cfd simulation of hydraulic jump in triangular channel seminar report

7/21/2019 CFD Simulation of Hydraulic Jump in Triangular Channel Seminar Report

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1. Introduction: CFD

Computational Fluid Dynamics, abbreviated as CFD, is the science of

predicting fluid flow, heat transfer, mass transfer, chemical reactions and related

phenomena by solving the mathematical equations which govern these processes.

For solving the problems of fluid mechanics, mainly Navier-Stoke's equations aresolved. If there is turbulence in the flow it becomes necessary to use turbulence

models. These equations are solved numerically over a discretized domain to

obtain a solution. These solutions may not be accurate due to errors induced due to

use of numerical methods. These errors can be reduced to certain extent by various

measures. It is necessary to validate the simulated results with experimental and

analytical results.

CFD has numerous applications in aerodynamics, hydraulics, thermodynamics,

animation, gaming, nuclear science, HVAC, automobile industry, manufacturingindustry etc.

2. Introduction: Hydraulic Jump

2.1 Definition:

Hydraulic Jump forms when supercritical flow gets converted into

subcritical flow. A dimensionless number known as Froude Number defines

whether flow is subcritical, critical or supercritical. A hydraulic jump will form in

a channel if Froude number F1, Upstream Depth y

1and downstream depth y

2

satisfy the equation-

This equation is derived by applying momentum principal to smooth horizontal

rectangular channels.

2.2 Hydraulic Jump in Triangular channel:

Hydraulic jump can also form in a triangular section channel. The hydraulic

jump in triangular open channel has not received much attention. Thus, relatively

scarce literature on hydraulic jumps in triangular channels is available to date.

Hager and Wanoschek declared that, regarding the sequent ratio and the relative

energy dissipation, trapezoidal and particularly triangular channels are much more

effective than rectangular channels, provided the inflow Froude number F1 is

fixed.

= 1 2 1 + 8 1


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By solving momentum and energy equations, relations for sequent depths

and energy loss can be derived.

5

=2 +

3 +

= =

+ +

Where, where yc=critical depth (2Q2/gz2)1/5, z =side slope; y1and y2=upstream

and downstream water depths; respectively.

2.3 Tools: Flowscience Flow 3D

Flow 3D is a RANS solver which specializes in solving free surface flows.Simple geometric shapes can be created for the modeling and complex models can

be imported as STL format. Pre-processing, solving and post-processing are done

in the same software environment. Meshing the domain is simple in Flow 3D. It

generates quadrilateral structured mesh for any type of geometry. The geometry is

modeled such that solid part represents obstacles in the flow field. The mesh

encompasses solid as well as liquid part. The solids are treated as no slip or wall

boundary type. The position of such wall is decided by checking how much

volume of a cell is occupied by the solid component. The algorithm used for this is

named as FAVOR (Fractional Area/Volume Obstacle Representation). Multi-block

meshing can be used where more resolution is required.

Single phase as well as multiphase flows can be solved. Flow with free

surface can be solved with Flow 3D with both the methods, but it is claimed to

give accurate results by utilizing volume of fluid method for a single fluid with

sharp interface. It neglects the forces due to rare medium (air) and instead applies a

free surface boundary condition by evaluating density of fluid in each cell. There

are five turbulence models available: the Prandtl mixing length model, the one-

equation, the two-equation k- and RNG models, and a large eddy simulation, LES,

model.


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3. Motivation:

CFD has been a field of interest and simulating multiphase flows is a

challenge in itself. Learning CFD is an insightful experience for understanding

fluid flows. Hydraulic jump is an intriguing phenomenon and had been of interestsince it was introduced. There has been some work done in the simulation of

hydraulic jump but no one has simulated steady hydraulic jumps. This attempt was

made to gain confidence in simulating multiphase flows and hydraulic jump in

particular. After simulating hydraulic jumps in 2D rectangular channel the problem

is taken into new dimension with hydraulic jumps in triangular channel.

4. Objectives:

Simulating hydraulic jump in 3D horizontal triangular channel

Validating the results with analytical solutions

Comparison of results of rectangular and triangular channels.

5. Literature Review:

V. T. Chow has discussed in detail about hydraulic jumps in horizontal and

sloping rectangular channel in his book- Open Channel Hydraulics. Some attempts

have been made to simulate hydraulic jumps. M. Javan & A. Eghbalzadeh (2011)

have simulated submerged hydraulic jump by solving RANS equation. A. M.

Gharangik and M. H. Chaudhry (1991) have simulated hydraulic jump by solving

Boussinesq equation. F. Rostami, M. Shahrokhi, Md. Saod, S. R. S. Yazdi (2012)

have simulated undular hydraulic jump using Flow 3D.

H. Chanson has done a review of current knowledge in hydraulic jumps in

his 2008 paper. W. H. Hager has discussed about hydraulic jump in U-shaped

channel (1989) and Impact Hydraulic Jump (1994). W. H. Hager and R. Gargano

(2002) have done experimental studies on undular hydraulic jumps in circular

conduit. S. A. Ead & H. K. Ghamry have discussed about hydraulic jumps incircular conduits in their 2002 paper. I. M. H. Rashwan has given analytical

solution to problem of hydraulic jump in horizontal triangular channel.


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6. Theory:

6.1 Hydraulic Jump:

Hydraulic Jump forms when supercritical flow gets converted into

subcritical flow. A dimensionless number known as Froude Number defines

whether flow is subcritical, critical or supercritical. A hydraulic jump will form in

a channel if Froude number , Upstream Depth and downstream depth satisfy the equation-

(1)

This equation is derived by applying momentum principal to smooth horizontal

rectangular channels. The momentum equation for hydraulic jump is given as

= (2)In Equation (2) the horizontal component of weight of water is neglected as the

channel is horizontal. Also the frictional force is considered to be insignificant and

it is neglected (i.e. channel bed is considered as smooth). and aremomentum at sections before jump and after jump respectively. and arehydrostatic pressures at sections 1 and 2 respectively. Substituting the respective

values in the above equation and using the relation = , equation (1) isobtained.

Equation (1) is verified by the experimental results. This equation is a basis

for validation of the results of this study.

Hydraulic jump only forms when there is some obstruction to the flow. In

many cases this obstruction is the weir constructed on downstream of a dam to

maintain tail water. To get a steady standing jump it is important that should beequal to tail water depth. If tail water depth is more than then submergedhydraulic jump is formed. If the tail water depth is more than the jump getswashed downstream up to the weir. Stilling basins are also designed to get astanding jump.

= 1 2 1 + 8 1


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6.2 Types of Hydraulic Jump:

a) Undular Jump: This type of jump occurs when the upstream Froude number is

between 1 and 1.7. No rollers are formed instead wavy surface is seen. Sequent

depth ratio is also very small

b) Weak Jump: for =1.7 to 2.5 a series of small rollers develop on the surface ofthe jump but downstream water surface remains smooth. energy loss is low.

Velocity throughout is uniform.

c) Oscillating Jump: For = 2.5 to 4.5 there is an oscillating jet entering the jumpbottom to the surface and back again with no periodicity. Each oscillation produces

a large wave of irregular period which, very commonly in canals can travel for

very long doing great damage to earth banks.

d) Steady Jump: For = 4.5 to 9 the jump is well balanced and performance is atits best. The energy dissipation ranges from 45 to 70%.

e) Strong Jump: For = 9 and higher rough jump action is generated. Waves aregenerated downstream. Energy dissipation may reach 85%.

In this study steady hydraulic jumps are chosen to simulate. The reason

being, these jumps are independent of tail water depth and remain steady. There

are no undulations like formation of waves. It is found that some turbulencemodels do not predict the formation of waves.

6.3 Basic Characteristics of Hydraulic Jump:

4.3.1 Energy Loss: The loss of energy in the jump is equal to the difference in

specific energies before and after the jump. It can be shown that the loss is

= = 4 (for rectangular channel) (3) = = 6 + + (for triangular channel) (4)

The ratio is known as the relative loss.4.3.2 Efficiency: The ratio of specific energy after the jump to that before the jump

is called as efficiency of the jump.


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= (

) 4() (5)

This equation indicates that the efficiency of the hydraulic jump is a dimensionless

function of Froude number of the approaching flow.

4.3.3 Height of Jump: The difference between the depths after and before the jump

is the height of jump. Expressing the terms as a ratio with respect to the initial

specific energy,

= (6)

is called as relative height and it can be shown that it is also a dimensionlessfunction of .Results of the simulation are to be compared with the characteristic curves forvalidation.

6.4 Basics of CFD:

The Navier-Stokes equations are the basic governing equations for a viscous, heat

conducting fluid. It is a vector equation obtained by applying Newton's Law of

Motion to a fluid element and is also called the momentum equation. It is

supplemented by the mass conservation equation, also called continuity equationand the energy equation. Usually, the term Navier-Stokes equations are used to

refer to all of these equations.

These are the simplified equations for Incompressible flow with Constant

Viscosity. Most of the fluids can be treated as incompressible without losing

accuracy in the results. (Mach number M< 0.2-0.3 then flow is incompressible)

(Mach number =velocity of body with respect to medium/velocity of sound in the

medium)


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This equation is impossible to solve analytically except for some basic cases where

we have simple boundaries and initial or boundary conditions. We are interested to

find out values of u, v, w and p which depend on x, y, z, and t, Together with

continuity equation we have to solve this when the flow is incompressible. It is a

nonlinear partial differential equation and it

is a very complicated to understand and solve. The equation is impossible to solve

analytically except for some basic conditions which may involve simple

boundaries and initial or boundary conditions.

Navier stokes Equations can be solved by using Numerical methods. The

Continuous domain is replaced by a discrete domain using a grid. In continuous

domain every flow variable is defined at every point in the domain. In discrete

domain each flow variable is defined only at grid points. (Grid points are the points

where grid lines cross). For every grid point the discrete equations are derived and

solved. There is always some error in the solution by Numerical methods. This

error can be minimized by increasing grid points to certain extent. Number of

iterations also makes the solution more accurate. Though the solution can never be

exact, it can be fairly accurate. Various numerical methods exist to solve the

differential equations. Fluent uses Finite Volume method for solving Navier-

Stokes equations.

+

+

+ =

+

+ +

+

+

+ =

+

+ +

+ + + = + + + ,,are velocities in ,, direction respectively

is density, is pressure, is viscosity, is the body force


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6.5 Turbulence Modeling:

Turbulence is the usual state of motion of fluids except at low Reynolds numbers.

Understanding its physics is essential in a wide range of scientific disciplines,

including engineering, progress in renewable energy, aerodynamics, astrophysics,geology or weather prediction. The governing equations for both laminar and

turbulent flows are the same (the Navier-Stokes equations), but the complexity of

turbulent flows is very high so huge computational resources are needed to proceed

to their direct solution without any model. This approach is known as Direct

Numerical Simulation (DNS). DNS is important to provide data for the

development and validation of turbulence models (both Large Eddy Simulation

(LES) and Reynolds Averaged Navier-Stokes (RANS) models) and also to be

directly applied to certain types of flows.

In this study RNG model is used to model the turbulence.


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7. Modeling:

7.1 Geometry, Boundary Conditions and Initial Condition:

Fig. 1: Geometry

The geometry is generated using Autodesk Inventor. Geometry of the

domain is a prism shape with a rectangular step on the end. Length is fixed such

that it will accommodate the jump length and some length is left out for

measurement of . The length for various cases is 7m. Height and width of thechannel is fixed and it is 0.7m. Only half of the channel is modeled to take

advantage of the symmetrical condition.

The geometry consists of a step at the far downstream end. This provides

necessary depth for the jump to occur by acting as an obstruction to the flow. It is

found that the flow adjusts to the height on its own independent of the height ofthe step provided at the outlet. But it should be noted that if the step height is not

sufficient then the jump is seen to get washed downstream near the step.

Boundary conditions are set as follows:

X min: Wall

X max: Symmetry


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Y min: Specified velocity

Y max: Outflow

Z min: Wall

Z max: Symmetry

At Y min boundary the velocity is specified accordingly for different cases

and fluid height is specified as 0.1m

Initial condition is the water level. Water level is set to be at the level of the

step. It is observed that the location of the jump is independent of initial condition.

Gravity is turned on in negative z direction.


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7.2 Meshing:

Fig. 2: Mesh XZ view

A structured uniform mesh is used for the domain. The spacing in y direction is

specified such that there will be 10 nodes occupied by water at the inlet. y is fixed

as 0.005m. x is taken 1.5 times y which comes out to be 0.0075m. this

maintains the cell aspect ratio to 1.5.


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Fig. 3: Mesh XY view

7.3 Solver Setup:

Double precision solver module in Flow 3D is used. Turbulence model used

is RNG. This is a two equation turbulence model. Volume of fluid method is used

to compute free surface location and it gives sharp location of the interface. VOFmethod checks which phase the cell contains. If the cell contains both the phases

this means there is interface present in the cell. Only one fluid with a sharp

interface is selected.

Other solver settings are maintained at default.


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8. Results and Analysis:

8.1 Graphs:

Graphs are plotted between various dimensionless quantities and Froude

number for the theoretically/analytically obtained values. Simulated values are

then overlaid on these plots. The simulated results are matching fairly with the

theoretical/analytical values. Energy is found out by post processing the simulated

results. The values found out are average velocity after the jump and height of free

surface. The velocity is used to find out the energy at section downstream of the

jump. Using these values various plots are prepared. First plot is for sequent depth.

This shows fairly matching values of simulated results with the analytical values.

This validates the software. Further the comparison is made between theoretically

obtained values of Energy and height of jump of rectangular channel to that ofsimulated values of triangular channel.

0

1

2

3

4

5

6

4 5 6 7 8 9 10 11

y2

/y1

F1

Sequent Depth Ratio

y2/y1 th

y2/y1 sim

Linear (y2/y1 th)


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

3 4 5 6 7 8 9 10

E2

/E1

F1

E2/E1

E2/E1 Rect

Analytical

E2/E1 Tri Simulated

Poly. (E2/E1 Rect

Analytical)

Poly. (E2/E1 Tri

Simulated)

0

0.1

0.2

0.3

0.4

0.5

3 4 5 6 7 8 9 10

hj/

E1

F1

hj/E1

hj/E1 Rect Analytical

hj/E1 Tri Simulated

Poly. (hj/E1 Rect

Analytical)

Poly. (hj/E1 Tri

Simulated)

0

0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

3 4 5 6 7 8 9 10

E/E1

F1

E/E1

dE/E1 Rect

Analytical

dE/E1 Tri

Simulated

Poly. (dE/E1 Rect

Analytical)

Poly. (dE/E1 Tri

Simulated)


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8.2 Visualizations:

Post processing is done for the case having Froude number 7 for displaying

vectors and contours of various quantities. Figure 4 shows the volume fraction of

water shaded with pressure contours. Formation of jump and the jump profile canbe seen in the figure.

Fig. 4: Volume fraction of water colored with Pressure Contours

Fig. 4 shows pressure distribution in the region occupied by water. Pressure

on the downstream of the jump is found out to be hydrostatic, but the pressure

distribution in jump region is not hydrostatic.

Fig. 5: Velocity magnitude


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Fig. 6: View of velocity vectors colored by velocity magnitude at different sections


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Fig. 8: Turbulent Dissipation

Fig. 5 shows the contours of velocity magnitude. The velocity of the jet

entering the jump is seen to be reducing significantly in the jump region itself.

Velocity distribution on the downstream of the jump is observed to be low and

fairly uniform. Fig. 6 shows velocity vectors. Spreading of the jet and formation of

jump rollers can be clearly seen. Fig. 7 shows contours of turbulent dissipation. It

can be seen that the turbulence is very high where the jet is entering into the jump.

There is presence of high turbulence in the whole jump region.


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9. Conclusion:

A steady hydraulic jump forms for the Froude number ranging from 4.5 to 9.

In this study steady hydraulic jump is simulated in a triangular channel with

smooth horizontal bed using VOF technique to predict free surface profile. TheRNG turbulence model is used to determine effect of turbulence in the flow field.

For validation of the numerical modeling the results are compared with

analytical data. Total ten cases are simulated for Froude number ranging from 4.5

to 9. The results showed that the numerical model is able to predict flow field of

hydraulic jump.

Height of jump is less in triangular channel than in rectangular channel.

Energy loss in hydraulic jump in triangular channel is more than that of rectangular

channel.


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10. References:

[1] V. T. Chow, Open-Channel Hydraulics, McGraw Hill, p.393-438

[2] Fox, Pritchard, McDonald, Introduction to Fluid Mechanics, Wiley-India,

2012, p.196[3] Rostami , Shahrokhi, Md Saod, Yazdi, Numerical simulation of undularhydraulic jump on smooth bed using volume of fluid method, Applied

Mathematical Modelling 37 (2013), p.1514-1522

[4] M. Javan, A. Eghbalzadeh, 2D numerical simulation of submerged

hydraulic jumps, Applied Mathematical Modelling 37 (2013), p.66616669

[5] I.M.H. Rashwan, Analytical solution to problems of hydraulic jumpin horizontal triangular channels, Ain Shams Engineering Journal, (2012)

[6] H. Chanson, Current knowledge in hydraulic jumps and related

phenomena, European Journal of Mechanics B/Fluids 28 (2009), p.191210

[7] C. W. Hirt and B. D. Nichols, Volume of Fluid (VOF) Method for the

Dynamics of Free Boundaries, Journal of computational physics 39(1981), p.201-225

[8] A. M. Gharangik and M. H. Chaudhry, Numerical Simulation of HydraulicJump, J. Hydraul. Eng, Vol.117, No. 9, (Sept 1991), p.1195-1211

[9] A. J. Peterka, Hydraulic Design of Stilling Basins and Energy Dissipators,USBR Engineering Monograph no 25 (May 1994)

cfd simulation of hydraulic jump in triangular channel seminar report

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