
Fluids in Motion Viscosity Continuity
Equation Flow in Open
Channels
Hydrostatics is very simple from the mathematical point of view and the ancient Greeks were familiar with the basic principles. Now that we are dealing with moving liquids, viscosity, turbulence and friction have to be considered but they are far too complex for simple mathematical treatment. It is important that you appreciate that we have a new ball game here. You probably have some idea what friction is; it is the resistance to motion experienced by a liquid flowing over a solid boundary like a pipe wall or sides of a culvert. It therefore must make sense that energy will be needed to overcome friction and keep the liquid moving. Turbulence is just a random motion, like eddies in the air when a large lorry drives past you at speed and energy is required to generate the turbulence, but viscosity is not quite so easy to understand. Viscosity is a measure of internal friction of the fluid, or its resistance to flow and movement. (More formally, viscosity is described as a measure of a liquid’s resistance to shear stress). For example, cold treacle is very stiff and does not flow easily, while water is thin and runny. The difference is that water has a low viscosity and treacle a high viscosity. If the treacle were heated up, it would flow much easier. So we can say that viscosity changes with temperature (which is why car engines use multigrade oils, for example, to cope with large changes in temperature). At 100° C the dynamic viscosity of water falls to 0.284 x 10^{3} kg/ms indicating that the liquid is getting thinner. At 20° C it is 1.005 x 10^{3} kg/ms. Furthermore the density changes from 998.2 kg/m^{3} at 20° C to 958.4 kg/m^{3} at 100° C, showing that it is lighter at higher temps. As a point of interest, why does ice float in water?? Water is the only material increases in density as it cools and then reduces its density when it solidifies or freezes. hence, ice is less denser than water and thus floats in water. Viscosity is the most important single property that affects the behaviour of a fluid. The more viscous the fluid, the thicker it is and the slower it deforms under stress. Whilst it is one of the most important factors controlling the flow and behaviour of a fluid, it does not appear in the equations that we will look at later. However, it is incorporated in one of the dimensionless parameters that will define and classify the type of flow and we call this Reynolds Number. Types of Fluid Flow. There are many different types of fluid flow. One of the first things you have to do when investigating a problem involving moving fluids is to define the type of flow that you are dealing with. Having done that, you will have an idea of which equations can be applied to the problem. The first step is to decide if the flow is Laminar or Turbulent. Laminar flow is usually associated with slow moving, viscous fluids. It is relatively rare in nature, although an example would be the flow of water through an aquifer. Groundwater velocities may be as little as a few metres per year. Turbulent flow is much faster and chaotic, and is the type usually encountered. A good example would be flow in a mountain stream. Between Laminar and turbulent flow is something we call Transistional Flow. Whether the flow is laminar, transitional or turbulent is very important with respect to the flow of liquid through pipelines since the characteristics of the three flow regimes are very different. Osborn Reynolds found that the type of flow is determined from the following equation: Re = rVD/m Where Re = Reynolds Number and has no dimensions r = Density of the liquid (kg/m^{3}) V = Mean or average velocity (m/s) D = The size of conduit (diameter of a pipe) (m) m = Dynamic viscosity Reynolds found from experimentation that as a general rule: Pipes Open Channels Laminar flow Re < 2000 Re < 500 Transistional flow Re = 2000 to 4000 Re = 500 to 2000 Turbulent flow Re > 4000 Re > 500 Steady and Unsteady flow. Another significant way of classifying the flow is to determine whether it is Steady or Unsteady. The key concept here is whether or not the discharge is changing with respect to time. In steady flow, the discharge Q is constant with respect to time. In unsteady flow, the discharge Q is not constant with respect to time. As explained before, turbulent flow is the most common type of flow and is characterised by fluctuations in velocity, so it could never truly be called steady. However, the definition is usually loosely interpreted so that if the mean velocity and discharge are not changing over a period of time, the flow is said to be steady. Minor fluctuations are ignored. Uniform and nonuniform flow The key concept here is whether or not the crosssectional area of flow and mean velocity change from one section to the next along the length of the conduit (pipe or channel) when the discharge is constant. For the flow to be Uniform the area (depth and width) and the mean pipe velocity must be the same at each successive crosssection. An example would be a pipe of constant diameter running full. It follows that Nonuniform flow occurs where the crosssectional area and mean velocity change from section to section, as would be the case with a pipeline of varying diameter. The continuity equation is derived from first principles. It is based on the concept of the conservation of mass between two crosssections of a continuous conduit. It is generally written as: Q = A_{1}V_{1} = A_{2}V_{2} Where A is the cross sectional are of flow and V is the mean velocity. The equation can be applied to as many sections as required. It is used whenever we need to calculate the mean velocity from a known discharge and area or to calculate the discharge from the known velocity and area. It can also tell us what happens to the velocity when the area changes (V_{2 }= V_{1}[A_{1}/A_{2}] ). For example, if the crosssectional area of a pipe that is running full is halved, then the mean velocity of flow must double in order to discharge the same quantity of water through the reduced section. Of course, both sections must have the same discharge (Q_{1 }= Q_{2}) otherwise water would be either disappearing or magically appearing from nowhere within the pipeline. The continuity equation is very simple, but it is one of the three most important equations in hydraulics. You should never forget it!!!! The Energy (or Bernoulli) Equation The energy equation, also known as the Bernoulli equation is another major tool that we can use to analyse a hydrodynamic system. Sometimes this and the continuity equation are needed to solve a particular problem. Energy is defined as the capacity for doing work. Work (done) is defined as a force multiplied by the distance moved in the direction of the force and consequently has the units Nm. Power is the rate of doing work, i.e. the product of a force and the distance moved per second in the direction of the force (Nm/s). There are three ways that something can possess energy. Perhaps the easiest to understand is that a body can have energy as a result of being raised to some height, z. Thus if a car is driven to the top of a hill, it can freewheel down again and do work by virtue of its elevation. This is called the Potential Energy of the body. Potential energy = Mgz Where M is the mass of the body, and g is the acceleration due to gravity. M x g is the weight of the body W. So ...... PE = Wz and if we relate this to unit weight of water, i.e. that of 1 cubic metre. (Because volumetric flow rate is cubic metres per second) It becomes PE = z and the units are metres. Another form of energy is Kinetic Energy. This is the energy possessed by a moving body. Kinetic energy = ^{1}/_{2} MV^{2}^{ } Where V is the velocity of the body. If W = Mg then M = W/g thus: KE = ^{1}/_{2} WV^{2}/g and if we relate this also to unit weight of water, It becomes KE = V^{2}/2g and the units are metres. The third form of energy will be less familiar since it has no direct equivalent in solid mechanics. It is the energy of a fluid when flowing under pressure, so it is referred to as Pressure Energy. If a liquid has a pressure P which acts over an area A then it is capable of exerting a force of P x A. In moving though a distance L, the flow work done is P x A x L or PAL. So, Pressure energy = PAL If we think of this equation as representing the pressure energy of a stream of moving liquid, then A x L represents the volume of the liquid. If this volume has a weight W, and the weight density of the liquid is rg then AL = W/rg. Substituting this in the PE equation, we get, PE = PW/rg and if we relate this again to unit weight of water, It becomes PE = P/rg and the units are metres If we now combine all three forms of energy by adding them together, we get, TE = z + V^{2}/2g + P/rg and the units are metres which is equal to the total energy per unit weight of the fluid. Or per cubic metre of water. This is the energy equation or Bernoulli equation. This equation is frequently used to investigate how the energy varies between two points in a fluid. Usually the centre line of a pipe for example. If we apply Bernoulli to two points on the centreline of a pipe, 1 and 2, we make the assumption that the total energy at points 1 and 2 is the same. (i.e. There is no loss of energy). Although energy may change from one form to another, the following is true, z_{1} + V_{1}^{2}/2g + P_{1}/rg = z_{2} + V_{2}^{2}/2g + P_{2}/rg With a real fluid, there will be a certain amount of energy lost due to friction and so the equation becomes, z_{1} + V_{1}^{2}/2g + P_{1}/rg = z_{2} + V_{2}^{2}/2g + P_{2}/rg + energy losses z is measured in metres, V in metres per second and P in N/m^{2}. However, each term in the Bernoulli equation has overall unit of metres. Thus they are often referred to as heads: elevation head, velocity head and pressure head. All three terms are measured in metres and can be called ‘Heads’. The sum of the three terms is often called the ‘Total Head’ as an alternative to the Total Energy’. Energy Losses is the ‘Head loss due to friction’ and also has the unit metres. The search for a friction loss formula Frictional losses are the most important features in pipe flow and pipeline design. By the 1850’s, designers had produced a number of purely empirical pipe flow equations. The most important of these, normally known as the D’Arcy Equation, relates the frictional head loss as follows: H_{loss} = 4 x f x L x V^{2}^{ } 2 x g x d or H_{loss} = 64 x f x L x Q^{2}^{ } 2 x g x d^{5} x p^{2} The bulk of the terms can be measured accurately but a factor f is included to make the measured head loss equate to the known length and diameter of pipe and the measured flow rate or average velocity. Initially, this equation gave quite accurate results. However, as engineers attempted to use it for pipe shapes, diameters and velocities other than those from which the equation had been developed, it was soon discovered that f was far from a simple constant. Today we know that f not only varies with the pipe diameter and the fluid velocity, but also with the Reynolds number of the flow. Flow in Open Channels Whilst less common than pressure pipes, open channels are still widely used and enter into the work of a variety of civil engineering specialists. The basic difference from a pressure pipe is simply that in an open channel, the fluid’s surface is exposed to atmospheric pressure. Thus the hydraulic grade line coincides with the surface of the fluid, and the crosssectional area of flow decreases and increases as the discharge rate varies. When water enters an open channel, the depth of flow gradually diminishes and then becomes constant throughout the channel, provided that its geometric crosssection and bed slope does not vary. This depth is known as Normal depth. Normal depth thus can only occur where a balance exists between acceleration down the channel and frictional retardation against the flow. In most real life cases, this is in the middle reaches of long straight lengths of channel of reasonably uniform crosssectional area. Various early engineers attempted to relate normal depth to the factors which obviously influenced it: the fluid’s average velocity, the geometry of the channel’s crosssection, the roughness of the bed materials and the slope of the channel’s bed. Chezy (in 1775) was the first to succeed, when he produced the following empirical equation. V = C x Ö(m x i) Where V = Average velocity C = Chezy constant m = Hydraulic radius or mean depth i = Slope of channel bed
m = A/P Where A = Area of flow
P = Wetted perimeter The constant C was included to give a measure of the roughness of the channel’s wall and bed material. However, this constant was soon found to suffer from the limitations that were found with D’Arcy’s f factor. To make matters worse, C was only suitable for the crosssectional geometries used by Chezy.
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