Open Channel Flow Over a Permeable River Bed
Chapter One
AIM AND OBJECTIVE OF THE STUDY
River as an example of open channel flow is very important; they provide us with food; sometimes source of drinking water; source of energy; transportation etc.
The main purpose of this study is to provide further insight into open channel flow over permeable beds and to enhance the understanding of the effect of bed permeability on the mean and instantaneous flow, as well as other flow parameters such as viscosity, pressure gradient and height of the channel.
CHAPTER TWO
LITERATURE REVIEW
As a result of importance of fluid flow in open channels a lot of work has been done on open channel flows but not much has been done on open channel flow through porous media. We also have much literature in porous media.
This section of our report aims at discussing the various research work done in recent times with regard to open channel flow through porous media. According to Rajput (1998), fluid may be defined as a substance which is capable of flowing, or a substance which deforms continuously when subjected to external shearing force. From the above definition it is clear that fluid has the characteristics of conforming to the shape of the containing vessel. Hence fluid has no definite shape and can undergo a deformation when a small amount of shear force is exerted on it. When fluid flows in a channel it takes the shape of the channel. An open channel is defined as a passage in which liquid flows with its upper surface exposed to the atmosphere, Rajput (1998). In order words open channel flows are characterized by a free surface which is exposed to the atmosphere. The pressure on this boundary thus remains approximately constant irrespective of any changes in the water depth and the flow velocity. In open channels the flow is due to gravity; thus the flow conditions are greatly influenced by the slope of the channel.
These free-surface flows occur commonly in engineering practice, and include both natural channels and artificial channels. The natural channels include large scale geophysical flows such as rivers, streams, estuaries, etc which have irregular sections of varying shapes, developed in a natural way, while artificial channels such as irrigational channels, drainage channels and sewers have cross-sections with regular geometrical shapes (which usually remain same throughout the length of the channel).
In general, channel flows may be steady or unsteady; uniform or varied; laminar or turbulent. Rajput(1998). The flow in an open channel may be laminar or turbulent depending upon the value of Reynolds number defined as
Re= rVR
m where Re is Reynolds number, is fluid density, is average velocity of flow in the channel, R is the hydraulic radius (defined as the ratio of the area of flow to the wetted perimeter) and is the dynamic viscosity of the fluid.
When Re < 500 flow is laminar Re > 2000 flow is turbulent
500 < Re < 2000 flow is transitional.
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The velocity of stream flow is controlled primarily by gravity and friction. Gravity pulls the water from high elevations at the river’s source to low c elevations at the river’s mouth. The steeper the elevation gradient, the greater the velocity. Friction tends to hold the water in place. Thus friction occurs in two places. First there is frictional drag between one layer of water with those above and below it. Second, there is frictional drag between the water and the bed and sides of the channel through which the water flows. The interaction of frictional and gravitational forces creates a boundary layer, where the force induced by gravity is retarded by forces induced by friction. The change in velocity between the bed and the top of the boundary layer is referred to as a velocity gradient.
The manner by which water flows in this boundary layer changes from lower velocities up stream to higher velocities downstream. Initially where water moves slowly, only the lower most layers of the water near the streambed show a velocity gradient. Viscous forces between water layers and between the lowest water layer and streambed cause a decrease in velocity of these layers towards the streambed. Each layer moves in parallel horizontal layers. This is called laminar flow. The boundary layer undergoing laminar flow at first thickens down flow. But as the velocity increases water layers become unstable [i.e. a vertical component becomes part of the direction of flow]. First the layers of water become more wave-like, rising and falling in the water column. Finally, the vertical component of flow result in the formation of eddies.
CHAPTER THREE
THEORY OF CHANNEL FLOW
FLUID
A fluid is a substance which deforms continuously when subjected to external shearing force. In particular, we are interested in water flow over an open channel.
CHARACTERISTICS OF A FLUID
- It has nodefinite shape of its own, but conforms to the shape of the containing
- A small amount of shear force exerted on a liquid/fluid will cause it toundergo a deformation which is continuous as long as the force continues to be
PROPERTIES OF FLUID
The properties of water are of much importance because the subject of hydraulics is mainly concerned with it.
Some important properties of water are: density, specific gravity, viscosity, vapour pressure, cohesion, adhesion, surface tension, capillarity, compressibility etc. fluid flow is affected by myriad of factors or variables such as:
- Velocity of flow
- Flowrate
- Density of fluid
- Temperature of the fluid
- Compressibility of the fluid
- Time of flow
- Pressure gradient
- Viscosity/kinematics viscosity
- Shear stress
- Volume of fluid
- Mass of fluid
- Gravity
- Inertia force
- Cohesion
- Adhesion
- Surface tension
- Capillarity
Velocity of flow:
The flow velocity u of a fluid is a vector field
u = u (x, t)
which gives the velocity of an element of fluid at a position x and time t . The flow velocity of a fluid effectively describes everything about the motion of a fluid. For the present work, we suppose that the velocity profile is the same at all distances down the channel; that is all x. This velocity profile
u ( y)
also known as distribution of velocity is a function of y only. Thus u (x, y) » u ( y) as will be seen in chapter four of this work. This means that the velocity is dependent on depth and not on distance from the source.
CHAPTER FOUR
MODELEQUATION ON OPEN CHANNEL FLOW OVER A PERMEABLE BED
Fluid motions in porous media are governed by the same fundamental laws that govern their motion in solid medium. These laws are based on the conservation of mass, momentum and energy. From a practical stand point, these basic laws are not applied directly to the problems of flows in porous media; instead, a semi empirical approach is used where Darcy’s law is employed instead of the momentum equation.
CHAPTER FIVE
ANALYSIS AND DISCUSSION OF RESULTS
The following data were generated from the analytical solutions presented here in chapter four
= cosh
+ sinh
– Gk (solution1).
mf
In table 5.1 below, velocity () was computed with varied values of permeability (k), while height of channel (y), porosity (f ), pressure
gradient (G) and viscosity ( ) were all kept constant. We also noted that if velocity is in cm/s, viscosity in centipoises and pressure gradient in atm/cm, then the unit of k is Darcy.
CONCLUSIONS
Open channel flow over a permeable riverbed is studied by analytical solution within the fully developed region. Solution (1) is for channel flow over the permeable bed.
Solution (2) is for open channel flow, while solution (3) is obtained by substituting the value of velocity vector in porous media, into the solution for open channel flow. It is observed that by decreasing the permeability in Darcy, due to increase in resistance against the flow in the porous region, the difference between the maximum of velocity profile in Porous region and open channel increases. This is as was predicted physically. It is also shown that there is a penetration of flow from porous media to open channel due to resistance against flow in the porous media and this penetration increase by decreasing the permeability. This penetration adjusts the mass flow in open and porous regions in order to have identical pressure difference in X-direction for both the open channel and the porous media. In the other hand, when permeability is high, there is a penetration from the open channel to porous media due to decrease in resistance against flow. Hence high permeability accounts for rivers meandering up and down. This becomes obvious in desert areas, where the river beds are sandy and drought is a problem.
As the level of a desert stream drops, places where it was at the bottom of its ‘wave’ dry up, and all you can see is sand in the river bed. But if you walk downstream a way, you will come to where it was at the top of its ‘wave’ and you can still see water in that section. Water is flowing in both places, of course. Where you saw it was dry, it is now flowing only underground.
RECOMMENDATIONS
When the permeability of a river is high, flooding in that area is not catastrophic. Rainfall and snow melt will cause temporary and gradual flooding.
Create ‘hard’ or impermeable, surfaces like roads, parking lots, sidewalks, patios, and roofs. This is called urban runoff. Instead of soaking into the ground and recharging the aquifer, rainfall begins to simply rush over these hard surfaces, running directly into streams.
The result is that aquifer levels go down, lowering the level of the streams, while run off during rainfall increases creating sudden, unseasonal flooding.
Wet lands like swamps and marshes should not be filled with dirt. Without wetlands, rain water enters the river directly and causes catastrophic flooding.
Dams or weirs may be built to control the flow, store water or extracts energy.
The management of river is a continuous activity as rivers tend to ‘undo’ the modifications made by people.
REEFERENCES
- Batchelor, G.K (1967). An Introduction to fluid Dynamics. Cambridge University Press New York Pp. 223-224.
- Calvert, J.B. (2003). Open channel flow http://mysite.du..edu/- etuttle/tech/opench.htm.
- How a river flows – http://chamisa.freeshell.org//htm.From Wikipedia, the free encyclopaedia. Accessed 24th September, 2002.
- Murray R. Spiege (1983). Advanced Mathematics for Engineers and Scientists. Mc Graw-Hill Book Company, Singapore Pp. 74-83.
- Nabovati A., Sousa, A.C.M. (2007). Fluid simulation in Random Porous Media at Pore level using the Lattice Boltzmann Method. Journal of Engineering Science and Technology pp. 226-237.
- Obikwelu, D.O. N (2003). Differential Equations and Application.
- AMDI-Nel publishers Ogidi, Nigeria.pp.60-61
- Parvazinia, M. Nassehi, V.and Wakeman, R.J. (2006) Multiscale finite element modelling of laminar steady flow through highly permeable media. Chemical Engineering Science, pp583-596
- Radiom, M. Khakbaz, M. and Kermani, M.J (2007). Numerical and Analytical Calculation of Air flow through an open channel linked to a porous media. Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran, 15875-4413.
- Rajput, R.K. (1998). Fluid Mechanics and Hydraulic Machines. Schand and Company, Ram Nagar, New Dehi.pp.3-29,867-871.
