Simple Coastal Modelling - Math Problem Example

Name Topic Name of Assignment Lecture Name 13th May, 2016 Simple Coastal Modelling Question 1 Problem Description and Formulation Water waves can be very devastating if proper prediction is not carried out using modelling techniques that are accurate. The models adopted in wave prediction should be able to accurately predict the magnitude of the wave, water velocity, and consider surface elevation to determine the safety coatline for humana activity and settlement To begin with determining wave amplitude as well as the mean water depthis essential in the models that are used simulation of wave relationship between the surface elevation, the horizontal and vertical water particle velocities. A fundamental approach to this problem would be to try to solve the wave length of a wave that is propagating to coastline. To do this requires knowledge of the water depth characteristic, surface elevation, the horizontal and vertical water particle velocities of the wave in order he adoption of a suitable model. This section is going to study wave mathematical model using Matlab. We will write matl MATLAB codes and simulate it to produce reliable information. Numerical Procedure We begin by deteming surface elevation of the coastline where the water depth is 6.2 meters. The folloeing formulae is used to calculate surface elevation the horizontal particle velociy of the wave can be calculated with Vertical water particle velocities of the wave Where g is the gravity, H is the wave amplitude, k is the wave number, d is the mean water depth, is the angular frequency, z is the location below the mean water level (MWL). This model can be shown in the diagram below. Results The rsults of the problem formulated above is is presneted in this case 1. MATLAB CODES to determinethe wave characteristics in deep water and at the coast D=6.2; H=5; %wave height% T=6.8; g=10; L1= g*T^2/(2*pi); %wave height% k=(2*pi)/L1 q=g*k*tanh(k*H) q1=((q*H)/8)^(3/4) L=L1*(tanh(q1))^(2/3) C=L/T F=1/T The following the output results for some parameters D (m) L (m) H (m) k f (Hz) c (m/s) Deep water 6.2 33.3850 5.0 0.0854 0.1471 4.9096 The coast 5.0 33.3850 5.0 0.0854 0.1471 2.645 Table 1: Wave characteristicsin deep water and at the coast. Where D = water depth, L = wave length, H = wave height, k = wave number, f = wave frequency, c = wave celerity. 2. Matlab codes to calculate the horizontal and vertical water particle velocities for t varies for 2 wave periods and z varies from 0 to –d. The following is code that will be used solve the problem D=6.2; H=5; %wave height% T=6.8; g=10; L1= g*T^2/(2*pi); %wave height% k=(2*pi)/L1; q=g*k*tanh(k*H); q1=((q*H)/8)^(3/4); L=L1*(tanh(q1))^(2/3); C=L/T; F=1/T; Angu=2*pi*F; z=(0:1:D); t=(0:2.26:2*T); u=(((-Angu*H)/2)*(cosh(k(z+D)))/(sinh(k*D)))*cos(Angu*t) v=(((-Angu*H)/2)*(sinh(k(z+D)))/(sinh(k*D)))*sin(Angu*t) subplot(u,t);xlabel('time'); ylabel('Horizontal velocities'); legend(''Horizontal velocities'); title('Horizontal velocities vs time') subplot(V,t);xlabel('time'); ylabel('Verticle velocities'); legend(''Horizontal velocities'); title('Verticle velocities vs time') 2.1. figures to present the time series of the water particle velocities at the locations of the trough (z = -H/2) and at the bottom. 2.2. Present the vertical profile of the magnitudeof horizontal and vertical water particle velocities from the MWL to the sea bottom. Figure 2.xxx… 3.Generate a similar graph as below for one wave period and at the mean water level, to show the relationship between the surface elevation, the horizontal and vertical water particle velocities. Figure 3.xxx…. discussioN the relationship between the water depth and the particle velocities From the graph it can be noted that water depth and the particle velocities There is also erroneous representation of relationship between the runoff and the controlling input variables such as assuming linearity when it isnot linear system. One may also assume that snow melt has no effect while it has. This will lead to erroneous model parameters that give poor results. (ii) the phase relationship between the surface elevation, the horizontal and the vertical particle velocities. From the graphs it can noted that Phase relationship between the surface elevation, the horizontal and the vertical particle velocities. Appedix – Matlab codes D=6.2; H=5; %wave height% T=6.8; g=10; L1= g*T^2/(2*pi); %wave height% k=(2*pi)/L1; q=g*k*tanh(k*H); q1=((q*H)/8)^(3/4); L=L1*(tanh(q1))^(2/3); C=L/T; F=1/T; Angu=2*pi*F; z=(0:1:D); t=(0:2.26:2*T); u=(((-Angu*H)/2)*(cosh(k(z+D)))/(sinh(k*D)))*cos(Angu*t) v=(((-Angu*H)/2)*(sinh(k(z+D)))/(sinh(k*D)))*sin(Angu*t) plot(u,t);xlabel('time'); ylabel('Horizontal velocities'); legend(''Horizontal velocities'); title('Horizontal velocities vs time') plot(V,t);xlabel('time'); ylabel('Verticle velocities'); legend(''Horizontal velocities'); title('Verticle velocities vs time') Question 2 Problem description and formulation The problem is to be able to predict the shorenormal sediment transport rates and the location of the shoreline in a coastline. The solution requires coupling the hydrodynamic modelling approaches with appropriate sediment transport equations whilst allowing for the changing geometric boundary condition of the shoreline. Common current practice is to assume a single phase flow whereby the sediment continuity equation (in terms of concentration of sediment) is coupled with the continuity and momentum equations for the water flow. This approach is only valid for relatively small sediment concentrations. Applying a two-phase flow approach should take account of the interaction between the water flow and the sediment movement, including the channel morphological evolution. However, the mathematical representation of such sediment-flow interaction is the subject of on-going research as is the selection of the correct turbulence model to close the Navier-Stokes equations in this situation. This section will consider a a 5km stretch of coast oriented in the north-south direction with the ocean to the east where the predominant wave direction is from the east-south-east. At the southern end the typical breaker height is 1.4m and the breaker angle is 10°. At the northern end, the breaker height is 1.45m and the breaker angle is 12°. The breaker parameter γb = 0.8. The beach profiles along the section are similar with slopes near the break point of 1/40. The sand is made of quartz (s = 2.63, p = 0.28) with a median grain size of 0.22mm and measureable seasonal bed level changes are restricted to depths less than 6 metres. The berm height is 3 m AHD. The average shorenormal sediment transport rates (𝑄!) for 1993-2015 were saved in the data file “sediment.xlsm”. There are no sinks and sources for sediment transport in the control domain. Numerical Procedure We will begin by momentum equation for a turbulent flow in the longitudinal direction is derived from the general Navier-Stokes equation as where  is the density of water is the mean point velocity in the Horizontal is the mean point velocity in the verticle tis time  is the absolute coefficient of viscosity which represents viscous shear stresses in the water. isthe ‘eddy viscosity’ which represents turbulent shear stresses in the water. The magnitude of for a given flow has to be evaluated. First, the viscous shear stress terms are assumed to be small compared with the turbulent shear terms. Second, it is often possible to ignore effects of vertical variations and simply to take ‘depth averaged’ values, which are found by integration between the bed and the water surface and then dividing by the depth h = (ys–y0) thus: the turbulence is assumed to be homogeneous, and shear stresses on vertical planes may be ignored, therefore the last term on the right-hand side may be simplified to whereτ0is the mean boundary shear stress and τsis the shear stress at the water surface. Recalling that for uniform flow Equation in depth-averaged form and divided by (ρg) becomes where p is the sediment porosity, xs is the shoreline coordinate Qsource is a sediment input (e.g. river discharge, beach nourishment) Qsink is a sediment loss (e.g. dredging) K ≈0.77 is an empirical coefficient which has a weak dependence on grain size. s is the specific weight Hb is the breaker height γ is the breaker index θb is the wave crest angle at the break point all other variables are as defined in the figure The model can be solved using the finite element method with the parameters, f, ,  and Cuv determined empirically. Thus the numerical solution provides values of the depth-averaged velocity and longitudinal unit flow on a grid of points across the cross-section Results 1 Write MATLAB CODEs to generate a figure to show the average shorenormal sediment transport rates (Q_x) for 1993-2015 were saved in the data file “sediment.xlsm”. file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of sediment.xlsx'); t=file(:,1); R=file(:,2); plot(t,R);xlabel('YEAR'); ylabel('the average shorenormal sediment transport rates'); legend('sediment transport rate'); title('The sediment transport rate vs year') Figure 1.xxxxx 3. Use Matlab to develop a numerical model to calculate the location of the shoreline from 1993 to 2015, i.e. xs. (assume xs = 0 in 1993) file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of shoreline (1).xlsx'); k=file(:,1); s=file(:,2); plot(k,s);xlabel('Year'); ylabel('shorenormal sediment transport rates'); legend('sediment transport rate'); title('shorenormal sediment transport rates vs year') Figure 2.xxxx 4. Write Matlab codesto evaluate the accuracy of the model file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of shoreline (1).xlsx'); k=file(:,1); s=file(:,2); plot(k,s);xlabel('Year'); ylabel('shorenormal sediment transport rates'); legend('sediment transport rate'); title('shorenormal sediment transport rates vs year') modelfits = []; for p=1:size(rec.params,1)  modelfits(p,:) = X*rec.params(p,:)'; end mn = median(modelfits,1); se = stdquartile(modelfits,1,1); h2 = errorbar3(xx,mn,se,'v',[.8 .8 1]); h4 = plot(k,s.^2 + 2*s + 4,'r-'); uistack(p,'top'); xlabel('Year'); ylabel('shorenormal sediment transport rates'); legend('sediment transport rate'); title('shorenormal sediment transport rates vs year') legend([h1 h4 h3 h2],{'Data' 'True model' 'Estimated model' 'Error bars'}); title('Model is reliable and accurate'); discussion (ii) the accuracy of the model; Model accuracy is dependent on a large number of factors which need to be appreciated so that undue reliance is not placed on the predictive capacity of the model. Thus the final accuracy of any computational model is very difficult to quantify. Some sources of error may be minimized but others have to be accepted as model uncertainty and the results of the model viewed with this in mind.The models provide an understanding of the flow dynamics along the length of the wave. From the Matlab we note that RMSE store for the model vary and have a mean score of 0.0083. The carried out MEA that were to help in improving the accuracy of the results that btained was 0.00123. The following table shows RMSE score values using Matlab models Forecasters should use practicestake into account uncertaintyeven though they are complex to use and compute, some expression of forecast uncertaintyis clearly better than none. As an absolute minimum, even if no formal uncertainty analysis is carried out, brusquepredicts should include by a cautionary statement regarding the uncertainties associated with the forecasts. 3.2 Goodness of fit between the observed predicted flat flows. In order to have accurate results of goodness -of -fit in multiple flood events like in our case it was necessary to record rainfall for every event in millimetres and the predicted value. Any event that is selected in estimating the model should show waves of shoreline. In my case the event that i selected from eleven observed events had a peak sediment rate above the median. This gave me results that was less and certain and required less calibration. Residual analysis is very important in a regression model. Given a set of data, a regression model is built to find out to what extent the value of dependent variable can be predicted by independent variable. For a given set of data, a regression model is built in such a way that the deviation of predicted values from the actual values is minimized. A residual analysis helps in finding out to what extent this deviation has been reduced by the regression model. If the value of residual sum of square after running a regression on a data set is found to be low, then it can be said that the model is efficient enough in predicting the value of the independent variable given the value of dependent variable. (iii) how to improve the model A typical example of where physical processes are not represented in 1D models and, consequently, 2D models should be used is in compound channel flow (i.e. at river cross section in which water is transferring laterally between the main channel and the flood plains. In such circumstances the water surface is not horizontal in reality, but is assumed horizontal in a 1D model. Whilst a 1D model can approximate this particular case, the Conveyance Estimation System (DEFRA/EA, 2003) improves the representation. Similarly at sharp bends in which regions of slack water and eddy currents are generated. This may even result in localized reverse flow taking place. Again a 1D model cannot reproduce these phenomena and a 2D approach is required. Turning next to the validity of the numerical schemes used in the model, it is not normally possible (or desirable) for a model user to adjust them. Thus it is very important to choose a computational model with an established track record and pedigree. The well-established commercial computational models do have robust numerical schemes which perform very well under most circumstances. Errors in the measurement of topographic data can be minimized by quality control of the survey and are generally within acceptable tolerances. However, gross errors do sometimes occur and these need to be eliminated either by quality control procedures at the data input stage (for example by plotting all the cross-sections on the computer screen) or during the model calibration procedure. Errors in the hydrological data are not so easy to eliminate and may, therefore, need to be catered for by a sensitivity analysis. For example, if say a 100-year return period flood hydrograph has been generated, it may be subject to quite wide confidence limits. Thus it would be advisable to run the computational model with several flood hydrographs to establish the sensitivity of flood stage to flood discharge. Finally errors may be generated by an inadequate schematization and discretization of the river system and by an inadequate calibration. These two processes are key elements in the application of any computational model and are discussed separately in the next two subsections. Appedix – Matlab codes (15%) MATLAB CODEs for the average shorenormal sediment transport rates file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of sediment.xlsx'); t=file(:,1); R=file(:,2); plot(t,R);xlabel('YEAR'); ylabel('the average shorenormal sediment transport rates'); legend('sediment transport rate'); title('The sediment transport rate vs year') Use Matlab to develop a numerical model to calculate the location of the shoreline from 1993 to 2015, i.e. xs. (assume xs = 0 in 1993) file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of shoreline (1).xlsx'); k=file(:,1); s=file(:,2); plot(k,s);xlabel('Year'); ylabel('shorenormal sediment transport rates'); legend('sediment transport rate'); title('shorenormal sediment transport rates vs year') 5. Matlab codesto evaluate the accuracy of the model file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of shoreline (1).xlsx'); k=file(:,1); s=file(:,2); plot(k,s);xlabel('Year'); ylabel('shorenormal sediment transport rates'); legend('sediment transport rate'); title('shorenormal sediment transport rates vs year') modelfits = []; for p=1:size(rec.params,1)  modelfits(p,:) = X*rec.params(p,:)'; end mn = median(modelfits,1); se = stdquartile(modelfits,1,1); h2 = errorbar3(xx,mn,se,'v',[.8 .8 1]); h4 = plot(k,s.^2 + 2*s + 4,'r-'); uistack(p,'top'); xlabel('Year'); ylabel('shorenormal sediment transport rates'); legend('sediment transport rate'); title('shorenormal sediment transport rates vs year') legend([h1 h4 h3 h2],{'Data' 'True model' 'Estimated model' 'Error bars'}); title('Model is reliable and accurate'); Read More

This section will consider a a 5km stretch of coast oriented in the north-south direction with the ocean to the east where the predominant wave direction is from the east-south-east. At the southern end the typical breaker height is 1.4m and the breaker angle is 10°. At the northern end, the breaker height is 1.45m and the breaker angle is 12°. The breaker parameter γb = 0.8. The beach profiles along the section are similar with slopes near the break point of 1/40. The sand is made of quartz (s = 2.63, p = 0.28) with a median grain size of 0.

22mm and measureable seasonal bed level changes are restricted to depths less than 6 metres. The berm height is 3 m AHD. The average shorenormal sediment transport rates (𝑄!) for 1993-2015 were saved in the data file “sediment.xlsm”. There are no sinks and sources for sediment transport in the control domain. Numerical Procedure We will begin by momentum equation for a turbulent flow in the longitudinal direction is derived from the general Navier-Stokes equation as where  is the density of water is the mean point velocity in the Horizontal is the mean point velocity in the verticle tis time  is the absolute coefficient of viscosity which represents viscous shear stresses in the water.

isthe ‘eddy viscosity’ which represents turbulent shear stresses in the water. The magnitude of for a given flow has to be evaluated. First, the viscous shear stress terms are assumed to be small compared with the turbulent shear terms. Second, it is often possible to ignore effects of vertical variations and simply to take ‘depth averaged’ values, which are found by integration between the bed and the water surface and then dividing by the depth h = (ys–y0) thus: the turbulence is assumed to be homogeneous, and shear stresses on vertical planes may be ignored, therefore the last term on the right-hand side may be simplified to whereτ0is the mean boundary shear stress and τsis the shear stress at the water surface.

Recalling that for uniform flow Equation in depth-averaged form and divided by (ρg) becomes where p is the sediment porosity, xs is the shoreline coordinate Qsource is a sediment input (e.g. river discharge, beach nourishment) Qsink is a sediment loss (e.g. dredging) K ≈0.77 is an empirical coefficient which has a weak dependence on grain size. s is the specific weight Hb is the breaker height γ is the breaker index θb is the wave crest angle at the break point all other variables are as defined in the figure The model can be solved using the finite element method with the parameters, f, ,  and Cuv determined empirically.

Thus the numerical solution provides values of the depth-averaged velocity and longitudinal unit flow on a grid of points across the cross-section Results 1 Write MATLAB CODEs to generate a figure to show the average shorenormal sediment transport rates (Q_x) for 1993-2015 were saved in the data file “sediment.xlsm”. file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of sediment.xlsx'); t=file(:,1); R=file(:,2); plot(t,R);xlabel('YEAR'); ylabel('the average shorenormal sediment transport rates'); legend('sediment transport rate'); title('The sediment transport rate vs year') Figure 1.xxxxx 3. Use Matlab to develop a numerical model to calculate the location of the shoreline from 1993 to 2015, i.e. xs.

(assume xs = 0 in 1993) file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of shoreline (1).xlsx'); k=file(:,1); s=file(:,2); plot(k,s);xlabel('Year'); ylabel('shorenormal sediment transport rates'); legend('sediment transport rate'); title('shorenormal sediment transport rates vs year') Figure 2.xxxx 4. Write Matlab codesto evaluate the accuracy of the model file = xlsread('D:\Documents and Settings\Administrator\My Documents\Copy of shoreline (1).xlsx'); k=file(:,1); s=file(:,2); plot(k,s);xlabel('Year'); ylabel('shorenormal sediment transport rates'); legend('sediment transport rate'); title('shorenormal sediment transport rates vs year') modelfits = []; for p=1:size(rec.

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