MIKE 21 ST. Non-Cohesive Sediment Transport Module. User Guide

Transcription

1 MIKE 21 ST Non-Cohesive Sediment Transport Module User Guide DHI Software 2007

2 Software development by : JAZ/HKJ Written by: JAZ G:\f-share\MikeZero\new\source\manuals\M21\m21st\Cover.fm 22 January :20 am 2

3 Please Note Copyright This document refers to proprietary computer software which is protected by copyright. All rights are reserved. Copying or other reproduction of this manual or the related programs is prohibited without prior written consent of DHI Water & Environment (DHI). For details please refer to your 'DHI Software Licence Agreement'. Limited Liability The liability of DHI is limited as specified in Section III of your 'DHI Software Licence Agreement': 'IN NO EVENT SHALL DHI OR ITS REPRESENTATIVES (AGENTS AND SUPPLIERS) BE LIABLE FOR ANY DAMAGES WHATSO- EVER INCLUDING, WITHOUT LIMITATION, SPECIAL, INDIRECT, INCIDENTAL OR CONSEQUENTIAL DAMAGES OR DAMAGES FOR LOSS OF BUSINESS PROFITS OR SAVINGS, BUSINESS INTERRUPTION, LOSS OF BUSINESS INFORMATION OR OTHER PECUNIARY LOSS ARISING OUT OF THE USE OF OR THE INA- BILITY TO USE THIS DHI SOFTWARE PRODUCT, EVEN IF DHI HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. THIS LIMITATION SHALL APPLY TO CLAIMS OF PERSONAL INJURY TO THE EXTENT PERMITTED BY LAW. SOME COUN- TRIES OR STATES DO NOT ALLOW THE EXCLUSION OR LIMITA- TION OF LIABILITY FOR CONSEQUENTIAL, SPECIAL, INDIRECT, INCIDENTAL DAMAGES AND, ACCORDINGLY, SOME PORTIONS OF THESE LIMITATIONS MAY NOT APPLY TO YOU. BY YOUR OPENING OF THIS SEALED PACKAGE OR INSTALLING OR USING THE SOFTWARE, YOU HAVE ACCEPTED THAT THE ABOVE LIMITATIONS OR THE MAXIMUM LEGALLY APPLICA- BLE SUBSET OF THESE LIMITATIONS APPLY TO YOUR PUR- CHASE OF THIS SOFTWARE.' Printing History June 2003 June 2004 August 2005 October

4 4 MIKE 21 ST

5 CONTENTS 5

6 1 ABOUT THIS GUIDE Purpose Prerequisites INTRODUCTION What is MIKE 21 ST? Application Areas Key Features EXAMPLES Introduction Corner Example Purpose Scenario Parameters Results Files Harbour Example Purpose Scenario Parameters Results Files Structures Example Purpose Scenario Parameters Results Files Nourishment Example Purpose Scenario Parameters Results Files REFERENCE MANUAL Ackers & White Transport Formulation Bed Load Transport Coefficient B Bed Roughness Bijker's Transport Method Critical Shields Parameter Engelund & Fredsøe Transport Theory MIKE 21 ST

7 4.7 Engelund & Hansen Transport Theory Grid Sizes Meyer-Peter & Müller Transport Formulation Morphological Parameters Morphological Schemes Filtering Methods Bed slope diffusivity effect Porosity Recovering From Errors During Simulation Relative Density of Sediments Sediment Size and Gradation Simulation Area Simulation Period and Time Step Size of Output Files STP Transport Model Water Temperature Wave Breaking Parameters Wave Interpolation Zyserman & Fredsøe Transport Formulation SCIENTIFIC BACKGROUND REFERENCES Non-Cohesive Sediment Transport in Pure Current Non-Cohesive Sediment Transport in Combined Waves and Current Computational Hydraulics

8 8 MIKE 21 ST

9 Purpose 1 ABOUT THIS GUIDE 1.1 Purpose This guide provides you with elaborate examples and reference material not found in the online help for the MIKE 21 ST software. 1.2 Prerequisites Though basic computer user skills are enough to operate the MIKE 21 ST software, you need some understanding of sediment transport concepts in order to validate results and understand all input parameters. In order to compute the sand transport based on pure current, MIKE 21 ST needs the output of a hydrodynamics simulation done by MIKE 21 HD. And if you want to take waves into consideration too you need a file with wave parameters calculated by MIKE 21 NSW or PMS. MIKE 21 ST ships with a handful of examples, including the output files needed to run simulations in both pure current and combined current and wave conditions. Please refer to "Examples" on page 15. 9

10 About This Guide 10 MIKE 21 ST

11 What is MIKE 21 ST? 2 INTRODUCTION 2.1 What is MIKE 21 ST? MIKE 21 ST is a module in the MIKE 21 application suite for calculating non-cohesive sediment (sand) transport rates. You can calculate sand transport based on pure current information, or you can take waves into consideration too. In addition to sand transport rates, a simulation will give you the initial rates of bed level changes. This is sufficient to identify potential areas of erosion or deposition, but can not take the place of a full morphological model. 11

12 Introduction 2.2 Application Areas MIKE 21 ST can simulate sand transport rates in a wide array of settings, including natural environments like tidal inlets, estuaries and coast lines, and man made constructions like harbours and bridges. Tide, wind, wave and current can all be taken into consideration for optimum precision in the simulations. 2.3 Key Features Here is a list of selected highlights in the MIKE 21 ST software: Constant or spatially varying bed material (i.e. median grain size and gradation) Five different sand transport theories available in pure current conditions: The Engelund and Hansen total-load transport theory The Engelund and Fredsøe total-load (determined as bed load + suspended load) transport theory The Zyserman and Fredsøe total-load (bed load + suspended load) transport formulation The Meyer-Peter and Müller bed-load transport theory The Ackers and White total-load transport formulation Two simulation methods available in combined current and wave conditions: Application of DHI's deterministic intra-wave sediment transport model, STP Bijker's total-load transport method The STP model provide these features to MIKE 21 ST: Arbitrary direction of wave propagation in respect to the current Distinguish between breaking and unbroken waves Geometric properties of bed material is described through a single grain size or a grain size distribution curve Plane/ripple-covered bed 12 MIKE 21 ST

13 Key Features When using the STP model in combined current and wave simulations, two-dimensional (2DH) or quasi three-dimensional (Q3D) sand transport methods are available Speed up simulations by using predefined sand transport tables Finite differences technique on space-staggered rectangular grid Courant-Friedrichs-Lewy stability criterion 13

14 Introduction 14 MIKE 21 ST

15 Introduction 3 EXAMPLES 3.1 Introduction To demonstrate the capabilities of MIKE 21 ST we have included four real-life examples of simulations. Corner (see "Corner Example" on page 15) Harbour (see "Harbour Example" on page 18) Structures (see "Structures Example" on page 22) Nourishment (see "Nourishment Example" on page 26) You can use the examples to get an understanding of the data you need to feed MIKE 21 ST for a successful simulation, and you are free to modify the input data and parameters and see how the results change accordingly. In a standard installation you will find the files needed to run the examples in C:\Program Files\DHI\MIKEZero\Examples\MIKE_21\ST\. 3.2 Corner Example Purpose Scenario We investigate a sharp channel corner with steady flow. The example will illustrate sand transport patterns and initial bed level change rates. The channel flow is forced from the southern open boundary to the western open boundary by a difference in water level of 0.05m. The bottom of the channel is horizontal and positioned 20m below datum. The bed material consists of uniform sand with median size d50 = 0.150mm and geometric standard deviation sg = Parameters The bathymetry is described using grid cells of 50 times 50 meters, and the hydrodynamic simulation from MIKE 21 HD is made with 30 second time slots. Adding it all up it correspond to a Courant number of eight. To avoid oscillations we increase the water level at the southern boundary gradually over 60 time slots, and we run the simulation for three hours to make sure that we arrive at a stable solution. 15

16 Examples The entire model area has a bed resistance specified by a constant Manning number of 32 m 1/3 /s, and a constant value of ε = 3 m 2 /s is used for the velocity-based eddy viscosity. Finally, the transport formulation of Engelund and Hansen is used to calculate the sand transport rates over the last two time slots in the MIKE 21 HD data file Results Based on data files and the parameters defined in the previous section the MIKE 21 ST software can now calculate a number of things for us and the Plot Composer can be used to plot some nice charts. Figure 3.1 below shows the initial rates of bed level changes after running the simulation (this is the file called dzdt.plc). The red area is where sand will aggregate, and the green areas are the ones prone to erosion. Figure 3.1 Initial rates of bed level change dz/dt (m/day) Figure 3.1 shows an area immediately upstream of the corner and adjacent to the wall where erosion will occur due to enhanced flow velocities and sediment transport capacity. 16 MIKE 21 ST

17 Corner Example As the velocity of the current drops due to flow expansion downstream of the corner, the transport capacity also decreases. As a result of these mechanisms, an area where sediment will tend to deposit can be seen immediately downstream of the corner in Figure 3.1. Other files of interest are Cornerflow.plc that gives you an overview of the current and transportrates.plc that shows the sand transport rates Files Here are the files provided with this example - in a standard installation they all reside in C:\Program Files\DHI\MIKEZero\Examples\MIKE_21\ST\Corner: Name: bathy.dfs2 Corner example, bathymetry The following specification files were used to run the simulations and plot the results: corner_hd.m21 Model: MIKE 21 Flow Model Corner example, hydrodynamic simulation corner_st.st2 Module: Mike 21 Non-Cohesive Sediment Transport Corner example, sediment transport simulation CornerBathy.plc MIKEZero Plot Composer Plot of model bathymetry CornerFlow.plc MIKEZero Plot Composer Plot of current field TransportRates.plc MIKEZero Plot Composer Plot of sediment transport rates Dzdt.plc 17

18 Examples MIKEZero Plot Composer Plot of initial rates of erosion/deposition 3.3 Harbour Example Purpose We investigate a harbour and the surrounding area on a sandy coast. Harbours tend to have an asymmetric design that protects them from the ocean, but this often lead to the formation of large-scale eddies in the separation area downstream of the harbour and subsequently sediments piling up right at the harbour entrance. This example looks at wave driven current and littoral transport patterns illustrating the phenomena just described Scenario The harbour is on a sandy coast consisting of sand with d 50 = 0.22mm and s g = 1.25 in the entire area. Irregular, directional waves (JONSWAP spectrum) move towards the coast from 102. Waves move over a water level corresponding to MWL = 1m. The wave characteristics at the offshore boundary of the model are H m0 = 2.11m and T p = 9.96s Parameters We are looking at an area that spans 1500m in the long-shore direction and 1625m in the cross-shore direction. The bathymetry is described with grid cells of five times five meters - this goes for both for the wave simulation with MIKE 21 PMS, and the hydrodynamic simulation with MIKE 21 HD. The model grid has an orientation of 201 measured from True North. Irregular, directional waves are added at the offshore boundary of the wave model. The frequency spectrum is of JONSWAP type, but around the mean direction of wave propagation (102 ) a cos n distribution of wave energy is used. At the other two open bounaries (left and right to the harbour) symmetry boundary conditions are used. The wave simulation takes wave breaking into consideration, using default values of the breaking parameters γ 1 and γ 2. Bed roughness is not considered. When simulating the wave driven currents with MIKE 21 HD we use constant values of velocity-based eddy viscosity ε = 1m 2 /s and Manning number M = 32m 1/3 /s all over the model area. The time slots are set to four 18 MIKE 21 ST

19 Harbour Example seconds and the simulation runs for four and a half hours to ensure stable conditions. The updrift boundary (left) is prescribed as flux, the downdrift boundary (right) is prescribed as level, and the offshore boundary is prescribed as zero flux. Prior to running the flow simulation the updrift and downdrift boundaries (prescribed as flux and level, respectively) was generated by use of the tool for calculation of Wave Generated Current and Setup of the MIKEZero system. The offshore boundary was prescribed as a zero flux one. Prior to running the sediment transport simulation, a sediment transport table (2DH) was generated with the tool for Generation of Sediment Transport Tables of the MIKEZero system Results In the present example, it is of interest to examine how the presence of the harbour influences the waves, the wave-driven current and the sediment transport patterns. Figure 3.2 below shows the wave-driven current calculated with MIKE 21 HD. The eddy on the right side of the harbour works with the strong offshore-directed current on the opposite side, moving any loose sediment available in the area towards the harbour entrance where it will settle and cause problems for the harbour basin. 19

20 Examples Figure 3.2 Calculated wave-driven current A number of other results are found in the plot files listed in the next section Files Here are the files provided with this example - in a standard installation they all reside in C:\Program Files\DHI\MIKEZero\Examples\MIKE_21\ST\Harbour: Name: Name: Name: ModelBathy.dfs2 Harbour example, bathymetry for wave and flow simulations IrrDirWaves.dfs2 Directional wave spectrum to be applied at offshore boundary of the wave model NorthBoundary.dfs1 Profile series of water levels and fluxes along northern boundary of flow model 20 MIKE 21 ST

21 Harbour Example Name: SouthBoundary.dfs1 Profile series of water levels and fluxes along southern boundary of flow model The following specification files were used to run the simulations and plot the results: WaveSimulation.PMS Model: MIKE 21 Parabolic Mild Slope Waves Harbour example, wave simulation HD_Simulation.M21 Model: MIKE 21 Flow Models Harbour example, hydrodynamic simulation TranspTable.21T Tool: Generation of Sediment Tables Harbour example, sediment transport table ST_Simulation.ST2 Module: Mike 21 Non-Cohesive Sediment Transport Structures example, sediment transport simulation Bathymetry.plc MIKEZero Plot Composer Plot of model bathymetry WaveField.plc MIKEZero Plot Composer Plot of calculated wave height and direction of propagation CurrentField.plc MIKEZero Plot Composer Plot of wave-driven current field TransportField.plc MIKEZero Plot Composer Plot of sediment transport rates 21

22 Examples 3.4 Structures Example Purpose Scenario We investigate a coastline with contructions placed in parallel with the coastline. This example will highlight how sand transport becomes very complex in such a setting and how sand tends to pile up behind the constructions. Two shore-parallel breakwaters with a length of 300 meters are placed 300 meters from the water line with 700 meters in between them. One of the breakwaters is 1.5 meters below the surface. Irregular, unidirectional waves with H rms = 1.65m and T p = 7.8s move towards the coast from 352. The normal to the beach points towards North. The bed consists of uniform sand with d 50 = 0.25mm and s g = Parameters We are investigating an area that spans 2700 meters along the shore and 840 meters into the ocean. For the hydrodynamic simulation with MIKE 21 HD we use grid cells of 10 times 10 meters, but for the wave simulation with MIKE 21 PMS we use grid cells of five times five meters to get at least six to seven grid points per wave length. We use constant values of velocity-based eddy viscosity e = 1.5m 2 /s and Manning number M = 28m 1/3 /s all over the model area. The time steps are set to four seconds and the simulation runs for four hours to ensure stable conditions. The wave simulation takes wave breaking into account with default values of the breaking parameters γ 1 and γ 2. Bed roughness is not considered. The model grid has an orientation of 90 measured from True North. The updrift boundary (bottom) is prescribed as flux, the downdrift boundary (top) is prescribed as level, and the offshore boundary is prescribed as zero flux. We use a sediment map to describe the bed and to make sure that no sand transport is calculated where the submerged construction sits. This isachieved by prescribing a value of d 50 = -99 at the relevant grid points. 22 MIKE 21 ST

23 Structures Example Prior to running the flow simulation we generate the upstream and downstream boundaries (prescribed as flux and level, respectively) by use of the tool for calculation of Wave Generated Current and Setup. The offshore boundary was prescribed as a zero-flux one. Before running the sediment transport simulation, a sediment transport table (2DH) was generated with the tool for Generation of Sediment Transport Tables of the MIKEZero system Results In the present case, it could be of interest to compare the wave, current and sediment transport fields calculated along the open coast (i.e. away from the structures) and in the vicinity of the submerged and surface-piercing breakwaters. Such comparison permits to assess the different ways in which the structures influence the hydrodynamics and the sediment transport depending on their freeboard. Specification files for the presentation of the results in graphical form have been provided together with your installation of MIKE 21. The sediment transport field computed for the stationary flow conditions of the present test is shown in Figure 3.3 below. The figure shows how the sand tends to pile up lee areas behind the construction because the ability to transport sand is weaker here: 23

24 Examples Figure 3.3 Calculated sediment transport rates in the vicinity of shore-parallel structures Files Here are the files provided with this example - in a standard installation they all reside in C:\Program Files\DHI\MIKEZero\Examples\MIKE_21\ST\Structures\: Name: Name: Name: PMSbathy.dfs2 Structure example, bathymetry for wave simulation Hdbathy.dfs2 Structure example, bathymetry for flow simulation NorthBND.dfs1 Line series of water levels and fluxes along northern boundary of flow model 24 MIKE 21 ST

25 Structures Example Name: Name: SouthBND.dfs1 Line series of water levels and fluxes along southern boundary of flow model Sediment.dfs2 Sediment map (d 50 and σ g ) for sediment transport run The following specification files were used to run the simulations and plot the results: WaveSimulation.PMS Model: MIKE 21 Parabolic Mild Slope Waves Structures example, wave simulation HD_Simulation.M21 Model: MIKE 21 Flow Models Structures example, hydrodynamic simulation TransportTable.21T Tool: Generation of Sediment Tables Structures example, sediment transport table ST_run.ST2 Module: Mike 21 Non-Cohesive Sediment Transport Structures example, sediment transport simulation ModelBathy.plc MIKEZero Plot Composer Plot of HD model bathymetry Wave Results.plc MIKEZero Plot Composer Plot of calculated wave height and direction of propagation HD_Results.plc MIKEZero Plot Composer Plot of wave-driven current field 25

26 Examples ST_Results.plc MIKEZero Plot Composer Plot of sediment transport rates 3.5 Nourishment Example Purpose Scenario We investigate a coast that has been subject to a shoreface nourishment on an otherwise uniform sea bed. Using Q3D sand transport patterns the example shows how the beach is protected behind the nourishment since waves break on it. You also see how return currents evolve around the nourishment from the excess water created by the breaking waves. The nourishment spans about 1450 meters along the coast. It is placed at a water depth of six meters and the water depth on top of the nourishments is 3.52 meters. The bed is made up of sand with d 50 = 0.20mm and s g = 1.5, whereas the sand for the nourishment is d 50 = 0.28mm. Irregular, directional waves with H m0 = 2.5m and T m = 7.47s (equal to T p = 9s) arrive from 260. Waves move over a water level of MWL = 0m Parameters We are looking at an area that spans 6000 meters along the coast and 1840 meters into the sea. The grid cell size for simulation of hydrodynamics with MIKE 21 HD is 10 times 10 meters, and for wave simulation with MIKE 21 NSW it is x = 2.5m and y = 10m. The waves coming from the offshore boundary (left) have a cos 8 directional distribution of wave energy. The simulation takes wave breaking into consideration, using default values of the parameters α, γ 1 and γ 2. Bed roughness, wind generation and wave-current interaction is not considered. When simulating the wave driven currents with MIKE 21 HD we use constant values of velocity-based eddy viscosity e = 0.75m 2 /s and Manning number M = 32m 1/3 /s all over the model area. The time step is set to five seconds and the simulation runs for six hours to ensure stable conditions. Furthermore we use a one hour warm hour period. Prior to running the flow simulation we generate the updrift and downdrift boundaries by use of the tool for calculation of Wave Generated Current 26 MIKE 21 ST

27 Nourishment Example and Setup. The updrift boundary (bottom) is prescribed as flux, the downdrift boundary (top) is prescribed as level, and the offshore boundary is prescribed as zero flux. Before running the simulation we use the tool Generation of Q3D Sediment Transport Tables to create the sediment transport tables used by this example Results Figure 3.4 below show MIKE 21 ST has calculated the sand transport levels in this setting. Figure 3.4 Q3D sediment transport rates in the area of the shoreface nourishment 27

28 Examples The figure shows how the presence of the nourishment reduces the magnitude of the littoral transport along the coast behind it; thus effectively protecting this stretch of coast against erosion. This protection is achieved at the expense of the erosion of the nourishment, which is indicated by the diverging transport field along it. The figure also shows how the sediment transport vectors within the surf zone point in offshore direction. This is due to the net offshore-directed transport component associated with the undertow that has been calculated by the Q3D version of MIKE 21 ST Files Here are the files provided with this example - in a standard installation they all reside in C:\Program Files\DHI\MIKEZero\Examples\MIKE_21\ST\Nourishment\: Name: Name: Name: Name: Name: Name: NSWbathymetry.dfs2 Nourishment example, bathymetry for wave simulation Hdbathymetry.dfs2 Nourishment example, bathymetry for flow simulation NorthBoundary.dfs1 Profile series of water levels and fluxes along northern boundary of flow model SouthBoundary.dfs1 Profile series of water levels and fluxes along southern boundary of flow model SedimentMap.dfs2 Sediment map (d 50 and σ g ) for sediment transport run Q3D_Transport_Table.lon and Q3D_Transport_Table.crs Q3D sediment transport table (longshore and crossshore transport rates) 28 MIKE 21 ST

29 Nourishment Example The following specification files were used to run the simulations and plot the results: WaveSimulation.NSW Model: MIKE 21 Nearshore Spectral Waves Nourishment example, wave simulation FlowSimulation.M21 Model: MIKE 21 Flow Models Nourishment example, hydrodynamic simulation Nourishment_Example.21T Tools: Generation of Q3D Sediment Tables and Wave Generated Current and Setup Nourishment example, Q3D sediment transport tables and boundary conditions for MIKE 21 HD STsimulation.ST2 Module: Mike 21 Non-Cohesive Sediment Transport Nourishment example, Q3D sediment transport simulation WaveResults.plc MIKEZero Plot Composer Plot of calculated wave height and direction of propagation FlowResults.plc MIKEZero Plot Composer Plot of wave-driven current field SedimentTransportResults.plc MIKEZero Plot Composer Plot of sediment transport rates 29

30 Examples 30 MIKE 21 ST

31 4 REFERENCE MANUAL In this section, additional information is provided on MIKE 21 ST. The topics in this section are arranged alphabetically. Ackers & White Transport Formulation (p. 32) Bed Load Transport Coefficient B (p. 33) Bed Roughness (p. 33) Bijker's Transport Method (p. 34) Critical Shields Parameter (p. 36) Engelund & Fredsøe Transport Theory (p. 37) Engelund & Hansen Transport Theory (p. 38) Grid Sizes (p. 39) Meyer-Peter & Müller Transport Formulation (p. 40) Morphological Parameters (p. 40) Porosity (p. 45) Recovering From Errors During Simulation (p. 45) Relative Density of Sediments (p. 47) Sediment Size and Gradation (p. 47) Simulation Area (p. 48) Simulation Period and Time Step (p. 48) Size of Output Files (p. 48) STP Transport Model (p. 49) Water Temperature (p. 50) Wave Breaking Parameters (p. 51) Wave Interpolation (p. 51) 31

32 Reference Manual Zyserman & Fredsøe Transport Formulation (p. 52) 4.1 Ackers & White Transport Formulation The dimensionless total-load transport rate G gr is calculated as G gr = C F gr A m (4.1) in which C, m and A are model parameters that depend on the dimensionless particle size, D gr, defined as D gr = d gs ( 1) 1 3 ν 2 (4.2) where d is the grain size, g is gravitational acceleration, s is the relative density of the bed sediment and ν is the kinematic viscosity of water. F gr is a general sediment mobility number defined as F gr = U f n gd( s 1) V log 10h d 1 n (4.3) where U f is the total shear velocity, h is the water depth, V is the depthaveraged current velocity. n is a constant that also depends on D gr and ranges from 0 for coarse material to 1 for fine material. The transport parameter G gr is defined as F gr = Xh U f sd V n (4.4) where X is the sediment mass flux per unit mass flow rate. 32 MIKE 21 ST

33 Bed Load Transport Coefficient B 4.2 Bed Load Transport Coefficient B The coefficient B is a proportionality constant in the expression for the total load transport in Bijker's Transport Method (p. 34), and can therefore be used as a calibration parameter. You will be asked to specify this parameter (together with the relative sediment density s and the water temperature T) only if Bijker's Transport Method (p. 34) has been selected. A value of the transport coefficient B has to be assigned to every grid point within the model area. This value can be specified in two different ways: a constant value for all grid points a map with a value for every grid point A value of B between 1 and 5 is expected by MIKE 21 ST. B = 1 is suggested for simulations outside the surf zone, whereas the use of B = 5 within the surf zone is advised. 4.3 Bed Roughness The bed roughness parameter is used in MIKE 21 ST to relate the depthaveraged current velocity V to shear velocity U f through U f = g V C (4.5) where g is gravity and C is the Chezy number. If the bed roughness has been expressed in terms of Manning numbers, they are internally converted to Chezy numbers through the relation C = Mh 1/6 (4.6) where h is the water depth. A value of bed resistance must assigned to every grid point in the model area. The bed resistance can be specified in two ways: 33

34 Reference Manual a constant value for all grid points a map with a value for every grid point If you specify the bed resistance by means of a map, the grid spacing and extent of the bed resistance data file must be the same as in the HD data file. The specification of bed resistance for your MIKE 21 ST simulation must be consistent with the specification used in the hydrodynamic simulation that constitutes the basis for the sand transport calculation. Check the specification file for MIKE 21 HD if you cannot remember precisely the type and values of bed resistance used. 4.4 Bijker's Transport Method According to Bijker, the total-load sediment transport, q t, is calculated as the sum of bed-load transport, q b, and suspended load transport, q s. qt = qb + qs = qb( Q) Q is a dimensionless factor defined as 33h Q = I 1 ln I r 2 (4.7) where h is the water depth, r is the bed roughness and I 1 and I 2 are Einstein's integrals, which must be evaluated numerically on the basis of the dimensionless reference level A = r/h and z*, defined as: z = w κu f,wc (4.8) w is the settling velocity of the suspended sediment, κ is von Karman's constant and U f,wc is the shear velocity under combined waves and current. The influence of the waves on the suspended-load transport is therefore taken into account through the shear velocity, U f,wc. 34 MIKE 21 ST

35 Bijker's Transport Method The roughness, r, can be related to the Chezy number, C, through 12h C = 18 log r (4.9) Following Bijker, the shear velocity in combined waves and current U f,wc is found as 1 -- ξ û b = = 2 V U f, wc U fc, 1 gv C 1 -- ξ û b V (4.10) where U f,c is the current-related shear velocity, V is the depth-averaged current velocity, û b is the amplitude of the wave-induced oscillatory velocity at the bottom, and ξ is a dimensionless factor that can be expressed in terms of the wave friction factor f w and Chezy's number C. ξ C f w = g (4.11) The wave friction factor f w is calculated according to Swart as f w a b = exp + r if 1.47 < ---- < 3000 r f w = 0.32 if a b r a b (4.12) a b is the amplitude of the wave motion at the bottom. a b = û b T π (4.13) a b and û b are evaluated using linear wave theory. 35

36 Reference Manual The bed load transport, q b, and suspended load transport, q s, are calculated according to q b = Bd 50 U exp 0.27 d 50 g f, c µu2 f, wc (4.14) It is interesting to note that the influence of the waves on the bed-load transport, q b, is accounted for through a "stirring term", i.e. the exponential in the previous equation. The "transporting term" is only related to the current by way of U f,c. B is a dimensionless bed load transport coefficient, is the relative density of sediments and µ is the so-called "ripple factor". and µ are defined as = s 1 = ρ s ρ (4.15) µ = C C (4.16) ρ s is the density of the sediment, ρ is the density of water and C' is the Chezy number related to the geometric characteristics of the bed material. It is calculated as 12h C = 18 log d 90 (4.17) d 90 is the sediment size for which 90% (in weight) of the bed material is finer. For uniform bed material d 90 becomes identical to the specified value of d Critical Shields Parameter The critical value of the dimensionless bed shear stress (the Shields parameter), θ c, is used to indicate the threshold value for initiation of sediment motion. MIKE 21 ST expects and will acccept values of θ c between and The user-selected value for θ c will not be used if you select either the Engelund & Hansen Transport Theory or the Zyserman & Fredsøe Trans- 36 MIKE 21 ST

37 Engelund & Fredsøe Transport Theory port Formulation. No threshold for initiation of motion is used in the first case, while θ c is assigned a value = in the second. 4.6 Engelund & Fredsøe Transport Theory The total-load transport rate q t is calculated as the sum of the bed-load transport q b and the suspended-load transport rate q s q t = q b + q s It is assumed that bed-load transport takes place in one single layer of thickness equal to one grain diameter d. The bed-load transport qb is calculated as q b = 5p( θ 0.07 θ c ) ( s 1)gd if θ > θ c (4.18) where p is the probability that all particles in a single layer will be in motion, θ is the dimensionless bed shear stress (Shields parameter) related to skin friction and θ c is the critical bed shear stress for initiation of motion. s is the relative density of the bed material. θ is defined as θ = U f ( s 1)gd (4.19) p is defined as p = 1 + π --β θ θ c (4.20) with β = the dynamic friction coefficient. Following the ideas of Einstein (1950), the suspended load q s is evaluated as 30h q s = 11.6U f c b a I 1 ln I 2 k N (4.21) 37

38 Reference Manual with c b = the bed concentration of suspended sediment, U f = the shear velocity related to skin friction, a = 2d = the reference level for c b, I 1 and I 2 = Einstein s integrals, h = the water depth and k N = Nikuradse s equivalent roughness = 2.5d. The integrals I 1 and I 2 are a function of the dimensionless reference level A = a/h and of the Rouse number z = w s /κu f, where w s is the settling velocity of the suspended sediment and κ = von Karman s constant ( 0.40). I 1 and I 2 are integrated between y = a and y = h, where y is measured upwards from the fixed bed level. Engelund and Fredsøe developed a semi-empirical relation for the value of c b at a = 2d 0.65 c b = ( λ) 3 (4.22) where the linear concentration λ is given by λ = πpβ θ θ c if θ > θ 0.027sθ c + πpβ 6 (4.23) Note that the transport formulation of Engelund & Fredsøe was developed on the basis of data obtained from experiments with bed material of sandfraction size. Therefore, you should make sure that the bed material used as input to the model falls within the proper range of grain sizes. 4.7 Engelund & Hansen Transport Theory The dimensionless rate of total-load transport Φ t is calculated as Φ t = θ C g (4.24) with C = Chezy s number and Φ t = q t ( s 1)gd 3 (4.25) 38 MIKE 21 ST

39 Grid Sizes with q t = the total-load sediment transport and g = gravitational acceleration. The dimensionless bed shear stress θ is defined as θ = U f ( s 1)gd (4.26) where U f is the shear velocity related to total friction, d is the grain diameter and s is the relative density of the bed material. This formulation assumes that the dimensionless bed shear stress θ is much larger than the Critical Shields Parameter for initiation of transport θ c. Therefore, the value of θ c specified by the user will not be applied if this formulation has been chosen. Something similar applies to the gradiation of the bed material, since Engelund & Hansen theory is based on the median grain size d Grid Sizes The grid used in the simulation of the sand transport rates and the rate of bed level changes is the same as that used to calculate the flow field using the hydrodynamic module, MIKE 21 HD. The wave grid may have the same grid spacing as the hydrodynamic one, or a smaller one. The sand transport calculations will only be performed in the area where wave and current data exist, i.e. in the region common to the wave and the hydrodynamic grids. It should be checked that the area of interest for the sediment transport computations be covered by both grids. The space resolution in the sand transport calculations is given by the spacing of the grid used for the hydrodynamic computations, so care must be taken that all the relevant details are covered when selecting the spacing of the hydrodynamic grid. The grid spacing of the hydrodynamic grid ( x HD, y HD ) must be equal to or an integral multiple of the grid spacing used to determine the wave characteristics ( x w, y w ), both for the x-and the y- directions. The proportionality factors need not to be equal. x HD = n x w with n 1 (4.27) y HD = m y w with m 1 (4.28) 39

40 Reference Manual 4.9 Meyer-Peter & Müller Transport Formulation The dimensionless bed-load transport rate Φ b is calculated as Φ b = 8( θ θ c ) 1.5 (4.29) with Φ b = q b ( s 1)gd 3 (4.30) θ is the Shields parameter related to skin friction and θ c is its critical value. s is the relative density of the bed sediment, g is the acceleration of gravity and d is the grain diameter. Note that for situations with fine sediment and/or high current speed, this formulation may underestimate the actual transport rates, since only the contribution from the bed load transport is accounted for. In case of doubt, you could repeat the simulation using another transport theory (e.g. Engelund & Fredsøe Transport Theory or Zyserman & Fredsøe Transport Formulation) in order to assess the importance of the suspended-load transport for the case being analysed Morphological Parameters Morphological Schemes MIKE 21 CAMS is used to solve the coupled governing equations for waves, flow, sediment transport and conservation of sediment mass. The last equation yields the bed level update scheme for computing the evolution of the bed levels (bathymetry) with time. The calculation of the rates of bed level changes is based on the calculated sediment transport rates at each morphological time step. The numerical method used in the bed level update scheme is specified in the MIKE 21 ST and described below. FTCS scheme This is the Forward Time, Centred Space scheme. Here, the dz/dt at each grid point reflects the local gradients in the sediment transport fluxes. Var- 40 MIKE 21 ST

41 Morphological Parameters iations in the initial dz/dt values are traceable to the sediment transport gradients. z n + 1 ( jk, ) z n ( j, k) t q x n ( jk, ) q n x ( j 1, k) q n y ( j, k) q n y ( jk, 1) n x y = 0 (4.31) This scheme is not recommended for morphodynamic simulations (using CAMS) because of the inherent limitation of this scheme for propagation problems, such as the propagation of bed disturbances. In this case, the scheme is guaranteed to result in instabilities. FTCS scheme may however be used in combination with including Bed slope diffusivity effect. The latter effect adds a diffusion term to the bed level update equation. Lax Wendroff scheme The initial dz/dt map reflects the gradients in the sediment transport pattern using a centred time-centred space scheme. The time centring is achieved by replacing the second order truncation error term in time (d2z/dt2) with equivalent spatial derivatives. It is assumed that the bed celerities vary slowly in time and space. z n + 1 ( jk, ) z n ( j, k) t (4.32) q x n ( jk, ) q n x ( j 1, k) q n y ( j, k) q n y ( jk, 1) n x y = t 2 2 z 2 2 z 2 z ---- C 2 x x 2 + C y y 2 + 2C x C y xy j, k This scheme suffers from high wave number oscillations (features resolved with less than 6 grid points) in the morphological evolution with time, since high wave number oscillations are not damped by the scheme. Hence, this scheme can only be used for short-term simulations (i.e. before the high wave number oscillations dominate the solution). The use 41

42 Reference Manual Filtering Methods of Filtering Methods removes some of the high wave number oscillations. Furthermore, including Bed slope diffusivity effect adds a diffusion term to the bed level update equation, which significantly stabilises the solution. Constant filter coefficients The filtering scheme is used to dissipate high wave number spatial oscillations in the bed level evolution. The reasons for this are described briefly: The conservation equation for sediment mass can be transformed into a propagation equation for bed level evolution. Analysis of the transformed equation show that the nonlinear celerity of bed level evolution leads to the generation of higher spatial harmonics of the bed level oscillations with time. Thus, any finite difference scheme with an amplification portrait of 1.0 for high wave numbers will keep the generated harmonics in the solution. In such a scheme, the harmonics may eventually (after a long enough time) destroy the morphological solution once the finite grid spacing becomes insufficient to properly resolve the harmonics. A two-dimensional filter (with constant coefficients) as described in Johnson & Zyserman (2002) is used to suppress the harmonics. In this case, the initial dz/dt map reflects the gradients in the sediment transport pattern as calculated by the selected morphological scheme AND it also reflects the effect of the filter in damping underlying high wave number bottom disturbances in the bathymetry. Thus, if the initial bathymetry has a lot of high wave number bottom irregularities, the initial dz/dt map will also reflect this, and may look rather shaky. As the high wave number bottom irregularities are damped, the predicted bathymetry from the morphodynamic simulation will be smooth with time. However, it should be noted that NOT all the bottom changes obtained using this scheme is due to sediment transport gradients, some of it is due to the damping of high wave number oscillations by the filter. This scheme is attractive, because it damps high wave number oscillations, allowing the modeller to focus on the morphological development of the main resolved morphological features in the area. Thus, the simulation can be carried out for a longer time period than without filtering. However, it has the side effect of damping high wave number oscillations also in areas with little or no sediment transport. This is not desirable, and may cause problems in a number of situations, e.g. navigation channels, river mouths or tidal inlets where the number of grid points typically used 42 MIKE 21 ST

43 Morphological Parameters to resolve the features is inadequate. This led to the modified version of the scheme, using Courant based filter coefficients. Courant based filter coefficients For this scheme, the filter factor (α in Johnson & Zyserman, 2002) is linked to the local Courant number of the bed level evolution, with a maximum value of α= This ensures that the filter has no effect in areas where the celerity of bed level evolution (or sediment transport is small), while damping high wave number oscillations in morphodynamically active areas. In this case, the initial dz/dt map reflects the gradients in the sediment transport pattern as calculated by the selected morphological scheme AND to a limited extent, it also reflects the effect of the filter in damping underlying high wave number bottom disturbances in the bathymetry Bed slope diffusivity effect The bed slope diffusivity effect is included in the Bed level update scheme. Two effects due to bed slope are included. These are the Longitudinal slope effect and the Transverse slope effect Longitudinal slope effect The longitudinal slope effect leads to an increase of the sediment transport in the down-slope direction. The converse is also true in the up-slope direction. The longitudinal slope diffusivity is quantified using the Meyer- Peter Muller bed load formula. q R0 = ( s 1)gd 3 8( θ' θ c0 ) 1,5 (4.33) Including the effect of a longitudinal bed slope, eq. (4.33) can be written as: q res ( s 1)gd 3 z = 8 θ' θ c0 µ ,5 s (4.34) where q res is the resulting sediment transport including the effect of bed slope, µ is a dynamic friction coefficient of the order of 1.0 and z s is the bed slope in the flow direction. The slope is assumed to be small. 43

44 Reference Manual By expanding eq. (4.34) using Taylor series the expression in eq. (4.35) is obtained: z q res = q R0 ε s q R s (4.35) The longshore slope diffusivity ε s is then calculated by ,5 8 ( s 1)gd 3 1 ε s = MPM_fac 1, q R0 (4.36) where MPM_fac is a user-defined scale factor between 0 and 2. Transverse slope effect This leads to a deflection in the sediment transport direction due to a transverse slope. This effect is quantified using the method of Engelund (1974), Fredsøe(1978). Basically, the transport due to transverse slope is proportional to transverse slope diffusivity, longitudinal flat bed sediment transport and the bed slope in the transverse direction. The transverse slope diffusivity ε n is calculated as 1 ε n = tanϕ (4.37) where ϕ is the dynamic friction angle. Bed level update scheme The equations for the bed slope diffusivity effect are re-formulated in such a way, so that it is transparently a diffusion term in the bed level update equation: z t 1 n n x q x x q y = y z Kxx ---- z Kxy ---- z Kxy ---- z Kyy ---- x x y y x y y (4.38) 44 MIKE 21 ST

45 Porosity where K xx = ε s q R0 cos 2 α + ε n q R0 sin 2 α (4.39) K xy = ( ε s q R0 ε n q R0 ) sinαcosα (4.40) K yy = ε s q R0 sin 2 α + ε n q R0 cos 2 α (4.41) Note that the output sediment transport fluxes are not updated to include the bed slope diffusivity effect Porosity The porosity of the bed material n must be specified as a constant value for the whole area where the transport rates of non-cohesive sediment are calculated. The usual value for sand (n = 0.40) is taken as default value by MIKE 21 ST Recovering From Errors During Simulation In most cases, the source of the error will have been written out to the log file generated during your MIKE 21 ST simulation. In order to recover from the error, follow the steps below. 1 Inspect the MIKE 21 ST log file that has been created in your work directory to check at what stage of the calculation the error occurred. The log file can be inspected using any text editor such as Notepad. 2 Inspect the MIKE 21 ST specification file for possible errors. The log file will give you an indication of what to check. The specification file can also be inspected using any text editor. 3 Run the sediment transport simulation again. If the problem persists, repeat steps 1 and 2 above until it has been solved. In most cases, the problem will be related to the fact that the axes of the sediment transport table(s) do not cover the whole range of variation of the parameters involved when calculating sediment transport under combined waves and current. 45

46 Reference Manual You can try to solve the problem by extending the sediment transport table(s). You can either use the tool for Extension of sediment tables (if you chose to use the 2DH version of the STP program to calculate the transport rates) or the tool for Extension of Q3D sediment tables (if you chose to use the Q3D version of the STP program). You can also use the facility for limiting the value of the relevant parameters within the Simulation Area (p. 48) to the minimum and/or maximum values along the corresponding axes of the sediment transport table(s). In order to do this, the options in Table 5.1 below can be used. The required option parameters have to be defined in the OPTION_PARAMETERS section of the specification file for your MIKE 21 ST run. By default, all parameters are internally set to 0, implying that an error message will be generated and the simulation will be stopped if the limits of the transport table are exceeded. You can change the defaults for some or all of the parameters to 1, and thus limit the parameter values for calculating sediment transport to the limits in the table. Table 4.1 Option parameters for avoiding search beyond the limits of the sediment transport table Option parameter (limit minimum) Limit_min_V Limit_min_HoD Limit_min_H Limit_min_T Limit_min_Sg Limit_min_d50 Limit_min_dddx Limit_min_dddy Limit_min_gam Option parameter (limit maximum) Limit_max_V Limit_max_HoD Limit_max_H Limit_max_T Limit_max_Sg Limit_max_d50 Limit_max_dddx Limit_max_dddy Limit_max_gam This facility should be used with utmost caution, since your sediment transport calculations will not be accurate beyond the limiting values of the parameters. Unless results have to be achieved for a quick preliminary assessment, it is always advisable to extend the sediment transport table(s). 46 MIKE 21 ST