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

Given water flows in a trapezoidal channel of known flow rate(Q), bed slope(S0), bed width(B), side slopes(m) and manning’s roughness(n).

We are expected to calculate Normal depth() and Critical depth() and specify the type of GVF profile obtained in the channel.

Plotting the GVF profiles of TYPE-1 and TYPE-2 using Direct step method and Runde-kutta method.

Typical examples:-

a) Backwater flow is observed before a dam and weir.

b) Drawdown flow is observed when there is a sudden drop in the channel.

INTRODUCTION➢ A steady non uniform flow in a prismatic channel with gradual

changes in its water surfaces elevation is termed as gradually varied flow(GVF).

Basic equation of GVF is = Assumptions:1)The pressure distribution at any section is assumed to be hydrostatic.2)For calculating friction slope Manning’s equation is used.

Literature work

Importance 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, and Estimation 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.2

Bresse (1860) – Direct integration for wide rectangular channels using the Chezy’s 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 Chow’s method. Venutelli (2004) – Direct integration for rectangular channels using

Manning’s equation

Literature work

Direct 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

Computation of normal depth and critical depth: Critical depth(): , equation for any open channel. , equation for trapezoidal channel. Normal depth():Q= , equation for solving for any open channel. , equation for trapezoidal channel.

Methodology

Classification of channels: Mild Channel (M- profile)             Yo>Yc Steep channel (S- profile)            Yc>Yo Critical channel (C- profile)         Yo=Yc Horizontal channel (H- profile)     So=0; 0=infinity Adverse channel (A- profile)        So=negative; 0= Does not exist

Methodology:

Computation of GVF: The GVF equation is given by We are using 2 methods to compute the GVF profile for a trapezoidal

channel.1. Direct Step Method2. Runga- Kutta method

Methodology: Direct step method: Equation for direct step method We will get E(energy) from equation Y We will get friction slope(Sf) from Q = In case of backwater curve we know that depth of flow(y) increase with

increasing direction of flow(x). In case of drawdown curve we know that depth of flow(y) decrease with

increasing direction of flow(x).

Methodology:

Runge-kutta method: This is a advanced numerical method to solve GVF profiles. The basic equation of GVF can be expressed as =F(y) We will get where is depth at )

Computational Procedure Matlab (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 Profile Direct Step Method Runge Kutta Method

Mild

Type-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 ProcedureType of Profile Direct Step Method Runge Kutta Method

STEEP

Type-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 ProcedureType of Profile Direct Step Method Runge Kutta Method

CRITICAL Type-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.

HORIZONTAL Type-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.

ADVERSE Type-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.

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:

ADVERSE SLOPE:

DISCUSSION

The 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.

DISCUSSION

Adverse Slope: The uniform flow condition is not possible for the adverse slope and hence the normal depth doesn’t 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.

CONCLUSION

From the graphs, it can be observed that the profiles obtained from both the methods are identical.

In the direct Step method we calculate Δx for predefined increments in the value of y. While in the Runge-Kutta method, we fix the value of Δx and obtain the value of y for each Δx. 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 Δx 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

3.http://www.academicjournals.org/article/article1380711438_Kaya.pdf

Thank You!