pRO

146 Project

19

Introduction

This report will consist of 4 parts. Part one is a summary of methods used to model a flood in a river. Each method is discussed qualitatively. Part two is a test run of the program explicengle. The test run is compared to the given pages for report document. Part three uses the same program as part 2 but for the Sacramento River. Runs are to be made and discussed in detail on how different variables affect the depth, velocity and flow of the river. Part four uses the program flow Pro to model the Sacramento River. Results and methods are discussed.

Part I Discussion1)a) Please discuss qualitatively how you can obtain the mass and momentum (Saint Venant) equations for flow in a river. i) Mass equationThe mass equation can be derived from the Continuity equation flor an unsteady variable-density flow

Inlet- There is a flow of Q entering the volume upstream. There is a flow qlat entering the volume on the side of the channel. qlat is in terms of flow/length. Density is assumed to stay constant therefore,

The integral of Volume with respect to cross sectional area is converted to flow + lateral flow multiplied by incremental length of channel. Outlet-

The outlet is the flow Q + the change in Q with respect to distance multiplied by incremental distance.We are holding Area constant so the integral of the incremental volume is equal to Adx

Therefore we get the following equation. Again density is constant throughout.

Divide by and some algebra you get

ii) Momentum

The sum of all the forces of gravity, expansion, pressure and contraction can be defined by

Again the integrals of volume with respect to area can be dealt with by inflow and out flows also the change in flow with respect to distance. After Divide by and some algebra, making h=y+z and we get the momentum equation below

b) What are the unknowns of those equations? The unknowns in the these equations are Q(x,t) and A(x,t)

c) What type of equations are they?Continuity equation is a PDEMomentum equation is also a PDE

2) Please explain the alternatives engineers have, when routing floods in rivers (in addition to the Muskingum method). Discuss the different levels of complexity versus physical processes left outside of the analysis. Indicate which method is fastest and less expensive. Please indicate also advantages and disadvantages of each method. Define the boundary and initial conditions for the computations.

Table 1: Routing Flood alternativesFully DynamicNon-InertiaKinematicHydrologic/Muskingum

equation

explanationEntire equation is used.Boundary conditions: A(x,t)Initial Conditions: A(x,t) when t=0Inertial terms are assumed to be 0.Boundary Conditions: A(x,t)Initial Conditions: A(x,t) when t=0

Uniform flow is assumed. Friction slope is equal to channel slope.Boundary Conditions: A(x,t)Initial Conditions: A(x,t) when t=0Flow is a function of only time.Boundary Conditions: NoneInitial Conditions: None

advantagesDynamic equations is valid for all flow scenarios and a good model of actual behavior of flow.Non-Inertia is cheaper than Fully dynamic and Faster than Fully Dynamic.Kinematic is even cheaper than Non-Inertia and even Faster than Non-InertiaHydrologic is the cheapest and Fastest

disadvantagesIt is Numerically Challenging to solve, Expensive and it takes time.Sometimes it does not give a good approximation. More assumptions can lead to even less reliable results.Dynamic effect of the flow is ignored. It Can be an inaccurate representation of the flow.

3) Please present a plausible explicit scheme to solve the 1D Saint-Venant (mass and momentum) equations.a) Mass

To make an explicit scheme the mass equation (1) needs to be discretized. The end result is the discretized equation (2). Each term is discretized. An explicit value of is found. i represents the spatial step and j represents the time step. Discretizations of y with respect to x and t were center and forward respectively.b) Momentum

(3)

To make an explicit scheme the momentum equation (1) needs to be discretized. The end result is the discretized equation (2). Mannings equation (3) is used to find equation (4) in (3) it is labeled as S. time step of is multiplied throughout the equation after discretization to create (2). Can be calculated from the corresponding depth . The quadratic equation can be used to solve for explicitly.

4) Discuss the potential implicit schemes to solve those equationsFor number implicit schemes Newton-Raphson, Regula-Falsi, Fixed-Point or Bisection methods can be used to approximate the values of Q and A.A weighted term is used to produce the following implicit formulas. When this term is greater than one then the equations stay implicit. Below are the implicit equations.a) Mass

b) Momentum

5) Please explain the origin of the HEC-1 scheme. How does it work? Prepare a plot in the space-time domain and indicate the nodes involved in the computational molecule.

The U.S. Army Corps of Engineers founded the Hydrologic Engineering Center (HEC). It models the change in flow in rivers due to rainfall.The HEC-1 method finds A(x,t) from initial and boundary conditions.

Figure 1: HEC-1 time vs distance with initial and boundary conditions

In this graph the distance between two points on the x axis is and on the t axis. will be given values. All boundary conditions and initial values will be given. With this given information we can find All A given the formula above. a and b are constants.Example. If we have the boundary conditions at points (0 ,0) & (1,0) we can find the point (1,1)Our equation would look like this

New graph after calculations would have all the boundary\initial conditions plus the new point (1, 1)

Figure 2: HEC-1 time vs distance with calculated.We can use this method to find all New graph after calculations

Figure 3: time vs distance all calculated.After this we can move on to row 2 and all the points needed can be found in this manner.

Part II Check Run1. Develop the run stated in the attached pages. In order to do that, please follow exactly the set of steps explained in the pages. You will need to select a time step based on the Courant criterion, and will need to select stations where the results will be displayed. Check that the results of your run match the results found in the attached pages. (There may be differences in the third digit in the numerical result.)

Table 2: InputsCHANNEL WIDTHB (m) = 5.000

MANNING COEFFICIENTn = 0.02

SLOPES OF SIDES m (-) = 0

DEPTH FOR UNIFORM FLOW hn (m) = 1.2

SLOPE OF THE CHANNEL BOTTOM Jf = 0.00100

TOTAL CHANNEL LENGTHLT (m) = 3000

INITIAL DISCHARGE Q0 (M3/S)= 8.249

INITIAL VELOCITY U0 (m/s) = 1.375

CELERITY OF GRAVITY WAVEC0 (m/s) = 3.431

TIME TO PEAK t' (s) = 1200

PEAK DISCHARGE QMAX (m3/s) = 50

TIME OF DECAY t'' (s) = 3600

NUMBER OF STEPS NDIV = 100

LENGTH OF STEPS DX (m) = 30

NUMBER OF NODES NN=NDIV+1 = 101

TIME STEP DT (s) = 0.1

DURATION OF COMPUTATIONS TMAX= 12500

FREQUENCY OF OUTPUT 1200

2. Plot the results in terms of water levels, discharges and velocities in the three stations selected as a function of time (i.e., reproduce all plots of the attached pages).

Figure 4: Compare h vs t

Figure 5: Compare u vs t

Figure 6: Compare Q vs t

Figure 7: Compare h,Q,u vs t

Figure 8: Compare h vs Q

Table 3: OutPut Data from check run

Table 4: Table from attached pages

DiscussionResults follow the run stated in the attached pages. As predicted the numbers may vary in the third digit. This can be seen in tables above.

Part III Sacramento River1) Once you have checked that the numerical results are fine, please develop numerical simulations for the propagation of a flood in the Sacramento River. Please estimate the width of the Sacramento River (you can use 200 m as a start), use a slope of 0.001 and a Manning's n of 0.025. Select a flood with a peak of 2,500 m3/s, which occurs after 18,000 seconds. Please simulate about 30 hours. Utilize a convenient time step based on what the program allows you to use. Objective. Model the Sacramento River with the same program.Table 5: Original Data set

Research:The given variables were B, Jf, n, t,QMAX. Side slope (m) of the Sacramento River was found to be 2/3 from http://www.water.ca.gov/levees/links/docs/Appendix-E.pdf. Length of the Sacramento river is over 700,000 meters long. Because of the large length, the run will be with only a section of the river. The LT value will be 100,000. The uniform depth of the river was found to be 2.43 meters from http://www.dbw.ca.gov/Pubs/Sacriver/SactoRiver.pdf. Decay time (t) should be 3x longer than the time to peak (t) so 54000 was chosen. DT, TMAX and Frequency were acquired by trial and error. Computational time (TMAX) was large to be conservative and not cut off the model before it finishes. The 300,000 was necessary for the second run. The TMAX was left at 300,000 for all 4 runs because this makes it easier to compare the end results. Frequency as well needed to be large as to not have an overwhelming amount of data points. Stations will be referred to as 1, 2 and 3 out of convenience.

2) Plot all results in terms of water levels, discharges and velocities in the three stations selected as a function of time, as in 4).

Figure 9: Original Data flow vs time

Figure 10: Original Data: Depth vs Time

Figure 11: Original Data: Velocity vs Time

Figure 12: Original Data h,u,Q vs t @station 1

Figure 13: Original Data depth vs flow @ station one

3) Perform other runs (three more) altering parameters of interest (for instance Manning's n, the time to the peak, etc.) and discuss your results.

Objective: Three more runs were to be obtained by changing one variable.

Figure 14: Flow vs Time @n=.1

Figure 15: Depth vs Time @n=.1

Figure 16: Velocity vs Time @n=.1

Figure 17: @ n=.1 depth vs flow @ station 1

Figure 18: @n=.1 h,u,Q vs t @ station 1

Figure 19: Flow vs Time @t=9000

Figure 20: Depth vs Time @t=9000

Figure 21: Velocity vs Time @t=9000

Figure 22: @ t=9000 h,u,Q vs t @ station 1

Figure 23: @t=9000 depth vs flow at station 1

Figure 24: Flow vs Time @m=0

Figure 25: Depth vs Time @m=0

Figure 26: Velocity vs Time @m=0

Figure 27: at m=0 h,u,Q vs t at station 1

Figure 28: at m=0 depth vs flow at station 1Discussion of Results:

Table 6: Tabulated results

Table 7: Peak and initial values and respective times

Final Discussion:Changing manning coefficient (n) had the most significant effect of the flood model of the Sacramento River. If we make sure that the river is clear of weeds, bushes, large rocks or other objects that impede flow then the rise in the river will decrease. As seen in the tabulated results the depth of the river greatly increased when n increased. We can expect the opposite to happen if n is decreased. I would recommend that no changes be made with regards to slope because it has very little effect on the behavior of the river. Rising time only made a significant difference when looking at times to peak. Another observation was that further down the river it had less influence. Any change in time to peak will only change when the peak occurs later down the river and will not affect the peak values greatly.

Part IV Flow Pro1) Install the code FLOW-PRO, also provided in the course. 2) Run the code with the following data:

Table 8: Input Data for flow pro

Table 9: OutPut Data for Flow ProDistanceDepthEnergyAreaVelocityTop WidthMomentum Distance Depth

03.53.567701.14320131.824145.3370.143

145.3373.3573.42967.1411.19220122.418145.6820.143

291.0193.2143.29364.2821.24520113.457146.1080.143

437.1273.0713.15861.4231.30220104.944146.6370.143

583.7642.9283.02358.5641.3662096.887147.3050.143

731.0692.7852.8955.7041.4362089.291148.1660.143

879.2352.6422.75952.8451.5142082.166149.2960.143

1028.5312.4992.6349.9861.62075.522150.8230.143

1179.3542.3562.50347.1271.6982069.373152.9510.143

1332.3052.2132.3844.2681.8072063.735156.0410.143

1488.3462.072.26141.4091.9322058.629160.7960.143

1649.1421.9272.14738.552.0752054.082168.7460.142

1817.8881.7852.04135.6912.2412050.132182.1120.142

20001.6431.94532.862.4352046.857

3) Please plot the backwater curve.

Figure 29: Depth and Velocity vs time4) What method does the code apply?Discussion:Flow Pro uses the center method for the backwater curve equation

The cheating method as described in class is used to make this equation explicit. Flow pro makes change in y (depth) fixed/constant. This means there is only one variable x. If you look at the table above you can see that depth varies by .143 meters every time.