# numerical analysis of gradually varied flow

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Guidelines on how to develop a MATLAB code for numerical analysis of Gradually Varied Flow

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Numerical analysis of gradually varied flow profiles

Numerical analysis of gradually varied flow profiles Group Members: Asritha Chillara (CE13B1006) Daitha Sai Charan (CE13B1007) Deepesh Sharma (CE13B1008) Dinesh Meena (CE13B1009) Saurabh Dongre (CE13B10010)

Problem Statement INTRODUCTIONLiterature workImportance of the GVF Profile computation:To understand how the construction of a weir/dam affects the flow properties in the channel.To check if the construction of a weir/dam causes inundation of land, andEstimation of the flood zone.The computational procedures available for the GVF profiles are broadly categorised as:Direct integration Numerical method Graphical method (no longer in use)

Literature workAs the differential equation of water surface written for Gradually Varied Flows (GVF) cannot be integrated directly, various numerical methods are used.2Bresse (1860) Direct integration for wide rectangular channels using the Chezys equation.Masoni (1900) Approximate direct integration method for rectangular channels.Bakhmeteff (1932) Direct integration method for all cross-sections.Chow (1955) Equation for direct integration.Kumar (1978) Direct integration for rectangular and triangular cross-sections.Patil (2001) Improved Chows method.Venutelli (2004) Direct integration for rectangular channels using Mannings equation

Literature workDirect Step Method: This is used for channels with uniform (prismatic) cross-section. For accurate results, the number of reaches to be considered should be large.Standard Step method: This method is mainly used for natural cross-sections, i.e. non-prismatic cross sections. Differential quadrature method (DQM): This method proposes solution for the equations of any system obtained in differential form by including the present boundary-initial conditions into the equation. In DQM, the partial derivative of a function with respect to a variable at a discrete point is approximated as a weighted linear sum of the function values at all discrete points in the region of that variable.3

Methodology Methodology Methodology: Methodology:Methodology: Computational ProcedureMatlab (R2015a) is used to do the computations and develop a GUI(Graphical User Interface) code for front end application.

Input data is entered by the user in the GUI interface in the appropriate places.

Initially, for the given inputs the value of Critical depth and Normal depth is calculated.Codes were developed individually(Direct Step Method and Runge Kutta method) for all the 5 profiles separately.

Finally the GUI was developed and all the codes were incorporated in the GUI code with some modifications.

Depending upon the data entered and the values of Yo and Yc, the code for the particular profile is selected for the computations.

Computational Procedure

Type of ProfileDirect Step MethodRunge Kutta MethodMildType-1(Backwater)The value of Y is increased from Yo. The condition is placed on x, such that when Sf becomes almost horizontal the code stops iteration.

The value of x is fixed suitably.The condition is placed on Y, such that when Sf becomes almost horizontal the code stops iteration.

Type-2(Drawdown)The difference between Yo and Yc is divided into certain number of steps.The value of x is calculated for each Y.

The value of x is fixed suitably.The value Y is calculated for each x, until Y reaches Yc. Computational Procedure

Type of ProfileDirect Step MethodRunge Kutta MethodSTEEPType-1(Backwater)The value of Y is increased from Yc.The condition is placed on x, such that when Sf becomes almost horizontal the code stops iteration.

The value of x is fixed suitably.The condition is placed on Y, such that when Sf becomes almost horizontal the code stops iteration.

Type-2(Drawdown)The difference between Yc and Yo is divided into certain number of steps.The value of x is calculated for each Y.

The value of x is fixed suitably.The value Y is calculated for each x, until Y reaches Yo.

Computational Procedure

Type of ProfileDirect Step MethodRunge Kutta MethodCRITICALType-1(Backwater)The value of Y is increased from Yc.The condition is placed on x, such that when Sf becomes almost horizontal the code stops iteration.

The value of x is fixed suitably.The condition is placed on Y, such that when Sf becomes almost horizontal the code stops iteration.

HORIZONTALType-2(Drawdown)

The value of Y is decreased until Yc is reached.The value of x is calculated for each Y.

The value of x is fixed suitably.The value Y is calculated for each x, until Y reaches Yc.

The value of Y is decreased until Yc is reached.The value of x is calculated for each Y.

The value of x is fixed suitably.The value Y is calculated for each x, until Y reaches Yc.

Computational Procedure

The GUI user interface is user friendly. One does not need to learn a language or type commands in order to run the application.

The code fails to return results for values of n > 0.1 .

The values of n are mostly less than 0.1, so it can be used for almost all surfaces except in certain cases.1

At times adjustment to the code may be necessary in few particular cases. RESULTS MILD SLOPE:

STEEP SLOPE:

HORIZONTAL SLOPE:

DISCUSSIONThe profiles obtained from the Direct Step method and Runge-Kutta method are approximately same.Mild Slope:The drawdown profile corresponds to the M-2 category and it is evident that the profile starts asymptotically from the normal depth y0 and ends perpendicular to the critical depth yc. The backwater profile of the Type 1 category or the M-1 profile starts asymptotically from the normal depth and continues till the slope approaches the bed slope S0.Steep Slope: The S-1 profiles are generally characterized by a hydraulic jump from the normal depth (Supercritical flow regime) to the sub critical regime and then extends till it reaches the bed slope value. Generally, S-2 profiles are of short length.Horizontal Slope: So, the only profile being considered is the H-2 profile (Drawdown). In this case, the profile ends perpendicular to the critical depth.

DISCUSSIONAdverse Slope: The uniform flow condition is not possible for the adverse slope and hence the normal depth doesnt exist. The A-2 profile is similar to the H-2 profile and exists for a very short length as can be observed from the graph.Critical Slope: The normal depth and the critical depth coincide for the critical slope. Even a small change in decimal values of Yo and Yc will develop a condition for Mild or Steep case, therefore to obtain a condition for critical slope is very difficult.CONCLUSIONFrom the graphs, it can be observed that the profiles obtained from both the methods are identical.In the direct Step method we calculate x for predefined increments in the value of y. While in the Runge-Kutta method, we fix the value of x and obtain the value of y for each x. Depending on the requirement, whether we want a more precise y or a more precise x, we can employ one of the methods. We control the input of x in case of Runge-Kutta method, so, it is more suitable if we want a precise value of x while in the direct step method, the control parameter is y and hence can be employed for precise y computations.To be more precise overall, one can use Direct Step Method to make computations for the portion where the value of slope changes at a higher rate and use Runge Kutta Method where the rate of change of slope is lower for the same profile. REFERENCES:

Flow in Open Channels by K. Subramanya

1.http://www.fsl.orst.edu/geowater/FX3/help/8_Hydraulic_Reference/Mannings_n_Tables.htm

2.http://people.eng.unimelb.edu.au/imarusic/proceedings/4/Apelt.pdf