3. Fluid Dynamics. 3.1 Uniform Flow, Steady Flow

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1 3. Flid Dynamics Objectives Introdce concepts necessary to analyse flids in motion Identify differences between Steady/nsteady niform/non-niform compressible/incompressible flow Demonstrate streamlines and stream tbes Introdce the Continity principle throgh conservation of mass and control volmes Derive the Bernolli (energy) eqation Demonstrate practical ses of the Bernolli and continity eqation in the analysis of flow Introdce the momentm eqation for a flid Demonstrate how the momentm eqation and principle of conservation of momentm is sed to predict forces indced by flowing flids This section discsses the analysis of flid in motion - flid dynamics. The motion of flids can be predicted in the same way as the motion of solids are predicted sing the fndamental laws of physics together with the physical properties of the flid. It is not difficlt to envisage a very complex flid flow. Spray behind a car; waves on beaches; hrricanes and tornadoes or any other atmospheric phenomenon are all example of highly complex flid flows which can be analysed with varying degrees of sccess (in some cases hardly at all!). There are many common sitations which are easily analysed. 3. Uniform Flow, Steady Flow It is possible - and sefl - to classify the type of flow which is being examined into small nmber of grops. If we look at a flid flowing nder normal circmstances - a river for example - the conditions at one point will vary from those at another point (e.g. different velocity) we have non-niform flow. If the conditions at one point vary as time passes then we have nsteady flow. Under some circmstances the flow will not be as changeable as this. He following terms describe the states which are sed to classify flid flow: niform flow: If the flow velocity is the same magnitde and direction at every point in the flid it is said to be niform. non-niform: If at a given instant, the velocity is not the same at every point the flow is non-niform. (In practice, by this definition, every flid that flows near a solid bondary will be non-niform - as the flid at the bondary mst take the speed of the bondary, sally zero. However if the size and shape of the of the cross-section of the stream of flid is constant the flow is considered niform.) steady: A steady flow is one in which the conditions (velocity, pressre and cross-section) may differ from point to point bt DO NOT change with time. nsteady: If at any point in the flid, the conditions change with time, the flow is described as nsteady. (In practise there is always slight variations in velocity and pressre, bt if the average vales are constant, the flow is considered steady. Combining the above we can classify any flow in to one of for type:. Steady niform flow. Conditions do not change with position in the stream or with time. An example is the flow of water in a pipe of constant diameter at constant velocity. CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 44

2 . Steady non-niform flow. Conditions change from point to point in the stream bt do not change with time. An example is flow in a tapering pipe with constant velocity at the inlet - velocity will change as yo move along the length of the pipe toward the exit. 3. Unsteady niform flow. At a given instant in time the conditions at every point are the same, bt will change with time. An example is a pipe of constant diameter connected to a pmp pmping at a constant rate which is then switched off. 4. Unsteady non-niform flow. Every condition of the flow may change from point to point and with time at every point. For example waves in a channel. If yo imaging the flow in each of the above classes yo may imagine that one class is more complex than another. And this is the case - steady niform flow is by far the most simple of the for. Yo will then be pleased to hear that this corse is restricted to only this class of flow. We will not be encontering any non-niform or nsteady effects in any of the examples (except for one or two qasi-time dependent problems which can be treated at steady). 3.. Compressible or Incompressible All flids are compressible - even water - their density will change as pressre changes. Under steady conditions, and provided that the changes in pressre are small, it is sally possible to simplify analysis of the flow by assming it is incompressible and has constant density. As yo will appreciate, liqids are qite difficlt to compress - so nder most steady conditions they are treated as incompressible. In some nsteady conditions very high pressre differences can occr and it is necessary to take these into accont - even for liqids. Gasses, on the contrary, are very easily compressed, it is essential in most cases to treat these as compressible, taking changes in pressre into accont. 3.. Three-dimensional flow Althogh in general all flids flow three-dimensionally, with pressres and velocities and other flow properties varying in all directions, in many cases the greatest changes only occr in two directions or even only in one. In these cases changes in the other direction can be effectively ignored making analysis mch more simple. Flow is one dimensional if the flow parameters (sch as velocity, pressre, depth etc.) at a given instant in time only vary in the direction of flow and not across the cross-section. The flow may be nsteady, in this case the parameter vary in time bt still not across the cross-section. An example of one-dimensional flow is the flow in a pipe. Note that since flow mst be zero at the pipe wall - yet non-zero in the centre - there is a difference of parameters across the cross-section. Shold this be treated as two-dimensional flow? Possibly - bt it is only necessary if very high accracy is reqired. A correction factor is then sally applied. Pipe Ideal flow Real flow One dimensional flow in a pipe. Flow is two-dimensional if it can be assmed that the flow parameters vary in the direction of flow and in one direction at right angles to this direction. Streamlines in two-dimensional flow are crved lines on a plane and are the same on all parallel planes. An example is flow over a weir foe which typical CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 45

3 streamlines can be seen in the figre below. Over the majority of the length of the weir the flow is the same - only at the two ends does it change slightly. Here correction factors may be applied. Two-dimensional flow over a weir. In this corse we will only be considering steady, incompressible one and two-dimensional flow Streamlines and streamtbes In analysing flid flow it is sefl to visalise the flow pattern. This can be done by drawing lines joining points of eqal velocity - velocity contors. These lines are know as streamlines. Here is a simple example of the streamlines arond a cross-section of an aircraft wing shaped body: Streamlines arond a wing shaped body When flid is flowing past a solid bondary, e.g. the srface of an aerofoil or the wall of a pipe, flid obviosly does not flow into or ot of the srface. So very close to a bondary wall the flow direction mst be parallel to the bondary. Close to a solid bondary streamlines are parallel to that bondary At all points the direction of the streamline is the direction of the flid velocity: this is how they are defined. Close to the wall the velocity is parallel to the wall so the streamline is also parallel to the wall. It is also important to recognise that the position of streamlines can change with time - this is the case in nsteady flow. In steady flow, the position of streamlines does not change. Some things to know abot streamlines Becase the flid is moving in the same direction as the streamlines, flid can not cross a streamline. Streamlines can not cross each other. If they were to cross this wold indicate two different velocities at the same point. This is not physically possible. The above point implies that any particles of flid starting on one streamline will stay on that same streamline throghot the flid. A sefl techniqe in flid flow analysis is to consider only a part of the total flid in isolation from the rest. This can be done by imagining a tblar srface formed by streamlines along which the flid flows. This tblar srface is known as a streamtbe. CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 46

4 A Streamtbe And in a two-dimensional flow we have a streamtbe which is flat (in the plane of the paper): A two dimensional version of the streamtbe The walls of a streamtbe are made of streamlines. As we have seen above, flid cannot flow across a streamline, so flid cannot cross a streamtbe wall. The streamtbe can often be viewed as a solid walled pipe. A streamtbe is not a pipe - it differs in nsteady flow as the walls will move with time. And it differs becase the wall is moving with the flid 3. Flow rate. 3.. Mass flow rate If we want to measre the rate at which water is flowing along a pipe. A very simple way of doing this is to catch all the water coming ot of the pipe in a bcket over a fixed time period. Measring the weight of the water in the bcket and dividing this by the time taken to collect this water gives a rate of accmlation of mass. This is know as the mass flow rate. For example an empty bcket weighs.0kg. After 7 seconds of collecting water the bcket weighs 8.0kg, then: mass of flid in bcket mass flow rate m & time taken to collect the flid kg / s ( kg s ) Performing a similar calclation, if we know the mass flow is.7kg/s, how long will it take to fill a container with 8kg of flid? CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 47

5 time mass mass flow rate s 3.. Volme flow rate - Discharge. More commonly we need to know the volme flow rate - this is more commonly know as discharge. (It is also commonly, bt inaccrately, simply called flow rate). The symbol normally sed for discharge is Q. The discharge is the volme of flid flowing per nit time. Mltiplying this by the density of the flid gives s the mass flow rate. Conseqently, if the density of the flid in the above example is 850 kg m 3 then: discharge, Q volme of flid time mass of flid density time mass flow rate density m / s ( m s ) m / s 008. l / s An important aside abot nits shold be made here: As has already been stressed, we mst always se a consistent set of nits when applying vales to eqations. It wold make sense therefore to always qote the vales in this consistent set. This set of nits will be the SI nits. Unfortnately, and this is the case above, these actal practical vales are very small or very large ( m 3 /s is very small). These nmbers are difficlt to imagine physically. In these cases it is sefl to se derived nits, and in the case above the sefl derived nit is the litre. ( litre m 3 ). So the soltion becomes 008. l / s. It is far easier to imagine litre than m 3. Units mst always be checked, and converted if necessary to a consistent set before sing in an eqation Discharge and mean velocity. If we know the size of a pipe, and we know the discharge, we can dedce the mean velocity Discharge in a pipe CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 48

6 If the area of cross section of the pipe at point X is A, and the mean velocity here is m. Dring a time t, a cylinder of flid will pass point X with a volme A m t. The volme per nit time (the discharge) will ths be Q volme A m t time t Q A m So if the cross-section area, A, is. 0 3 m and the discharge, Q is 4 l / s, then the mean velocity, m, of the flid is Q A m / s m Note how careflly we have called this the mean velocity. This is becase the velocity in the pipe is not constant across the cross section. Crossing the centreline of the pipe, the velocity is zero at the walls increasing to a maximm at the centre then decreasing symmetrically to the other wall. This variation across the section is known as the velocity profile or distribtion. A typical one is shown in the figre below. 3 3 A typical velocity profile across a pipe This idea, that mean velocity mltiplied by the area gives the discharge, applies to all sitations - not jst pipe flow. 3.3 Continity Matter cannot be created or destroyed - (it is simply changed in to a different form of matter). This principle is know as the conservation of mass and we se it in the analysis of flowing flids. The principle is applied to fixed volmes, known as control volmes (or srfaces), like that in the figre below: An arbitrarily shaped control volme. For any control volme the principle of conservation of mass says CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 49

7 Mass entering per nit time Mass leaving per nit time + Increase of mass in the control volme per nit time For steady flow there is no increase in the mass within the control volme, so For steady flow Mass entering per nit time Mass leaving per nit time This can be applied to a streamtbe sch as that shown below. No flid flows across the bondary made by the streamlines so mass only enters and leaves throgh the two ends of this streamtbe section. A streamtbe We can then write mass entering per nit time at end mass leaving per nit time at end Or for steady flow, This is the eqation of continity. ρ δa ρ δa ρδa ρ δa Constant m& The flow of flid throgh a real pipe (or any other vessel) will vary de to the presence of a wall - in this case we can se the mean velocity and write ρ A m ρ Am Constant m& When the flid can be considered incompressible, i.e. the density does not change, ρ ρ ρ so (dropping the m sbscript) A A Q This is the form of the continity eqation most often sed. This eqation is a very powerfl tool in flid mechanics and will be sed repeatedly throghot the rest of this corse. CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 50

8 Some example applications We can apply the principle of continity to pipes with cross sections which change along their length. Consider the diagram below of a pipe with a contraction: Section Section A liqid is flowing from left to right and the pipe is narrowing in the same direction. By the continity principle, the mass flow rate mst be the same at each section - the mass going into the pipe is eqal to the mass going ot of the pipe. So we can write: A ρ A ρ (with the sb-scripts and indicating the vales at the two sections) As we are considering a liqid, sally water, which is not very compressible, the density changes very little so we can say ρ ρ ρ. This also says that the volme flow rate is constant or that Discharge at section Discharge at section Q A Q A 3 3 For example if the area A 0 0 m and A 3 0 m and the pstream mean velocity,. m / s, then the downstream mean velocity can be calclated by A A 7. 0m / s Notice how the downstream velocity only changes from the pstream by the ratio of the two areas of the pipe. As the area of the circlar pipe is a fnction of the diameter we can redce the calclation frther, A A π d / 4 d π d / 4 d d d Now try this on a diffser, a pipe which expands or diverges as in the figre below, CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 5

9 Section Section If the diameter at section is d 30mm and at section d 40mm and the mean velocity at section is 30. m / s. The velocity entering the diffser is given by, m / s Another example of the se of the continity principle is to determine the velocities in pipes coming from a jnction. Total mass flow into the jnction Total mass flow ot of the jnction ρ Q ρ Q + ρ 3 Q 3 When the flow is incompressible (e.g. if it is water) ρ ρ ρ Q Q + Q 3 A A + A 3 3 If pipe diameter 50mm, mean velocity m/s, pipe diameter 40mm takes 30% of total discharge and pipe 3 diameter 60mm. What are the vales of discharge and mean velocity in each pipe? d Q A π m / s 3 Q 03. Q m / s Q Q + Q 3 Q Q 03. Q 0. 7Q m / s Q A 0 936m s CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 5

10 Q A m s CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 53

11 3.4 The Bernolli Eqation - Work and Energy Work and energy We know that if we drop a ball it accelerates downward with an acceleration g 9. 8m / s (neglecting the frictional resistance de to air). We can calclate the speed of the ball after falling a distance h by the formla v + as (a g and s h). The eqation cold be applied to a falling droplet of water as the same laws of motion apply A more general approach to obtaining the parameters of motion (of both solids and flids) is to apply the principle of conservation of energy. When friction is negligible the Kinetic energy mv sm of kinetic energy and gravitational potential energy is constant. Gravitational potential energy m g h (m is the mass, v is the velocity and h is the height above the datm). To apply this to a falling droplet we have an initial velocity of zero, and it falls throgh a height of h. We know that so Initial kinetic energy 0 Initial potential energy m g h Final kinetic energy mv Final potential energy 0 kinetic energy + potential energy constant Initial kinetic energy + Initial potential energy Final kinetic energy + Final potential energy so mgh v mv gh Althogh this is applied to a drop of liqid, a similar method can be applied to a continos jet of liqid. The Trajectory of a jet of water CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 54

12 We can consider the sitation as in the figre above - a continos jet of water coming from a pipe with velocity. One particle of the liqid with mass m travels with the jet and falls from height z to z. The velocity also changes from to. The jet is travelling in air where the pressre is everywhere atmospheric so there is no force de to pressre acting on the flid. The only force which is acting is that de to gravity. The sm of the kinetic and potential energies remains constant (as we neglect energy losses de to friction) so As m is constant this becomes mgz + m mgz + m + gz + gz This will give a reasonably accrate reslt as long as the weight of the jet is large compared to the frictional forces. It is only applicable while the jet is whole - before it breaks p into droplets. Flow from a reservoir We can se a very similar application of the energy conservation concept to determine the velocity of flow along a pipe from a reservoir. Consider the idealised reservoir in the figre below. An idealised reservoir The level of the water in the reservoir is z. Considering the energy sitation - there is no movement of water so kinetic energy is zero bt the gravitational potential energy is mgz. If a pipe is attached at the bottom water flows along this pipe ot of the tank to a level z. A mass m has flowed from the top of the reservoir to the nozzle and it has gained a velocity. The kinetic energy is now m and the potential energy mgz. Smmarising Initial kinetic energy 0 Initial potential energy m g z Final kinetic energy m Final potential energy mgz We know that kinetic energy + potential energy constant so CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 55

13 so mgz m + mgz mg( z + z ) m g( z z ) We now have a expression for the velocity of the water as it flows from of a pipe nozzle at a height ( z z ) below the srface of the reservoir. (Neglecting friction losses in the pipe and the nozzle). Now apply this to this example: A reservoir of water has the srface at 30m above the otlet nozzle of a pipe with diameter 5mm. What is the a) velocity, b) the discharge ot of the nozzle and c) mass flow rate. (Neglect all friction in the nozzle and the pipe). g( z z ) g m / s Volme flow rate is eqal to the area of the nozzle mltiplied by the velocity Q A π m 3 The density of water is 000 kg / m so the mass flow rate is mass flow rate density volme flow rate ρq s kg / s In the above examples the resltant pressre force was always zero as the pressre srronding the flid was the everywhere the same - atmospheric. If the pressres had been different there wold have been an extra force acting and we wold have to take into accont the work done by this force when calclating the final velocity. We have already seen in the hydrostatics section an example of pressre difference where the velocities are zero. p z p z CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 56

14 The pipe is filled with stationary flid of density ρ has pressres p and p at levels z and z respectively. What is the pressre difference in terms of these levels? or p p ρ g( z z ) p p + gz + gz ρ ρ This applies when the pressre varies bt the flid is stationary. Compare this to the eqation derived for a moving flid bt constant pressre: + gz + gz Yo can see that these are similar form. What wold happen if both pressre and velocity varies? 3.4. Bernolli s Eqation Bernolli s eqation is one of the most important/sefl eqations in flid mechanics. It may be written, p g g z p g g z ρ + + ρ + + We see that from applying eqal pressre or zero velocities we get the two eqations from the section above. They are both jst special cases of Bernolli s eqation. Bernolli s eqation has some restrictions in its applicability, they are: Flow is steady; Density is constant (which also means the flid is incompressible); Friction losses are negligible. The eqation relates the states at two points along a single streamline, (not conditions on two different streamlines). All these conditions are impossible to satisfy at any instant in time! Fortnately for many real sitations where the conditions are approximately satisfied, the eqation gives very good reslts. The derivation of Bernolli s Eqation: B B Cross sectional area a z mg A A An element of flid, as that in the figre above, has potential energy de to its height z above a datm and kinetic energy de to its velocity. If the element has weight mg then CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 57

15 potential energy mgz potential energy per nit weight z kinetic energy m kinetic energy per nit weight g At any cross-section the pressre generates a force, the flid will flow, moving the cross-section, so work will be done. If the pressre at cross section AB is p and the area of the cross-section is a then force on AB pa when the mass mg of flid has passed AB, cross-section AB will have moved to A B therefore volme passing AB mg ρg m ρ distance AA m ρa work done force distance AA m pm pa ρa ρ work done per nit weight p ρg This term is know as the pressre energy of the flowing stream. Smming all of these energy terms gives Pressre nit weight Kinetic nit weight Potential nit weight Total energy per + energy per + energy per energy per nit weight or p ρ g + g + z H As all of these elements of the eqation have nits of length, they are often referred to as the following: pressre head p ρg velocity head g potential head z total head H CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 58

16 By the principle of conservation of energy the total energy in the system does not change, Ths the total head does not change. So the Bernolli eqation can be written p z H ρ g + g + Constant As stated above, the Bernolli eqation applies to conditions along a streamline. We can apply it between two points, and, on the streamline in the figre below or or Two points joined by a streamline total energy per nit weight at total energy per nit weight at total head at total head at p g g z p g g z ρ + + ρ + + This eqation assmes no energy losses (e.g. from friction) or energy gains (e.g. from a pmp) along the streamline. It can be expanded to inclde these simply, by adding the appropriate energy terms: Total energy per nit weight at Total energy per nit + per nit + per nit weight at Loss weight Work done weight Energy spplied per nit weight p g g z p g g z h w q ρ + + ρ CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 59

17 3.4. An example of the se of the Bernolli eqation. When the Bernolli eqation is combined with the continity eqation the two can be sed to find velocities and pressres at points in the flow connected by a streamline. Here is an example of sing the Bernolli eqation to determine pressre and velocity at within a contracting and expanding pipe. A contracting expanding pipe A flid of constant density ρ 960 kg / m 3 is flowing steadily throgh the above tbe. The diameters at the sections are d 00mm and d 80mm. The gage pressre at is p 00kN / m and the velocity here is m / s. We want to know the gage pressre at section. 5 We shall of corse se the Bernolli eqation to do this and we apply it along a streamline joining section with section. The tbe is horizontal, with z z so Bernolli gives s the following eqation for pressre at section : ρ p p + ( ) Bt we do not know the vale of. We can calclate this from the continity eqation: Discharge into the tbe is eqal to the discharge ot i.e. A A A A d d 7 85m s. / Notice how the velocity has increased while the pressre has decreased. The phenomenon - that pressre decreases as velocity increases - sometimes comes in very sefl in engineering. (It is on this principle that carbrettor in many car engines work - pressre redces in a contraction allowing a small amont of fel to enter). /. CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 60

18 Here we have sed both the Bernolli eqation and the Continity principle together to solve the problem. Use of this combination is very common. We will be seeing this again freqently throghot the rest of the corse Pressre Head, Velocity Head, Potential Head and Total Head. By looking again at the example of the reservoir with which feeds a pipe we will see how these different heads relate to each other. Consider the reservoir below feeding a pipe which changes diameter and rises (in reality it may have to pass over a hill) before falling to its final level. Reservoir feeding a pipe To analyses the flow in the pipe we apply the Bernolli eqation along a streamline from point on the srface of the reservoir to point at the otlet nozzle of the pipe. And we know that the total energy per nit weight or the total head does not change - it is constant - along a streamline. Bt what is this vale of this constant? We have the Bernolli eqation p z H p z ρg + g + ρg + g + We can calclate the total head, H, at the reservoir, p 0 as this is atmospheric and atmospheric gage pressre is zero, the srface is moving very slowly compared to that in the pipe so 0, so all we are left with is total head H z the elevation of the reservoir. A sefl method of analysing the flow is to show the pressres graphically on the same diagram as the pipe and reservoir. In the figre above the total head line is shown. If we attached piezometers at points along the pipe, what wold be their levels when the pipe nozzle was closed? (Piezometers, as yo will remember, are simply open ended vertical tbes filled with the same liqid whose pressre they are measring). Piezometer levels with zero velocity As yo can see in the above figre, with zero velocity all of the levels in the piezometers are eqal and the same as the total head line. At each point on the line, when 0 CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 6

19 p z H ρ g + The level in the piezometer is the pressre head and its vale is given by What wold happen to the levels in the piezometers (pressre heads) if the if water was flowing with velocity? We know from earlier examples that as velocity increases so pressre falls p ρ g. Piezometer levels when flid is flowing p ρ g + g + z H We see in this figre that the levels have redced by an amont eqal to the velocity head,. Now as g the pipe is of constant diameter we know that the velocity is constant along the pipe so the velocity head is constant and represented graphically by the horizontal line shown. (this line is known as the hydralic grade line). What wold happen if the pipe were not of constant diameter? Look at the figre below where the pipe from the example above is replaced be a pipe of three sections with the middle section of larger diameter Piezometer levels and velocity heads with flid flowing in varying diameter pipes The velocity head at each point is now different. This is becase the velocity is different at each point. By considering continity we know that the velocity is different becase the diameter of the pipe is different. Which pipe has the greatest diameter? Pipe, becase the velocity, and hence the velocity head, is the smallest. CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 6

20 This graphical representation has the advantage that we can see at a glance the pressres in the system. For example, where along the whole line is the lowest pressre head? It is where the hydralic grade line is nearest to the pipe elevation i.e. at the highest point of the pipe Energy losses de to friction. In a real pipe line there are energy losses de to friction - these mst be taken into accont as they can be very significant. How wold the pressre and hydralic grade lines change with friction? Going back to the constant diameter pipe, we wold have a pressre sitation like this shown below Hydralic Grade line and Total head lines for a constant diameter pipe with friction How can the total head be changing? We have said that the total head - or total energy per nit weight - is constant. We are considering energy conservation, so if we allow for an amont of energy to be lost de to friction the total head will change. We have seen the eqation for this before. Bt here it is again with the energy loss de to friction written as a head and given the symbol h f. This is often know as the head loss de to friction. p g g z p g g z h f ρ + + ρ CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 63

21 3.5 Applications of the Bernolli Eqation The Bernolli eqation can be applied to a great many sitations not jst the pipe flow we have been considering p to now. In the following sections we will see some examples of its application to flow measrement from tanks, within pipes as well as in open channels Pitot Tbe If a stream of niform velocity flows into a blnt body, the stream lines take a pattern similar to this: Streamlines arond a blnt body Note how some move to the left and some to the right. Bt one, in the centre, goes to the tip of the blnt body and stops. It stops becase at this point the velocity is zero - the flid does not move at this one point. This point is known as the stagnation point. From the Bernolli eqation we can calclate the pressre at this point. Apply Bernolli along the central streamline from a point pstream where the velocity is and the pressre p to the stagnation point of the blnt body where the velocity is zero, 0. Also z z. p p + + z + + z ρg g ρg g p p + ρ ρ p p + ρ This increase in pressre which bring the flid to rest is called the dynamic pressre. Dynamic pressre ρ or converting this to head (sing h Dynamic head g p ) ρ g The total pressre is know as the stagnation pressre (or total pressre) Stagnation pressre p or in terms of head + ρ Stagnation head p ρ g + g CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 64

22 The blnt body stopping the flid does not have to be a solid. I cold be a static colmn of flid. Two piezometers, one as normal and one as a Pitot tbe within the pipe can be sed in an arrangement shown below to measre velocity of flow. Using the above theory, we have the eqation for p, A Piezometer and a Pitot tbe p p + ρ ρgh ρgh + ρ g( h h ) We now have an expression for velocity obtained from two pressre measrements and the application of the Bernolli eqation. CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 65

23 3.5. Pitot Static Tbe The necessity of two piezometers and ths two readings make this arrangement a little awkward. Connecting the piezometers to a manometer wold simplify things bt there are still two tbes. The Pitot static tbe combines the tbes and they can then be easily connected to a manometer. A Pitot static tbe is shown below. The holes on the side of the tbe connect to one side of a manometer and register the static head, (h ), while the central hole is connected to the other side of the manometer to register, as before, the stagnation head (h ). A Pitot-static tbe Consider the pressres on the level of the centre line of the Pitot tbe and sing the theory of the manometer, p A p + ρgx p p + ρg( X h) + ρ gh B man p A pb p + ρgx p + ρg( X h) + ρ gh We know that p pstatic p + ρ, sbstitting this in to the above gives ρ p + hg( ρman ρ) p + man gh( ρm ρ) ρ The Pitot/Pitot-static tbes give velocities at points in the flow. It does not give the overall discharge of the stream, which is often what is wanted. It also has the drawback that it is liable to block easily, particlarly if there is significant debris in the flow. CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 66

24 3.5.3 Ventri Meter The Ventri meter is a device for measring discharge in a pipe. It consists of a rapidly converging section which increases the velocity of flow and hence redces the pressre. It then retrns to the original dimensions of the pipe by a gently diverging diffser section. By measring the pressre differences the discharge can be calclated. This is a particlarly accrate method of flow measrement as energy loss are very small. A Ventri meter Applying Bernolli along the streamline from point to point in the narrow throat of the Ventri meter we have p g g z p g g z ρ + + ρ + + By the sing the continity eqation we can eliminate the velocity, Q A A A A Sbstitting this into and rearranging the Bernolli eqation we get CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 67

25 p p ρg A + z z g A g p p ρg A A + z z To get the theoretical discharge this is mltiplied by the area. To get the actal discharge taking in to accont the losses de to friction, we inclde a coefficient of discharge Q A Q C Q C A Q ideal actal d ideal d actal C A A d g p p + z z ρg A A This can also be expressed in terms of the manometer readings p p ρg p + ρgz p + ρ gh + ρg( z h) man ρman + z z h ρ Ths the discharge can be expressed in terms of the manometer reading:: Q actal C A A d ρman gh ρ Notice how this expression does not inclde any terms for the elevation or orientation (z or z ) of the Ventrimeter. This means that the meter can be at any convenient angle to fnction. The prpose of the diffser in a Ventri meter is to assre gradal and steady deceleration after the throat. This is designed to ensre that the pressre rises again to something near to the original vale before the Ventri meter. The angle of the diffser is sally between 6 and 8 degrees. Wider than this and the flow might separate from the walls reslting in increased friction and energy and pressre loss. If the angle is less than this the meter becomes very long and pressre losses again become significant. The efficiency of the diffser of increasing pressre back to the original is rarely greater than 80%. A A CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 68

26 3.5.4 Flow Throgh A Small Orifice We are to consider the flow from a tank throgh a hole in the side close to the base. The general arrangement and a close p of the hole and streamlines are shown in the figre below Tank and streamlines of flow ot of the sharp edged orifice The shape of the holes edges are as they are (sharp) to minimise frictional losses by minimising the contact between the hole and the liqid - the only contact is the very edge. Looking at the streamlines yo can see how they contract after the orifice to a minimm vale when they all become parallel, at this point, the velocity and pressre are niform across the jet. This convergence is called the vena contracta. (From the Latin contracted vein ). It is necessary to know the amont of contraction to allow s to calclate the flow. We can predict the velocity at the orifice sing the Bernolli eqation. Apply it along the streamline joining point on the srface to point at the centre of the orifice. At the srface velocity is negligible ( 0) and the pressre atmospheric (p 0).At the orifice the jet is open to the air so again the pressre is atmospheric (p 0). If we take the datm line throgh the orifice then z h and z 0, leaving h g This is the theoretical vale of velocity. Unfortnately it will be an over estimate of the real velocity becase friction losses have not been taken into accont. To incorporate friction we se the coefficient of velocity to correct the theoretical velocity, gh C actal v theoretical Each orifice has its own coefficient of velocity, they sally lie in the range( ) To calclate the discharge throgh the orifice we mltiply the area of the jet by the velocity. The actal area of the jet is the area of the vena contracta not the area of the orifice. We obtain this area by sing a coefficient of contraction for the orifice So the discharge throgh the orifice is given by A C A actal c orifice CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 69

27 Q A Q A actal actal actal C C A C A C A c v orifice theoretical d orifice theoretical d orifice gh Tank emptying from level h to h. The tank has a cross sectional area of A. In a time dt the level falls by dh or the flow ot of the tank is (-ve sign as δh is falling) Q Av Q δ A h δt Rearranging and sbstitting the expression for Q throgh the orifice gives A δh δt C A g h d This can be integrated between the initial level, h, and final level, h, to give an expression for the time it takes to fall this distance A t C A g d o o A C A g d o A C A g d o h h δh h [ h] h h [ h h ] CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 70

28 3.5.6 Sbmerged Orifice We have two tanks next to each other (or one tank separated by a dividing wall) and flid is to flow between them throgh a sbmerged orifice. Althogh difficlt to see, carefl measrement of the flow indicates that the sbmerged jet flow behaves in a similar way to the jet in air in that it forms a vena contracta below the srface. To determine the velocity at the jet we first se the Bernolli eqation to give s the ideal velocity. Applying Bernolli from point on the srface of the deeper tank to point at the centre of the orifice, gives p g g z p + + g g z + + ρ ρ ρgh h ρg g g( h h ) i.e. the ideal velocity of the jet throgh the sbmerged orifice depends on the difference in head across the orifice. And the discharge is given by Q C A d o C A g h h d o Two tanks of initially different levels joined by an orifice By a similar analysis sed to find the time for a level drop in a tank we can derive an expression for the change in levels when there is flow between two connected tanks. Applying the continity eqation δ Q A h A h δ δt δt Qδt A δh A δh Also we can write δh + δh δh So CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 7

29 Then we get A δh A δh A δh δh A A δh + A C A g( h h ) δt d o Re arranging and integrating between the two levels we get Qδt A δh A A δh δt ( A + A ) C A g h A A t ( A + A ) C A g A A ( A + A ) C A g A A ( A + A ) C A g d d d d o o o o h h [ h] A A δ A + A h final initial δh h h h final initial [ hinitial hfinal ] (remember that h in this expression is the difference in height between the two levels (h - h ) to get the time for the levels to eqal se h initial h and h final 0). Ths we have an expression giving the time it will take for the two levels to eqal. CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 7

30 3.5.8 Flow Over Notches and Weirs A notch is an opening in the side of a tank or reservoir which extends above the srface of the liqid. It is sally a device for measring discharge. A weir is a notch on a larger scale - sally fond in rivers. It may be sharp crested bt also may have a sbstantial width in the direction of flow - it is sed as both a flow measring device and a device to raise water levels Weir Assmptions We will assme that the velocity of the flid approaching the weir is small so that kinetic energy can be neglected. We will also assme that the velocity throgh any elemental strip depends only on the depth below the free srface. These are acceptable assmptions for tanks with notches or reservoirs with weirs, bt for flows where the velocity approaching the weir is sbstantial the kinetic energy mst be taken into accont (e.g. a fast moving river) A General Weir Eqation To determine an expression for the theoretical flow throgh a notch we will consider a horizontal strip of width b and depth h below the free srface, as shown in the figre below. Elemental strip of flow throgh a notch velocity throgh the strip, gh discharge throgh the strip, δq A bδh gh integrating from the free srface, h 0, to the weir crest, h H gives the expression for the total theoretical discharge Q g bh dh theoretical This will be different for every differently shaped weir or notch. To make frther se of this eqation we need an expression relating the width of flow across the weir to the depth below the free srface Rectanglar Weir For a rectanglar weir the width does not change with depth so there is no relationship between b and depth h. We have the eqation, b H 0 constant B CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 73

31 Sbstitting this into the general weir eqation gives A rectanglar weir Q B g h dh theoretical 3 B H 0 gh To calclate the actal discharge we introdce a coefficient of discharge, C d, which acconts for losses at the edges of the weir and contractions in the area of flow, giving Qactal Cd B gh V Notch Weir For the V notch weir the relationship between width and depth is dependent on the angle of the V. 3/ 3/ V notch, or trianglar, weir geometry. If the angle of the V is θ then the width, b, a depth h from the free srface is b ( H h) tan θ So the discharge is θ / Q g ( H h) theoretical tan h dh θ g tan Hh h 5 5 H 8 g tan H 5 θ 0 5/ 3/ 5/ And again, the actal discharge is obtained by introdcing a coefficient of discharge 8 Qactal Cd g tan H 5 θ 5/ H 0 CIVE 400: Flid Mechanics Flid Dynamics: The Momentm and Bernolli Eqations 74

32 3.6 The Momentm Eqation We have all seen moving flids exerting forces. The lift force on an aircraft is exerted by the air moving over the wing. A jet of water from a hose exerts a force on whatever it hits. In flid mechanics the analysis of motion is performed in the same way as in solid mechanics - by se of Newton s laws of motion. Accont is also taken for the special properties of flids when in motion. The momentm eqation is a statement of Newton s Second Law and relates the sm of the forces acting on an element of flid to its acceleration or rate of change of momentm. Yo will probably recognise the eqation F ma which is sed in the analysis of solid mechanics to relate applied force to acceleration. In flid mechanics it is not clear what mass of moving flid we shold se so we se a different form of the eqation. Newton s nd Law can be written: The Rate of change of momentm of a body is eqal to the resltant force acting on the body, and takes place in the direction of the force. To determine the rate of change of momentm for a flid we will consider a streamtbe as we did for the Bernolli eqation, We start by assming that we have steady flow which is non-niform flowing in a stream tbe. A streamtbe in three and two-dimensions In time δt a volme of the flid moves from the inlet a distance δ t, so the volme entering the streamtbe in the time δt is this has mass, and momentm volme entering the stream tbe area distance A δt mass entering stream tbe volme density ρ A δt momentm of flid entering stream tbe mass velocity ρ A δt Similarly, at the exit, we can obtain an expression for the momentm leaving the steamtbe: momentm of flid leaving stream tbe ρ A δt CIVE 400: Flid Mechanics Momentm Eqation and its Application 75

33 We can now calclate the force exerted by the flid sing Newton s nd Law. The force is eqal to the rate of change of momentm. So Force rate of change of momentm F ( ρ Aδt ρ Aδt ) δt We know from continity that Q A A, and if we have a flid of constant density, i.e. ρ ρ ρ, then we can write F Qρ( ) For an alternative derivation of the same expression, as we know from conservation of mass in a stream tbe that we can write The rate at which momentm leaves face is The rate at which momentm enters face is mass into face mass ot of face rate of change of mass &m dm ρ A ρ A dt ρ A m & ρ A m Ths the rate at which momentm changes across the stream tbe is i.e. & ρ A ρ A m & m & Force rate of change of momentm F m& ( ) F Qρ ( ) This force is acting in the direction of the flow of the flid. This analysis assmed that the inlet and otlet velocities were in the same direction - i.e. a one dimensional system. What happens when this is not the case? Consider the two dimensional system in the figre below: CIVE 400: Flid Mechanics Momentm Eqation and its Application 76

34 Two dimensional flow in a streamtbe At the inlet the velocity vector,, makes an angle, θ, with the x-axis, while at the otlet make an angle θ. In this case we consider the forces by resolving in the directions of the co-ordinate axes. The force in the x-direction F x Rate of change of momentm in x - direction Rate of change of mass change in velocity in x - direction ( θ θ ) m& cos cos ( ) m& And the force in the y-direction x ( cos cos ) ρq θ θ ( ) ρq x x ( θ θ ) F m& sin sin y x ( y y ) m& ( sin sin ) ρq θ θ ( y y ) ρq CIVE 400: Flid Mechanics Momentm Eqation and its Application 77

35 We then find the resltant force by combining these vectorially: Fresltant Fx + F y And the angle which this force acts at is given by F φ tan y F For a three-dimensional (x, y, z) system we then have an extra force to calclate and resolve in the z- direction. This is considered in exactly the same way. In smmary we can say: The total force exerted on the flid rate of change of momentm throgh the control volme ( ot in ) Q( ) F m& ρ ot Remember that we are working with vectors so F is in the direction of the velocity. This force is made p of three components: F R Force exerted on the flid by any solid body toching the control volme F B Force exerted on the flid body (e.g. gravity) F P Force exerted on the flid by flid pressre otside the control volme So we say that the total force, F T, is given by the sm of these forces: in FT FR + FB + FP The force exerted by the flid on the solid body toching the control volme is opposite to F R. So the reaction force, R, is given by R F R x CIVE 400: Flid Mechanics Momentm Eqation and its Application 78

36 3.7 Application of the Momentm Eqation We will consider the following examples:. Force de to the flow of flid rond a pipe bend.. Force on a nozzle at the otlet of a pipe. 3. Impact of a jet on a plane srface. 4. Force de to flow rond a crved vane The force de the flow arond a pipe bend Consider a pipe bend with a constant cross section lying in the horizontal plane and trning throgh an angle of θ. Flow rond a pipe bend of constant cross-section Why do we want to know the forces here? Becase the flid changes direction, a force (very large in the case of water spply pipes,) will act in the bend. If the bend is not fixed it will move and eventally break at the joints. We need to know how mch force a spport (thrst block) mst withstand. Step in Analysis:. Draw a control volme. Decide on co-ordinate axis system 3. Calclate the total force 4. Calclate the pressre force 5. Calclate the body force 6. Calclate the resltant force CIVE 400: Flid Mechanics Momentm Eqation and its Application 79

37 . Control Volme The control volme is draw in the above figre, with faces at the inlet and otlet of the bend and encompassing the pipe walls. Co-ordinate axis system It is convenient to choose the co-ordinate axis so that one is pointing in the direction of the inlet velocity. In the above figre the x-axis points in the direction of the inlet velocity. 3 Calclate the total force In the x-direction: In the y-direction: ( ) F ρq T x x x x x ( cosθ ) F ρq T x cosθ ( ) F ρq F T y y y sin0 0 y y T y sinθ ρq sinθ 4 Calclate the pressre force F P F p A cos0 p A cosθ p A p A cosθ P x F p A sin0 p A sinθ p A sinθ P y pressre force at - pressre force at 5 Calclate the body force There are no body forces in the x or y directions. The only body force is that exerted by gravity (which acts into the paper in this example - a direction we do not need to consider). CIVE 400: Flid Mechanics Momentm Eqation and its Application 80

38 6 Calclate the resltant force F F + F + F T x R x P x B x F F + F + F T y R y P y B y ( cos ) F F F 0 ρq θ p A + p A cosθ R x T x P x F F F 0 ρq sinθ + p A sinθ R y T y P y And the resltant force on the flid is given by And the direction of application is F F F R R x R y F φ tan F the force on the bend is the same magnitde bt in the opposite direction 3.7. Force on a pipe nozzle R F R Force on the nozzle at the otlet of a pipe. Becase the flid is contracted at the nozzle forces are indced in the nozzle. Anything holding the nozzle (e.g. a fireman) mst be strong enogh to withstand these forces. The analysis takes the same procedre as above:. Draw a control volme. Decide on co-ordinate axis system 3. Calclate the total force 4. Calclate the pressre force 5. Calclate the body force 6. Calclate the resltant force & Control volme and Co-ordinate axis are shown in the figre below. R y R x CIVE 400: Flid Mechanics Momentm Eqation and its Application 8

39 Notice how this is a one dimensional system which greatly simplifies matters. 3 Calclate the total force By continity, Q A A, so ( ) F F Q T T x ρ F T Q x A A ρ 4 Calclate the pressre force F P F pressre force at - pressre force at We se the Bernolli eqation to calclate the pressre Is friction losses are neglected, h f 0 the nozzle is horizontal, z z and the pressre otside is atmospheric, p 0, and with continity gives P x p p + + z + + z + ρg g ρg g p ρq A A h f 5 Calclate the body force The only body force is the weight de to gravity in the y-direction - bt we need not consider this as the only forces we are considering are in the x-direction. CIVE 400: Flid Mechanics Momentm Eqation and its Application 8