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The purpose of construction of open channel is transported water from a location to another for irrigation, navigation, flood control to minimize water time of residence, etc... The cross-section of open channels at the top can be open or closed. Channels called closed conduit if the top is closed examples tunnels, and pipes etc., otherwise is called open channels examples, rivers, streams, estuaries etc.. A free-surface flow or open-channel flow referred to the flow in an open channel or in a closed conduit having a free surface. Fig. (1-1)
Fig. 1-1. Free-Surface flow
The close closed conduit is possible to have both free-surface flow and pressurized flow at different times. (Fig. 1-2), it is also possible to have these flows at a given time in different reaches of a conduit, For example, the flow in a storm sewer section (Fig. 1-3)
Fig. 1-2. Pipe or pressurized flow
Fig. 1-3. Combined free-surface and pressurized flow
Structures such as bends, obstructions (bridge piers), abutments, junctions of tributary with main channels, and transitions such as (contraction, and expansion) are causes flow to choke and form jumps. These hydraulic conditions generally necessitate higher walls, bridges and other costly containment structures. The bank erosion, damaged equipment, increased operating expense, and reduction of efficiency are caused by poorly designed channel. (Berger et al 1995). This thesis is a study combination on both experimental and numerical modelling of three dimension flow 3D free surface turbulent for open channels with different cross-sections.
2 flow classification
As explained in the following paragraphs free-surface flow may be classified into different types based on different criteria. (Fig. 1-5)
Fig. 1-4. Free-surface and pressurized flows (Courtesy, Professor C. S. Song
2.1 Steady and Unsteady Flows
This classification is based on the time variation of velocity at a specified location. The flow is called steady flow if the velocity at a given point does not change with respect to time the local acceleration, ∂,. However, the flow is called unsteady flow if the velocity at a given location changes with respect to time, . (Fig. 1-6). In case steady flows of dimensions other than one dimension, the time variation of all components of velocity is zero.
Fig. 1-5. Classification of flows
By having coordinates with respect to a moving reference sometimes it's possible to transform unsteady flow into steady flow. This oversimplification is supportive in the derivation of governing equations and in the visualization of flow (Chaudhry, M. H. 2008). A transformation is possible only if the wave shape does not vary as the wave propagates. In a smooth channel the shape of surge wave moving does not change and as a result the propagation of a surge wave unsteady flow may be transformed into steady flow by moving the reference coordinates at the absolute surge velocity. And the surge wave appears to an observer to be stationary; thus the flow may be considered as steady flow.
2.2 Uniform and Nonuniform flows
This classification based on the variation of flow velocity with respect to space at a specified instant of time. The flow is called uniform flow if the flow velocity at a given instant of time does not change within a given length in other words the convective acceleration is zero. In mathematical expression , . And it's called non-uniform flow if the flow velocity at instant of time changes with respect to distance in other words the convective acceleration is not equal to zero. In mathematical expression , .
Also flows may be classified as gradually varied if the flow depth varies at a slow rate with respect to distance or rapidly varied flow if the flow depth varies significantly in a short distance. Note that the steady and unsteady flows are characterized by the variation with respect to time at a given location, whereas uniform or varied flows are characterized by the variation at a given instant of time with respect to distance.
2.3 Laminar and Turbulent Flows
If the flow particles appear to move in an explicit smooth paths and the flow appears to be as a movement of thin layers on top of each other the flow is called laminar flow. And if the flow particles move in irregular paths the flow is called turbulent, which are not fixed with respect to either time or space.
The ratio of viscous to inertial forces determines whether the flow is laminar or turbulent: if the viscous force is dominant the flow is laminar. Otherwise the flow is turbulent. The ratio of viscous and inertial forces is defined as the Reynolds number.
In which = Reynolds number; = mean flow velocity; = a characteristic length (either hydraulic depth or hydraulic radius may be used as the characteristic length in freesurface Flows); and = kinematics' viscosity of the liquid. Hydraulic depth is equal to the flow area divided by the water-surface width, and the hydraulic radius equal to the flow area divided by the wetted parameter. The transition from laminar to turbulent flows occurs for, in which is based on the hydraulic radius as the characteristic length.
Laminar free-surface flows are awfully uncommon in natural applications. A smooth and glassy flow surface may be due to surface velocity being less than that essential to form capillary waves and may not automatically be due to the fact that the flow is laminar. For this purpose care should be taken at the same time as selecting geometrical scales for the hydraulic model studies so that the flow depth on the model is not very small. Because very small depth may turn out laminar flow on the model whereas the prototype flow to be modelled is turbulent. The results of such a model are not dependable.
2.4 Subcritical, Supercritical, and Critical Flows
If the flow velocity equal to velocity of gravity the flow is called critical flow approximately have a small fluctuation. A gravity wave may be produced by a change in the flow depth. If the flow velocity is less than the critical velocity the flow is called subcritical flow, and if the flow velocity is greater than the critical velocity the flow is called supercritical flow. The Froude number, Fr, is equal to the ratio of inertial and gravitational forces and, for a rectangular channel, it is defined as
In which = flow depth. Depending upon the value of , flow is classified as subcritical if < 1; critical if = 1; and supercritical if > 1.
Channels may be natural or artificial. Various names have been used for the
artificial channels: A long channel having mild slope usually excavated in the
ground is called a canal. A channel supported above ground and built of
wood, metal, or concrete is called a flume. A chute is a channel having very
steep bottom slope and almost vertical sides. A tunnel is a channel excavated
through a hill or a mountain. A short channel flowing partly full is referred
to as a culvert.
A channel having the same cross section and bottom slope throughout
is referred to as a prismatic channel, whereas a channel having varying cross
1-5 Velocity Distribution 9
section and/or bottom slope is called a non-prismatic channel. A long channel
may be comprised of several prismatic channels. A cross section taken normal
to the direction of flow (e.g., Section BB in Fig. 1-8) is called a channel section.
The depth of flow, y, at a section is the vertical distance of the lowest point of
the channel section from the free surface. The depth of flow section, d, is the
depth of flow normal to the direction of flow. The stage, Z, is the elevation
or vertical distance of free surface above a specified datum (Fig. 1-8). The
top width, B, is the width of channel section at the free surface. The flow
area, A, is the cross-sectional area of flow normal to the direction of flow. The
wetted perimeter, P is defined as the length of line of intersection of channel
wetted surface with a cross-sectional plane normal to the flow direction. The
hydraulic radius, R, and hydraulic depth, D, are defined as
R = A
D = A
(1 − 3)
Expressions for A, P, D and R for typical channel cross sections are presented
in Table 1-1.
Fig. 1-8. Definition sketch
Numerical modelling of turbulent flows
Representation of mathematical statements of the conservation laws of physics is governed equations.
The mass of fluid is conserved matter cannot be created or destroyed in other words what goes in must come out.
Conservation of momentum is the rate of change of momentum equals the sum of the forces on a fluid particle (Newton's second law).
Conservation of energy is the rate of change of energy is equal the sum of the rate of heat addition to and the rate of work done on a fluid particle (first law of thermodynamics). Energy cannot be created or destroyed.
Mass conservation in three dimensions
For steady flow
Mass flow rate in = Mass flow rate out
Imagine a small cube shaped element within a fluid flow:
In the X-direction, net outflow (i.e. flow out minus flow in)
From the product rule of differentiation
Similarly, net outflow in the y-direction
And net outflow in the z-direction
In case of compressible fluid density is changing with time, the resulting change in mass of fixed volume element will be:
Change in density, volume
Now, as mass cant be created or destroyed:
For incompressible flow, expression for change is become zero:
, , divided through by volume and density
Conservation of momentum
Net force acting on fluid=rate of change of momentum
(Where is the mass flow rate ))
The rate change of the x-momentum entering through the front, side and bottom=
And from the product rule of differentiation,
For steady incompressible flow the above equation can be written:
So, assuming steady, incompressible 3-d flow, rate of change of x-momentum is:
(i.e. sum of forces in x-direction)
We stated that
Net force acting on fluid=rate of change of momentum
The forces acting on fluid particles can be sort as follow.
We can expression these forces mathematically, so for our steady incompressible 3-d case, witting (i.e. the element volume):
Rate of change of momentum in x-direction,
Body force in x-direction, pressure force in x-direction
And viscous force in x-direction,
In the same way we can write a similar expression in the y, and z-directions. If we had carried out the previous derivation in three-dimensions, and had not assumed steady state, we would have arrived at:
Conservation of momentum in x-direction
Conservation of momentum in y-direction
Conservation of momentum in z-direction
These are the Navier-Stockes equations (simplified to the in compressible, constant viscosity case.) they where first derived by Claude-Luis Navier in France in 1822, and independently by Sir George Gabriel Stokes in England in 1845. The equations are 2nd order non-linear partial differential equations so frightening that nobody could solve them (apart from very simple cases) for over 100 years! Luckily, the computer was invented.
Conservation of energy
In computational fluid dynamics (CFD) simulations, its not necessary to consider conservation of energy -we assume isothermal fluid. If we are considering conservation of energy, we can derive a formula in a similar way to the formulae we found for conservation of mass and momentum. We won't go into the derivation here, but we will look at the terms that are involved. Conservation of energy is derived in dependent on (1st law of thermodynamics) that's energy cannot created or destroyed.
Energy added=heat added + work done
The Navier-Stokes equations
So far we have assumed that the fluid density is constant, but in practical flows the mean density may vary and the instantaneous density always exhibits turbulent fluctuations. Bradshaw et al. (1981) state that small density fluctuations do not appear to affect the flow significantly. If rms velocity fluctuations are on the order 5% of the mean speed they show that density fluctuations are unimportant up to Mach numbers around 3 to 5. In free turbulent flows we shall that velocity fluctuations can easily reach value around 20% of the mean velocity. In such circumstances density fluctuations start to affect the turbulence around Mach number of 1. To summarised the results of the current section we quote, without proof, in table 3.1 the density-weighted averaged
Characteristics of simple turbulent flows
Most of the theory of turbulent flow and its modelling was initially developed by careful examination of the turbulence structure of thin shear layers. In such flows large velocity changes are concentrated in thin regions. Expressed more formally, the rates of changes of flow variables in the x-direction of the flow are negligible compared to the rates of change in the cross-stream y-direction . Furthermore, the cross-stream width of the region over which changes take place is always small compared to any length scale L in the flow direction
Given an engineers recognised interest in mean quantities we review data for the mean velocity distribution U=U(y) and the pertinent Reynolds stresses,, and . Local values of the above mentioned quantities can be measured very effectively by means of hote-wire anemometry (Comte-Bellote, 1976). More recently laser Doppler anemometers have been widely used for mean flow and turbulence measurement (Buchhave et al, 1979)
The most common turbulence models are classified
Based on time-averaged Reynolds equations
Zero equation model-mixing length model
Two equation model-k-ε model
Reynolds stress equation model
Algebraic stress model
Large eddy simulation
Based on space-filtered equations
Mixing length model
Assessment of the mixing length model
Easy to implement and cheap in terms of computing resources
Good prediction for thin shear layers: jets, mixing layers, wakes, and boundary layers
Completely incapable of describing flows with separation and recirculation
Only calculates mean flow properties and turbulent shear stress
The k-ε model
In two dimensional thin layers the changes in the flow direction are always so slow that the turbulence can adjust itself to local conditions. If the convection and diffusion of turbulence
Direct Numerical simulation (DNS)
The result of a DNS contain very detailed information about the flow. This can be very useful but, on the one hand, it is far more information than any engineer needs and, on the other, DNS is too expensive to be employed very often Peric (1996).
Large Eddy Simulation (LES)
The turbulent flow contain a wide range of length and time scales. The large scale motions usually more energetic than the small scale ones. The small scales are generally weaker, and then provide little transport of those properties. Large eddy simulations are three dimensional, time dependent and expensive but much less costly than a DNS of the same flow Peric (1996).
Reynolds Averaged Navier-Stockes (RANS) Equations
In case of statically steady flow, each variable can be written as the sum of average value.
Here is the time and the averaging interval.
If the flow is unsteady, time averaging cannot be used and must be replaced by ensemble averaging.
Where is the number of members of the ensemble, which must be large enough to eliminate the effects of the fluctuations?
From equation (2) it follows that. Thus, averaging any linear term in the conservation equations simply gives the identical term for the averaged quantity.
The averaged continuity and momentum equations can, for incompressible flows without body forces, be written in tensor notation in Cartesian coordinates as:
Where the are the mean viscous stress tensor components:
Finally the equation for the mean of a scalar quantity can be written:
……. (7) Peric (1996).
properties can be neglected it's possible to express the influence of turbulence on the mean flow in terms of the mixing length. If convection and diffusion are not negligible-as in the case of recalculating flows-a compact algebraic prescription for the mixing length is no longer feasible. The mixing length model lacks this kind of generality. The way forward is to consider statements regarding to dynamics of turbulent. The k-ε model focuses on the mechanisms that affect the turbulent kinetic energy.
The k-ε model equations
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