. The layer at the wall is called the viscous sublayer, the momentum equations are governed by viscous effects. At a certain distance from the wall turbulence becomes the 55

71 Erosion model governing process. Near a wall, in the viscous sublayer, the k-ω turbulence model is not able to calculate the velocity profile because the viscous effects are governing. The inability of the turbulence model to predict the velocity at the wall leads to instability. This problem is solved using wall functions, readily available in OpenFOAM. The exact workings of these functions remain part of the black-box approach, it is merely stated that this problem is taken care of. Boundary condition for turbulence at inlet Considering the patch Inlet in table 8.1, two different types are given: fixed- Value and zerogradient. The type fixedvalue demands user defined input e.g. a value for the velocity at the inlet. The values for pressure and velocity are input parameters for the model, those for turbulence and turbulent dissipation must be calculated 2. In order to calculate the turbulence at the inlet, first the turbulence intensity is needed. The intensity at the core of fully developed duct flow is defined as: With: The turbulence is calculated by: With: I = 0, 16.Re 1 8 (8.1) I = Turbulence intensity [ ] k = Turbulence The turbulent dissipation becomes: k = 3 2. (ū.i)2 (8.2) [ m 2 s 2 ] u = Mean free stream velocity [ ] ω = 0, 09.k β.ν ω = Turbulent dissipation [ ] 1 s β = Turbulent viscosity ratio [ ] (8.3) The turbulent viscosity ratio depends on the Reynolds number. Generally it holds β = 100 for Re > 100, parameters/ 56

72 8.4 Erosion calculations Data gathering Based on the output generated by OF the erosion can be calculated. OF calculates the velocity, pressure etc. throughout the entire domain but the area of interest lies with erosion of the bed. To calculate erosion, fluid motions near the bed surface are used. The fluid motions are taken from sample points at an user-specified distance from the bed. Each iteration a new bed level is created for the next timestep. After a bed level has been made, the coordinates of the sample points are defined with the help of this bed level. Consider the first time step of the simulation in which the created bed profile is perfectly flat. In this bed a number of datapoints is distributed, see the black dots in figure 8.3. Per datapoint, perpendicular to the bed, a sample point is created at a user specified distance to the bed, see figure 8.4. By default this distance is the characteristic length of a cell. Figure 8.4: Samplepoints The locations of the samplepoints are stored in a file. After the CFD software is executed, the data at all samplepoints is gathered. When a samplepoint and a grid point do not have the same coordinates, in other words there is no data at the location of the samplepoint, OF interpolates the data from the surrounding known grid points. The result consists of two separate files. One with the velocities in x-,y- and z-direction and another with turbulence intensities Flow driven erosion Flow velocities over the bed are known in x-,y- and z-direction. Using the Van Rhee stability parameter (paragraph 3.4) the erosion can be calculated. Erosion is calculated using the velocity component horizontal to the surface. Since the bed is not horizontal for the majority of data points, the velocity parallel to the bed must be calculated. 57

73 Erosion model For each sample point OF generates the velocity components with the corresponding coordinates. With the help of the coordinates the slope angle of the bed at each datapoint can be determined. Considering figure 8.5, the slope angle α in data point 2 is calculated as the angle of the dotted line (connecting datapoints 1 and 3) with the horizontal. The velocity at the samplepoint is converted to a velocity parallel to the surface in datapoint 2. Figure 8.5: Calculation of slope angle in datapoint 2 As explained in the previous paragraph the velocities are sampled at a certain distance from the bed. To calculate the erosion, the shear stress at the bed must be determined instead of the velocity at a distance from the bed. In figure 8.6 this process is shown. The velocity profile near the bed is described by a logarithmic wall function. Using this wall function the flow velocity in a samplepoint is converted to a shear stress at the bed. Figure 8.6: Graphic interpretation of the wall function 58

74 In the EM this process is described with the following equation [13]: With: τ b = ρ w.u p 2 [ 1 κ. ln ( 32.Y k s )] 2 (8.4) u p = Velocity parallel to bed [ ] m s κ = Von Kármán constant [ ] Y = Distance to bed [m] In the numerator of equation (8.4) the density is that of water. The reason is that the erosion model is a single phase model and can only calculate with one density. Normally this density should be the mixture density leading to higher shear stress over the bed and consequently higher erosion velocities. On the other hand the near bed concentration should then be added leading to smaller erosion velocities, see equation (3.1). With the formulation of the shear stress over the bed, the Shields parameter is defined as follows: θ = τ b (ρ s ρ w ).g.d 50 = Turbulence driven erosion u p 2 g..d 50. [ 1 κ. ln ( 32.Y k s )] 2 (8.5) Characteristic for a jet, impinging on a bed, is the creation of a stagnation point. In this stagnation point there are turbulent fluctuations but on average there is no or little flow velocity, see appendix F. When merely using velocity to calculate erosion, no erosion takes place in the stagnation point. To account for erosion in regions with high turbulence, the Hofland stability parameter was incorporated in the erosion model. Since the Hofland stability parameter was introduced to calculate erosion in areas of high turbulence and little average flow velocities, the term for average flow velocity is removed from equation (3.37): θ H = [ α ] 2 k. Lm y (8.6) g..d 50 With: θ H = Modified Hofland stability parameter [ ] 59

75 Erosion model In the EM the value for y is equal to the sample distance. The EM calculates the Shields parameter twice, once using the average velocity and once using turbulence. In the stagnation point the value calculated with the modified Hofland stability parameter will be governing, outside this area the Shields parameter will be larger. The erosion model uses the largest of both parameters: θ EM = max[θ, θ H] (8.7) With: θ EM = Shields parameter used in erosion model [ ] Computation of the bed level for the next time step The erosion model simulates a translating jet. One way to model this is to make a long domain over which the jet will move. The downside of this approach is that a large domain requires lots of computing power. Instead of moving the jet over the domain it was decided to move the bed under the jet. This results in a smaller domain and shorter simulation times per iteration. Two operations must be completed before the bed level in the next timestep is defined. The deformation of the bed as a result of erosion must be determined and secondly the bed must be translated under the jet. It was explained how the Shields parameter for the erosion model θ EM was determined. With the theory presented in chapter 3 this parameter can be converted into an erosion velocity. In each datapoint the erosion velocity, as well as the direction of this velocity is known. Given that erosion takes place during a certain time t, the new bed level is calculated. This process is clarified in figures 8.7a and 8.7b. 60

76 (a) Initial bed (b) Eroded bed Figure 8.7: Erosion in the EM The bed moves with a certain trail velocity under the jet. So after the eroded bed profile has been determined this profile is translated, see figures 8.8a and 8.8b. (a) Translation of eroded bed (b) Translated bed Figure 8.8: Translation in the EM Each timestep the datapoints move in horizontal direction as a result of the translation due to trailing. This means that datapoints near the end of the domain leave the computational domain with the speed of the trail velocity. This is no problem as long as new datapoints are created at the beginning of the domain at the same rate. This is exactly wat the EM does, see figure 8.9. A new horizontal bed level and new datapoints are created at the begin of the domain. Going back to the flow chart in paragraph 8.2, the third an last step has been finalized. The bed level for the next timestep is generated and acts as input for the next iteration step. 61

77 Erosion model Figure 8.9: Bed level for a translating jet, the black dots are the datapoints 62

78 Chapter 9 Results erosion model 9.1 Introduction A description of the numerical erosion model was given in the previous chapter. It became obvious that the erosion could not be generated by flow alone and a turbulence erosion term was incorporated in the model. To validate this approach simulations were done for both stationary and translating jets. Initially the model could only handle low flow velocities. Since post-trenching is a process of high speed erosion, the model could not produce satisfying results. The next paragraph explains why the the simulations were restricted to low flow velocities. The model was validated using a data-set for impinging jets with low flow velocities [9]. Both stationary and trailing jets were simulated and compared with the available data. Further development of the EM made it possible to run simulations for high flow velocities. This gave the opportunity to compare the results of the large model tests with the results of the simulations. The validation of the model was improved since all input parameters were defined more accurately. This is described in the penultimate paragraph after which conclusion are drawn in the last. 9.2 Stability The cause of the convergence problems, mentioned in paragraph 8.3.1, is a result of the lay-out of the domain. To explain this problem the focus is laid on the connection between the cilindrical inlet and the rectangular domain, see figure

79 Results erosion model (a) Velocity at transition (cross-section) (b) Turbulence at transition (topview) Figure 9.1: domain Area of transition between cilindrical jet inlet and rectangular As soon as the flow leaves the cilindrical enclosure it will start to entrain surrounding fluid. This means that fluid directly next to the cilindrical inlet undergoes a large acceleration. However, exactly at this point there is a boundary. Now consider the first grid cell at the boundary. At the top side of this grid cell is a boundary, the velocity is unknown and the pressure is zero. Furthermore the turbulence and the dissipation are unknown. At the bottom side of the cell the velocity, turbulence and turbulent dissipation are at a maximum. Over the length of a single cell the variables have to develop from arbitrary to maximum values. Figure 9.1b shows that the turbulence reaches a maximum value next to the jet inlet on the topside of the rectangular domain. For low velocities the simulation remains stable, but for large velocities the differences over the length of a cell become too large and the simulation becomes unstable. A possible solution is to position the end of the cilindrical inlet at a distance from the topside of the rectangular domain. More grid cells are then present between the boundary and the end of the inlet, the variables can establish more gradually. An example with a different positioning of the inlet is given in figure 9.2. With this lay-out the solution is stable even for high flow velocities. Unfortunately the progress in the lay-out of the domain was made in a late stadium. Therefore only a single simulation for a trailing jet with high flow velocities could be done. This simulation is described in paragraph The other results described in the next paragraphs were obtained with the former lay-out of the domain. 64

80 (a) Walls of jet inlet (b) Development of velocity Figure 9.2: Cross section jet inlet 9.3 Stationary jet The lack of erosion capacity in the stagnation point led to the incorporation of turbulence driven erosion in the EM. To validate this approach and fit the value for α (see equation (8.6)) several simulations were done. Starting point was the value α = 6 as proposed by Hofland although with the remark that this value is quite high. The simulation produced an erosion profile with largely overestimated dimensions of the trench. The same simulation was executed using lower values for α (2 1 2, 3 and 4). It appeared that the EM kept overestimating the dimensions of the trench. However, by using lower values for α the erosion in the stagnation point again became too small. The results shown in figures 9.3 to 9.5 were produced for α = 3. Figure 9.3: Velocity distribution stationary jet (α = 3) 65

81 Results erosion model Figure 9.4: Pressure distribution stationary jet (α = 3) Figure 9.5: Turbulence stationary jet (α = 3) 66

82 These figures again show the effect of too little erosion in the stagnation point. For the value α = 3 the turbulence driven erosion in the stagnation point is less than the flow driven erosion in the area surrounding this point. It was mentioned that the erosion depths were overestimated. In addition to this, during none of the simulations an equilibrium scour profile was found. This was not expected since the results of the data-set [9] showed that equilibrium was reached after 10 seconds whereas the EM simulations covered a time-span of 60 seconds. So the erosion depth was overestimated although the turbulence driven erosion was less than the flow driven erosion. For a stationary jet, the flow by itself already leads to an overestimation of the scour depth. Increasing the turbulent erosion would enlarge the overestimation of scour depth. 9.4 Trailing jet The simulations for trailing jets were divided into cases with low and high flow velocities. This paragraph is divided in two parts accordingly Low flow velocity The experiments of Yeh [9] provided a usefull, though small set of data. Validation of the EM for a trailing jet with low flow velocities was done with two of his experiments. During both experiments the flow velocity was 2,05 m s and the trail velocities were 0,09 and 0,11 m s. This results in values for R 1 = v jet v trail of respectively 24 and 18. The results of the simulations are given in figures 9.6 and 9.7. Figure 9.6: Results of simulations compared with data from Yeh [9] 67

83 Results erosion model Figure 9.7: Cross section of domain after simulation (R 1 = 18), the very shallow trench is hardly visible Some remarks have to be made with respect to these results. The domain which was chosen for the erosion model was smaller than in the tests of Yeh. The domain was limited to the trench itself and not the ridges at both sides. Beforehand it was known that the EM could not recreate the ridges since sedimentation was not incorporated. Secondly the domain became much smaller thereby decreasing the amount of time needed for the simulation. The factor α from the Hofland parameter had to be fitted again since the fit with the stationary jet had not been satisfying. After some test runs it appeared that the best relation with the Yeh data set was found for α = 3. Furthermore it appeared that the turbulence induced erosion was needed to bring the trench at depth. When a simulation was done only using the van Rhee stability parameter it appeared that the scour depth was far too little. It was not clear if the turbulence induced erosion was only needed to penetrate the soil and start up the trenching. It could be possible that the erosion process was governed by flow induced erosion once the first part of the trench was created. Simulations for horizontal flow over a bed were performed after which the erosion velocities calculated with the model were compared with analytic calculations. This gave matching results, the erosion equations in the EM were correct. Therefore it was tried to run simulations without turbulence induced erosion but with a pre-defined trench. The idea was that if a trench already 68

84 existed the front would remain intact if the erosion velocity exceeded the trail velocity. However, this approach gave poor results. The trench would not remain at the pre-defined depth unless the jet itself was able to penetrate into the soil. This can be explained by the fact that after a certain number of timestep the pre-defined trench has left the domain, from this point on the trench is fully defined by the properties of the jet. This proved that the turbulence term was needed to simulate the post-trenching process. Taking a look at figure 9.6 it is clear that the erosion profiles of the EM and the data-set do not match. This can be explained by the large amount of undocumented parameters used by Yeh. The input of the erosion model consists of a list of parameters among which are the porosity, water temperature, bed roughness and others. All these parameters are used in the Van Rhee stability parameter. Not knowing the exact values used by Yeh leads to an accumulation of inaccuracies. In the stagnation point where erosion is determined by the Hofland parameter, these uncertainties are collected in the factor α. Due to fitting of this parameter the erosion depth shows resemblance with the data-set. Outside the stagnation point erosion is calculated with the Van Rhee stability parameter and such a factor is absent. The erosion velocity is not corrected for the inaccuracies. These inaccuracies combined with the simplifications in the EM result in a different scour profile compared with reality High flow velocity Due to the continuing development of the model it became possible to simulate high flow velocities. This made it possible to simulate two of the tests described in chapter 7. The majority of parameters used in the erosion equations were determined for these tests. So for validation of the high velocity simulations a complete and accurately documented data-set was available. Only two tests could be simulated because the EM can only handle jets at a 90 degree angle. At the time of writing this thesis development of the model had not progressed up to a point where inclined jets could be modeled. Test three of the large scale test was simulated by the EM, the settings can be found in table (not available). The results of this simulation are given in figures 9.8 and

85 Results erosion model Figure 9.8: This figure is not available Figure 9.9: Cross section of domain after simulating test 3 Figure 9.8 shows the bed level measured after the execution of test three and the bed level as generated by the EM. The shape of the measured trench is approached by the simulation. The blue circles mark the exact data points generated by the EM. These datapoints are connected by a spline to recreate the exact scour profile. The usage of this spline is valid since the mesh generator of the EM does exactly the same. This becomes clear in figure 9.9 where the bottom level in both x- and z-direction is represented by a spline as well. Although the shape of the trenches is similar, the EM overestimates the erosion depth. This is no surprise considering the assumptions that were made to simplify the model. After all, the model makes use of a steady 70

86 state solver and does not include differences in density or the transport of particles. An important parameter is the aforementioned variable α = 3. The value for this parameter was obtained by fitting it with the help of an incomplete data-set. The results could possibly be improved by re-fitting the parameter α using the data of test three. After all, the differences of the simulated and measured bed profiles in figures 9.6 and 9.8 show the importance of accurate and complete input. 9.5 Conclusion Simulations showed that jet erosion cannot be described with the Van Rijn and Van Rhee erosion equations. A reason could be that the equations only use average flow velocities. Visual observations led to the assumption that erosion can be caused by turbulent fluctuations in the stagnation point of a jet, see figure F.3. Therefore the (modified) Hofland stability parameter was incorporated in the model. Calculating erosion using both flow velocities and turbulence intensities led to more accurate results. Comparison of the model with a data-set for low flow velocities did not provide satisfactory results due to a lack of data. After adjustments to the domain, the erosion model was able to handle large flow velocities. A simulation was executed with the data of large scale test three. The simulated data showed a large resemblance with the maesured data of test three. Nevertheless comparison with a single test is not enough to validate the model. It is important to realise that results presented by the EM completely depend on the modeled processes. Since there is no full comprehension of the processes that occur it is dangerous to draw conclusions based on the model. However it can be concluded that the addition of a turbulence term in the erosion equations deserves further investigation. Considering the above and having stated that further validation is needed, the current version of the model provides a first estimate for the erosion of 90 degrees inclined deflective jets. 71

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88 Chapter 10 Conclusions and recommendations 10.1 Conclusions based on model tests This paragraph is not available Conclusions based on numeric erosion model Erosion equations calculating with average velocities are not suitable to model erosion in the stagnation point of a jet. In this point the average velocity is very low or zero. The addition of a turbulence term in the erosion equations gives a better approach of reality. Several simulations were done and compared with an available data-set. The erosion model could not be validated because too many parameters of the data-set were unknown. After further development of the erosion model, one of the executed model test was simulated. This simulation led to an approach of the measured bed profile and showed much resemblance with respect to the shape of the trench. However, the erosion model can not be validated on the basis of a single test Recommendations All tests described in this thesis were executed using a single type of sand. Additional model tests for different types of sand should be done to complete the data-set. This would also provide extra data to accurately determine the aformentioned boundary. 73

89 Conclusions and recommendations It is recommended to further develop the erosion model and make it suitable for inclined jets. This allows the total data-set, generated during the large scale model tests, to be simulated. The results can be used to validate the model and improve the fit of α. Secondly the model can be developed to simulate multiple runs over the same trench. This would further expand the possiblities to validate the model with the generated data-set. Finally it is recommended to further investigate erosion as a result of turbulent fluctuations. 74

90 List of symbols c = Coefficient for void and grain shape, approx. 0, 005 0, 010 [ ] c b = Near bed concentration [ ] c µ = Coefficient [ ] C D = Drag coefficient D = Particle parameter [ ] D r = Relative density [ ] D 0 = Diameter of jet at nozzle [m] D 50 = Median grain diameter [m] e = Void ratio [ ] e max = Maximum void ratio [ ] e min = Minimum void ratio [ ] E = Pick up flux [ kg m 2.s f 0 = Darcy Weisbach friction coefficient [ ] F c = Centrifugal force [N] F D = Drag force [N] F r = Froude number [ ] F r d = Densimetric Froude number [kg] ] g = Gravitational acceleration [ ] m s 2 h = Water depth [m] H cl = Distance between nozzle and bed along centerline [m] h max = Max height the waterjet reaches [ ] m s I = Turbulence intensity [ [ ] ] I 0 = Impulse at nozzle kg.m s 2 I s = Impulse at distance s from nozzle [ m 2 s 2 ] [ ] kg.m s 2 k = Turbulence k 0 = Permeability of settled original bed [ ] m s k l = Permeability at loose state [ ] m s k s = Effective bed roughness [m] L = Length [m] L m = Length scale of turbulence [m] m = Mass of water [kg] 75

91 Conclusions and recommendations m s = Mass [kg] n 0 = Porosity of the settled original bed [ ] n l = Porosity at loose state [ ] q c = Resistance of cone [ MN Q 0 = Flow at jet nozzle [ m 3 s ] m 2 ] Q s = Flow at distance s from nozzle r = Distance to center [m] R = Coefficient [ ] Re = Reynolds number [ ] [ ] S = Settling flux kg m 2.s u = Flow velocity [ ] m s ū = Average flow velocity [ ] m s u = Mean free stream velocity [ ] u 0 = Velocity at jet nozzle [ ] m [ ] m 3 s s u 0,perp = Velocity perpendicular to the bed [ ] m s u m,s = Maximum core velocity [ ] m s u p = Velocity parallel to bed [ ] m s u (s, r) = Velocity at distance s from nozzle and distance r from centerline [ ] m s u v = Vertical velocity component of waterjet at height zero [ ] m s u = Shear velocity [ ] m s s = Position along flow trajectory [m] T = Temperature [Celsius] v e = Erosion velocity [ ] m s V ero = Volume of eroded sand [ m 3] w c = Settling velocity due to centrifugal force [ ] m s w s = Settling velocity [ ] m s w s,c = Total settling velocity [ ] m s Y = Distance to bed [m] 76

92 α 4 for average sand of 120 < D < 300 µm α = Angle between trail direction and erosion direction [ ] α = Coefficient [ ] β = Slope angle [degrees] β = Turbulent viscosity ratio [ ] = ρs ρw ρ w [ ] n = n l n 0 1 n l [ ] θ = Shields parameter [ ] θ b = Angle between jet and surface in x, y plane [degrees] θ cr = Critical Shields parameter [ ] θ EM = Shields parameter used in erosion model [ ] θ H = Hofland stability parameter [ ] θ H = Modified Hofland stability parameter [ ] θ cr = Modified Shields parameter [ ] κ = Von Kármán constant [ [ ] ] µ = Dynamic viscosity kg m.s [ ] ν = T = Kinematic viscosity m 2 s ρ = Density of jetted medium ρ bed = Density of sand bed ρ s = Density of the grains [ ] kg m 3 [ ] kg [ m 3 kg [ ] kg m 3 m 3 ] ρ w = Density of water τ = Bed shear stress [ ] N m 2 φ = Pick up function [ ] φ = Modified pick up flux [ ] φ b = Angle between jet and surface in x, z plane [degrees] ϕ = Angle of internal friction [degrees] ψ Shape factor (0, 7 for sand) ω = Turbulent dissipation [ 1 s ] 77

93

94 Appendix A Small scale tests This appendix is not available. 79

95

96 Appendix B Conversion of CPT to density This appendix is not available. 81

97

98 Appendix C Mass balance This appendix is not available. 83

99

100 Appendix D Test results This appendix is not available. 85

101

102 Appendix E Boundary conditions The input for the simulation of scale test three is given in this appendix. Figure E.1: File boundary conditions pressure 87

103 Boundary conditions Figure E.2: File boundary conditions velocity 88

104 Figure E.3: File boundary conditions turbulence 89

105 Boundary conditions Figure E.4: File boundary conditions turbulent dissipation 90

106 Appendix F Stagnation point F.1 Introduction As described in chapter 2 a stagnation point is present where the jet hits the bed. In this point the jet stagnates, it builds up pressure and there is no or little flow. Initially the computational model calculated erosion by means of the flow velocity. Therefore no erosion occured in the stagnation point. In order to model the erosion of a bed under a jet an additional method was sought after. Three phenomena were looked into, pressure gradients, turbulence and LES. Figure F.1: Pressure in stagnation point F.2 Pressure gradient Figure F.1 shows a pressure gradient. The gradient over a single particle was examined to determine whether or not it is capable to accelerate a particle, 91

107 Stagnation point see figure F.2. Figure F.2: Pressure gradient The accelerating force F p is counteracted by the force of gravity F g and the force as a result of inflow of pore water F s multiplied with the internal angle of friction: F p = tan(φ).(f g + F s ) (F.1) Elaboration of this equation led to a negligible erosion velocity. Although pressure plays no role for the onset of erosion, it can remove soil by shearing once erosion has started. F.3 Turbulence Secondly the option of erosion due to turbulence was examined. Visual observations of an impinging jet led to the conclusion that there is highly turbulent flow in the stagnation point, see the movie still in figure F.3. As stated in paragraph the EM makes use of the RANS-equations to model turbulent flow. The disadvantage of Reynolds averaging is that the turbulent fluctuations are averaged over time. So, the turbulent fluctuations clearly visible on film are averaged out in the EM. If it is true that the turbulent fluctuations lead to erosion, which seems likely, this is not accounted for in the EM. To solve this problem a solution was sought in which erosion could be expressed as a function of turbulence. 92

108 Figure F.3: Turbulent flow for penetrating jet F.3.1 Friction velocity The turbulence as calculated by OpenFOAM could be converted into a shear velocity with the following equation [13]: k = u 2 cµ (F.2) Application of this formula led to erosion in the stagnation point. However, this equation is normally used as a boundary condition at a surface in which flow over a hydraulically rough surface creates turbulence. Reversing the usage of this equation led to uncertainties and the physical validity of this approach could not be given. Instead, the option of a Large Eddy Simulation (LES) was explored in order to model the turbulence. F.3.2 LES A LES-solver models turbulent flow (figure F.4), very unlike the steady state solver used in the EM. Since there was no interaction between the bed and the flow calculated by OpenFOAM it was chosen to record the flow over the bed for a hundred time steps during a LES simulation. For each time step the erosion was calculated without actually implementing this in the bed level. Due to the irregular flow pattern the expectation was that, although the average velocity in the stagnation point is zero, fluctuations of high velocity would be 93

109 Stagnation point (a) t=50 s (b) t=75 s Figure F.4: Flow pattern for LES solver present. These fluctuations could be the source of erosion in the stagnation point. The result of the method is shown in figure F.5, although there is some erosion directly underneath the jet there still is a peak in the stagnation point 1. This result does not correspond with reality. Figure F.5: Erosion profile with LES solver 1 Note: The label at the y-axis shows large values. The reason is that the erosion of a few milimeters each time step is superimposed leading to a large depth over a hundred timesteps. The bed level was not altered after each time step. 94