“It is EXACT, Jane” – a story told to the lecturer by a botanist colleague. The most important river in Australia is the Murray River, 2 375 km (Danube 2 850 km), maximum recorded Àow 3 1 3 1 3 950 m s (Danube at Iron Gate Dam: 15 400 m s ). It has many tributaries, Àow measurement in the system is approximate and intermittent, there is huge biological and Àuvial diversity and irregularity. My colleague, non-numerical by training, had just seen the demonstration by an hydraulic engineer of a one-dimensional computational model of the river. She asked: “Just how accurate is your model?”. The engineer replied intensely: "It is EXACT, Jane". River Engineering Nothing in these lectures will be exact. We are talking about the modelling of complex physical systems. John Fenton A further example of the sort of thinking that we would like to avoid: in the area of palaeo- Institute of Hydraulic Engineering and Water Resources Management hydraulics, some Australian researchers made a survey to obtain the heights of Àoods at individual Vienna University of Technology, Karlsplatz 13/222, trees. This showed that the palaeo-Àood reached a maximum height on the River Murray at a 1040 Vienna, Austria certain position of 18=01 m (sic), Having measured the cross-section of the river, they applied the Gauckler-Manning-Strickler Equation to determine the discharge of the prehistoric Àood, stated to URL: http://johndfenton.com/ 3 1 be 7 686 m s ... URL: mailto:[email protected] William of Ockham (England, c1288-c1348): Ockham’s razor is the principle that can be popularly stated as “when you have two competing theories that make similar predictions, the simpler one is the better”. The term razor refers to the act of shaving away unnecessary assumptions to get to the simplest explanation, attributed to 14th-century English logician and Franciscan friar, William of Ockham. The explanation of any phenomenon should make as few assumptions as possible, eliminating those that make no difference in the observable predictions of the explanatory 3

1. Introduction hypothesis or theory. When competing hypotheses are equal in other respects, the principle recommends selection of the hypothesis that introduces the fewest assumptions and postulates the 1.1 The nature of what we will and will not do – illuminated by some fewest entities while still suf¿ciently answering the question. That is, we should not over-simplify aphorisms and some people our approach. “There is nothing so practical as a good theory” – stated in 1951 by Kurt Lewin (D-USA, In general, model complexity involves a trade-off between simplicity and accuracy of the model. 1890-1947): this is essentially the guiding principle behind these lectures. We want to solve Occam’s Razor is particularly relevant to modelling. While added complexity usually improves the practical problems, both in professional practice and research, and to do this it is a big help to have ¿t of a model, it can make the model dif¿cult to understand and work with. a theoretical understanding and a framework. The principle has inspired numerous expressions including “parsimony of postulates”, the “The purpose of computing is insight, not numbers” – the motto of a 1973 book on numerical “principle of simplicity”, the “KISS principle” (Keep It Simple, Stupid). Other common methods for practical use by the mathematician Richard Hamming (USA, 1915-1998). That restatements are: statement has excited the opinions of many people (search any three of the words in the Internet!). Leonardo da Vinci (I, 1452–1519, world’s most famous hydraulician, also an artist): his However, numbers are often important in engineering, whether for design, control, or other aspects variant short-circuits the need for sophistication by equating it to simplicity “Simplicity is the of the practical world. A characteristic of many engineers, however, is that they are often blinded ultimate sophistication”. by the numbers, and do not seek the physical understanding that can be a valuable addition to the Wolfang A. Mozart (A, 1756–1791): “Gewaltig viel Noten, lieber Mozart”, soll Kaiser Josef II. numbers. In this course we are not going to deal with many numbers. Instead we will deal with über die erste der großen Wiener Opern, die “Entführung”, gesagt haben, und Mozart antwortete: the methods by which numbers could be obtained in practice, and will try to obtain insight into “Gerade so viel, Eure Majestät, als nötig ist.” (Emperor Joseph II said about the ¿rst of the great those methods. Hence we might paraphrase simply: "The purpose of this course is insight into the Vienna operas, “Die Entführung aus dem Serail”, “Far too many notes, dear Mozart”, to which behaviour of rivers; with that insight, numbers can be often be obtained more simply and reliably". Mozart replied “Your Majesty, there are just as many notes as are necessary”). The truthfulness of the story is questioned – Josef was more sophisticated than that ...

2 4 Albert Einstein (D-USA,1879-1955): “Make everything as simple as possible, but not simpler.” 1.3 Types of channel Àow to be studied This is a better and shorter statement than Ockham! An important part of this course will be the study of different types of channel Àow. Karl Popper (A-UK, 1902-1994) argued that we prefer simpler theories to more complex ones “because their empirical content is greater; and because they are better testable”. In other words, a (a) Steady uniform Àow Case (a) – Steady uniform Àow: simple theory applies to more cases than a more complex one, and is thus more easily falsi¿able. Normal depth kq Steady Àow is where there is no change kq Popper coined the term critical rationalism to describe his philosophy. The term indicates his with time, C@Cw 0. Distant from control kq rejection of classical empiricism, and of the classical observationalist-inductivist account of structures, gravity and resistance are in science that had grown out of it. Logically, no number of positive outcomes at the level of (b) Steady gradually-varied Àow balance, and if the cross-section is constant, experimental testing can con¿rm a scienti¿c theory (Hume’s “Problem of Induction”), but a single the Àow is uniform, C@C{ 0.Thisisthe  counterexample is logically decisive: it shows the theory, from which the implication is derived, to kq simplest model, and often is used as the be false. For example, consider the inference that “all swans we have seen are white, and therefore basis and a ¿rst approximation for others. all swans are white”, before the discovery of black swans in Australia. Popper’s account of the (c) Steady rapidly-varied Àow Case (b) – Steady gradually-varied Àow: logical asymmetry between veri¿cation and falsi¿ability lies at the heart of his philosophy of science. It also inspired him to take falsi¿ability as his criterion of demarcation between what is Where all inputs are steady but where k and is not genuinely scienti¿c: a theory should be considered scienti¿c if and only if it is falsi¿able. q channel properties may vary and/or a control may be introduced which imposes This led him to attack the claims of both psychoanalysis and contemporary Marxism to scienti¿c (d) Unsteady Àow status, on the basis that the theories enshrined by them are not falsi¿able. a water level at a certain point. The height of the surface varies along the channel. Thomas Kuhn (USA, 1922-1996): In The Structure of Scienti¿c Revolutions argued that scientists For this case we will study the governing work in a series of paradigms, and found little evidence of scientists actually following a Figure 1.1: Different types of Àow in an open channel differential equation that describes how falsi¿cationist methodology. Kuhn argued that as science progresses, explanations tend to become conditions vary along the waterway, and we more complex before a sudden paradigm shift offers radical simpli¿cation. For example Newton’s will obtain an approximate mathematical solution to solve general problems approximately. 5 7 classical mechanics is an approximated model of the real world. Still, it is quite suf¿cient for (a) Steady uniform Àow Case (c) – Steady rapidly-varied Àow: most ordinary-life situations. Popper’s student Imre Lakatos (H-UK, 1922-1974) attempted to Normal depth kq kq Figure (c) shows three separate gradually- reconcile Kuhn’s work with falsi¿cationism by arguing that science progresses by the falsi¿cation varied Àow regions separated by two kq of research programs rather than the more speci¿c universal statements of naive falsi¿cationism. rapidly-varied regions: (1) Àow under a Another of Popper’s students Paul Feyerabend (A-USA, 1924-1994) ultimately rejected any (b) Steady gradually-varied Àow sluice gate and (2) a hydraulic jump. The prescriptive methodology, and argued that the only universal method characterising scienti¿c basic hydraulic approximation that variation progress was "anything goes!" kq is gradual breaks down in those regions. 1.2 Summary We can analyse them by considering energy (c) Steady rapidly-varied Àow or momentum conservation locally. In this We will use theory, but we will try to keep things simple, rather simpler than is often the case in course we will not be considering these – • this ¿eld, especially in numerical methods. earlier courses at TUW have. kq Often our knowledge of physical quantities is limited, and approximation is justi¿ed. • Case (d) – Unsteady Àow: We will recognise that we are modelling. (d) Unsteady Àow • Here conditions vary with time and position An approximate model can often reveal to us more about the problem. as a Àood wave traverses the waterway. We • It might be thought that the lectures show a certain amount of inconsistency – in occasional will consider Àood wave motion at some • places the lecturer will develop a more generalised and “accurate” model, paradoxically to Figure 1.2: Different types of Àow in an open channel length. emphasise that we are just modelling. We will attempt to obtain insight and understanding – and a sense of criticality. •

6 8 1.4 Some possibly-surprising results Consider ¿gure 1.3 showing an open channel Àow with forces per unit mass acting on a particle. The ¿gure is drawn, showing that in general, the depth is not constant, and the bed is not parallel Effects of turbulence on dynamics to the free surface. The surface is an isobar, a line of constant pressure, s =0.IntheÀow, other Where the Àuid Àow Àuctuates in time, apparently randomly, about some mean condition, e.g. isobars will approximately be parallel to this, while the channel bed is not necessarily an isobar. the Àowofwind,waterinpipes,waterinariver.InpracticewetendtoworkwithmeanÀow We consider the vector Euler equation for the motion of a Àuid particle 1 properties, however in this course we will adopt empirical means of incorporating some of the Acceleration = Pressure gradient + Body forces per unit mass effects of turbulence. Consider the { component of velocity at x a point written as a sum of the  × mean (x¯)andÀuctuating (x0) components: In a direction parallel to the free surface, the pressure is constant and there is no pressure gradient. The acceleration of the particle will be given by the difference between the component of gravity x =¯x + x0= j sin  and the resistance force per unit mass. We usually do not know the details of that, so there is By de¿nition, the mean of the Àuctuations, which we write as x0,is little that we can say. Now considering a direction perpendicular to that, given by the co-ordinate W q on the ¿gure, there is very little acceleration, so we assume it to be zero, and so we obtain the 1 x0 = x0 gw =0> (1.1) result W 1 Cs Z0 0= j cos . where W is some time period much longer than the Àuctuations. Cq  Now integrating this with respect to q between a general point, such as q = g at the particle Now let us compute the mean value of the square of the velocity, such as we might ¿nd in shown, and q =0on the surface where s =0we obtain  s = j cos  g. × It is much more convenient to measure all elevations vertically, and so we use k, such that g = k cos , and we obtain the general expression for pressure s = jk cos2 = (1.3) 9 11 computing the mean pressure on an object in the Àow: Hydrostatic approximation

2 2 2 2 x = (¯x + x0) = x¯ +2¯xx0 + x0 > expanding, That result is really only important on structures such as spillways and block ramps. In almost 2 2 2 all open channels the slope is small enough such that cos  1,andwecanusethehydrostatic = x¯ + 2¯xx0 + x0 , considering each term in turn,  2 2 approximation, obtained from a static Àuid, where the surface is horizontal, =¯x +2¯x x0 + x0 ,but,as x0 =0from(1.1), 2 2 s = jk= (1.4) =¯x + x0 = (1.2) hence we see that the mean of the square of the Àuctuating velocity is not equal to the square of the Substituting k =  },where is the free surface elevation and } is the elevation of an arbitrary  2 point in the Àuid, mean of the Àuctuating velocity, but that there is also a component x0 , the mean of the Àuctuating components. We will need to incorporate this. s = j ( }) = (1.5)  Pressure in open channel Àow – effects of resistance on Àows over steep slopes From this we have at a speci¿c vertical cross-section, s + j} = j> (1.6) q An almost-universal assumption in river en- gineering is that the pressure distribution is so that anywhere on a vertical section s + j} is constant, given by the free surface elevation. Isobars hydrostatic, equivalent to that of water which The general result of equation (1.3) for Àow on a ¿nite slope seems to have been k  Resistance g is not moving, such that pressure s at a point is forgotten by many. In general, pressure in Àowing water is not “hydrostatic”. j sin  given by the height of water above, s = jk, However in this course, bed slopes are small enough that we will use it. 3 where  is Àuid density ( 1000 kgm for 2 j cos  fresh water), j 9=8ms is gravitational  Figure 1.3: Channel Àow showing isobars and forces per acceleration, and k is the vertical height of the unit mass on a Àuid particle surface above the point. This is not necessarily thecaseinÀowing water, and needs to be known for cases such as spillways or block ramps, which are steep. 10 12 2. Resistance in river and other open channel Àows One of the most famous series of experiments in Àuid mechanics was } performed by Johann Nikuradse at Göttingen in the 1930s, who studied the The resistance to the Àow of a stream is probably the most important problem in river mechanics. Àow of Àuid over uniformly-rough sand grains. The Àuid was actually air, Page 14: We consider a simple theory based on force balance and some classical Àuid mechanics and the sand grains were actually in circular pipes, but the results are still experiments to obtain a Àow formula for a wide channel. valid enough. With those results, for a wide channel of depth k with sand grains of size Page 16: To obtain the equivalent formula for channels of any section we consider velocity x distributions in real streams and develop an approximation giving a general Àow formula. ns, the velocity distribution for fully rough Àow (no effects of viscosity), the universal velocity distribution can be written: Page 20: We consider an approximation to that formula and ¿nd that we have obtained the Figure 2.2: Idealised x 30} logarithmic velocity Gauckler-Manning-Strickler formula, including a theoretical prediction of Strickler’s formula for x =  ln > (2.2)  n pro¿le in turbulent Àow the effect of boundary grain size. s over rough bed Page 25: Comparison with a series of experiments validates the approach, giving an explicit Àow intermsoftheshearvelocityx = @, the von Kármán constant  0=4, the vertical co-ordinate }, and where the factor of 30 is for closely-packed uniform sand grains. formula for a variety of channel boundaries. p It varies with other types of boundary roughness. The mean velocity X is obtained by integrating Page 31: For more general river problems, considering the nature of the bed particles and bed between 0 and k, such that forms, vegetation, meandering, and possibly obstacles, it is better to use a formulation in which 1 k x 30@e forces and the mechanics are clearer: the Chézy-Weisbach Àow formula. X = xg} =  ln = (2.3) k  n @k Z0 s Page 33: A large number of stream-gauging results are considered and the values of the Weisbach where e=exp(1)=2=718 ===is Euler’s number. The result has been obtained in terms of relative resistance coef¿cient, its dependence on grain size, and on the state of the bed are obtained. roughness [email protected] = @ using equation (2.1) to give Empirical formulae are considered.  p 1 D 30@e Page 38: The common problem of calculating the water depth for a given Àow rate is considered. X = j V ln = (2.4)  S n @k A computational method is developed and applied. r μ s ¶ 13 15

2.1 The channel Àow formula We have obtained something very useful – a formula for the mean Àow velocity in a wide channel of constant depth k, slope V, and relative roughness ns@k. We have used simple mechanics plus an Here, the fundamental Àow formula for steady uniform Àow in channels is developed from theory empirical laboratory result. Surprisingly, the formula is explicit in terms of physical quantities – we and experimental results. We show that the traditional Àow formulae of Gauckler-Manning and have not had to assume a value like the Manning coef¿cient q or Strickler coef¿cient nSt =1@q! Chézy-Weisbach are simply different approximations to that. That was for a wide channel with an idealised O Consider a horizontal length O of uniform logarithmic velocity distribution. In nature, for channel Àow, inclined at a small angle  to channels of any general cross-section there is j sin  the horizontal, with cross-sectional area D. D the problem that the velocity has a maximum j The volume of the element is DO,thevertical somewhere below the surface, and in general  gravitational force on the water is jDO, Resistance S the isovels are something like Figure 2.3. where  is Àuid density and j is gravitational Even in straight channels there are longitudinal Figure 2.1: Uniform Àow in a channel, showing resistance and gravity forces on a ¿nite length, plus cross-section acceleration. The component of this along the vortices such that in the centre of the channel Figure 2.3: Cross-section of Àow showing isovels and, for quantities slope is jDO sin . The resistance force along a number of points on the bed, where the fastest-likely the maximum in velocity, which would be Àuid comes from and how far it travels, the effective the slope, of length O@ cos  is SO@cos ,where is the mean resistance shear stress, assumed expected to be at the surface, is actually at a length scale for resistance calculations. uniformly distibuted around the wetted perimeter S around which it acts. Equating gravitational lower position. and resistance components gives SO@cos  = jDO sin . Tohighaccuracyforsmall, cos  1 and sin  tan  = V, the slope, giving To obtain a Àow formula for channels of any cross-section, we hypothesise that the effective   depth k for resistance calculations is the typical distance from points with the highest velocity D = j V= (2.1) to the nearest point on the bed, as suggested by the red arrows on the ¿gure. The Àuid Àow  S on the boundary, where resistance occurs, would be similar to that in a channel, not of the Our problem is now to express shear stress in terms of Àow quantities. actual mean depth, but the mean length of the red arrows.Typical length scales as shown by the arrows are somewhat smaller than the overall mean depth of Àow. Our problem is then, how to 14 16 approximate that distance? We examine the approach suggested by the lecturer (Fenton 2011). grain size ns. For example, if we were studying the Danube, we might simply use the typical We consider the experimental data for the vertical position of grain size ns = G =0=02 m. Traditional practice, however is often to use Chézy-Weisbach the locus of velocity maxima in rectangular channels from and Gauckler-Manning-Strickler formulae, which introduce resistance coef¿cients which, while Yang, Tan & Lim (2004). They presented a formula for the k more general, and allow for other forms of resistance such as vegetation and bed forms, are more } height above the bed of the velocity maximum as a function max empirical and their physical signi¿cance less clear. of position across the channel. If the mean value of this is Our Àow formula (2.6) can be written calculated by integration, a formula for the mean elevation E of the velocity maximum } @k is obtained as a function of Figure 2.4: Experimental determination of T D max velocity maxima in rectangular channel X = =  j V> where (2.7) aspect ratio (channel width E divided by depth k). D r S 1 30@e 1 30@e 1 12= There is another length scale in equation (2.4) which is  = ln = ln ln > (2.8) 1=0  n @ (D@S )  %   % the ratio of area to perimeter D@S ,which,asSAE, s should be smaller than the mean depth D@E, so it might 0=8 where we have introduced the symbol % for the relative roughness be a candidate for the depth scale as experienced by the }max@k 0=6 ns Equivalent grain size Grain size and % = = > (2.9) Experimental D@S Hydraulic mean depth  Depth bed. We calculate the ratio: (D@S )@k0=4 range D@S Ek E@k and, although 30@e 11=0, it is usually written as 12, quite acceptably because our knowledge = = = (2.5) 0=2 }max@k — experiment  k (E +2k)k E@k +2 (D@S )@k,eqn(2.5) of ns is usually not good. We now have a Àow formula for steady uniform Àow in a channel 0=0 1 2 10 100 based on simple theory, experimental observations, and a bold approximation. We will soon see Both this and the experimental formula for }max@k are Aspect ratio E@k plotted in Figure 2.5. Remarkably, the two coincide how the most common Àow formulae in practice are an approximation to this, but this has the advantage that it is an explicit formula for the resistance coef¿cient in terms of the equivalent closely over a wide range of aspect ratios, so that we have Figure 2.5: Rectangular channels: dimension- relative roughness of boundary grains. found the quantity D@S mimics the behaviour of }max, less mean elevation of }max@k and the effective which we have suggested is, instead of k, the apparent depth (D@S )@k 17 19 depth that the Àow on the bed experiences. This suggests that in equation (2.4), instead of k in the Relative unimportance of grain size term from the velocity distribution, we can use D@S . We cannot claim that this is a justi¿cation as strong as it looks, but we have seen that D@S already appears in the equation, appearing naturally In fact, , although all-important for us, is relatively slowly varying with grain size. Consider a in the simple mechanical equilibrium calculation. small change in the relative roughness % (1 + {). The relative change in the factor  is  ln (12@(% (1 + {))) { For D@S we will not use the conventional and misleading term “hydraulic radius” (German after = 1  > Strickler – “Pro¿l- oder hydraulischer Radius”).  ln (12@%)   ln (12@%) having expanded the logarithm as a power series ln (1 + {)={ + === . Now for a value of For channels that are not rectangular we have presented no results. Our suggestion is that D@S % =0=001 (a 1mmgrain in 1mof water), a relative change of { = 50% gives a relative change in will still be a plausible approximation, and it already appears in equation (2.4). The use of D@S the factor  in the equation of only 5%. Even for a much rougher case of % =0=1, the same relative was justi¿ed by Keulegan (1938), however while that work mathematically correctly integrated change of 50% in grain size changes the left side by just 10%. It seems that it does not matter so logarithmic velocity distributions perpendicular to parts of various shapes of cross-section, it did much if we cannot specify the bed conditions all that accurately. not give any attention to the real velocity distributions, especially ignoring the phenomenon of the velocity maximum being below the surface. 2.2 The Gauckler-Manning-Strickler formula We have shown that D@S is approximately equal to the mean distance of the maximum velocity The Gauckler-Manning formula from the bed, so that the Àow on the bed is similar to that of a channel of depth D@S . In channels that are wide, which is most, S E and D@S is about the same as the geometric mean depth The most widely-used resistance formula in river engineering is the Gauckler-Manning formula.  D@E. Our suggested channel Àow formula, replacing k by D@S in equation (2.4) is T 1 D 2@3 D 2@3 X = = sV = n sV> (2.10) T 1 D 30@e D q S St S X = = j V ln = (2.6) μ ¶ μ ¶ D  S n @ (D@S ) where q is the Manning coef¿cient and n =1@q is the Strickler coef¿cient, used in German- r μ s ¶ St If we knew an accurate value of ns, this is probably the formula that we should use, as it is speaking countries. This was originally suggested by Gauckler and then by Manning in the in terms of phyical quantities that we know or we can approximate, including the equivalent nineteenth century, having observed that variation with D@S seemed to be of this simple power 18 20 form, compared with our equation (2.6) obtained from theory and experiments of the early which is simply the Gauckler-Manning equation but with an explicit expression for the Strick- twentieth century, which we wrote as equations (2.7) and (2.8) as ler/Manning coef¿cient: 1 7=8sj T D 1 12=D@S 1 12= n = = = (2.15) X = =  j V where  ln = ln = (2.11) St q 1@6 D S   ns  % ns r A similar result was obtained by Strickler (nearly a century ago, without optimising software!), An approximation to our formula based entirely on experiment on boundary roughnesses of equivalent mean diameter from We now show that the Gauckler-Manning formula is an approximation to the theoretical and G =0=1mmto G = 300 mm, (where that diameter was sometimes calculated from alluvial gravel experimental expression (2.11). with relative lengths of the three axes 1:2:3). For the numerical coef¿cient he obtained a value of 4=75s2 6=7, giving his expression On Figure 2.6 is shown how the dimension-  25 6=7sj less factor  varies as a function of relative n = = (2.16) St G1@6 roughness %, given by equation (2.8) from 20 The expression (2.15) here has been obtained by a quite different route, and the agreement between experimental Àuid mechanics. It is actually the two expressions, one based on sand grains glued to the inside of a circular pipe carrying air, is possible to approximate that curve closely 15 (%) encouraging. Of course, for river engineering purposes, Strickler’s result (2.16) is to be preferred. using a monomial function d@%. The best 10 2@3 values of d and  can be found by performing Logarithmic function, eqn (2.8) We call the Àow formula in terms of (D@S ) and coef¿cients written simply as 1@q or nSt a least-squares ¿t. Using 11 points equally- 5 Power function, eqn (2.12),  =1@7 the Gauckler-Manning (GM) formula, however with the expression (2.16) for nSt, we call it the spaced in log % between % =0=001 and 0=1, Power function, eqn (2.13),  =1@6 Gauckler-Manning-Strickler (GMS) formula. 0 the result obtained is  =1@7=00 (which is 0.001 0.01 0.1 a surprising coincidence), and d 8=9.This Relative roughness % = n@(D@S ) gives close agreement with the expression from Figure 2.6:

21 23 the logarithmic velocity distribution shown in the ¿gure, giving Sensitivity to boundary particle size D@S 1@7  =8=9 = (2.12) As earlier, we examine the effect of uncertainty or variability in the size of the boundary particles n μ s ¶ (and any perceived ambiguity between ns and G), using a power series expansion 1@6 Using this gives us another Àow formula {nSt {G  1{G 9@14 = 1+ 1= + ===> T 8=9sj D nSt G  6 G X = = sV> μ ¶ D 1@7 S and so a fractional change in boundary particle size gives a relative change of 1@6 of that amount ns μ ¶ where D@S appears simply raised to a power, similar to the Gauckler-Manning formula (2.10). in G. Again for this form the exact particle size is actually not so important. However unlike that, this gives an explicit formula for the coef¿cient in terms of bed grain size. We might consider this to be an advance, however if we modify our approach slightly we obtain a similar approximation, which actually gives the well-known and widely-used Gauckler-Manning formula. The logarithmic function is now approximated again, this time by a function e@%1@6, where e is a constant. It can also be determined by performing a least-squares ¿t, giving a value of e 7=8 such that  D@S 1@6  =7=8 > (2.13) n μ s ¶ with results shown in Figure 2.6, showing that this is also quite a good approximation to the logarithmic function. Substituting into the Àow formula, equation (2.11) and re-writing, we obtain @ T 7=8sj D 2 3 X = = sV> (2.14) D 1@6 S ns μ ¶ 22 24 Test of logarithmic and GMS formulae 2.3 Boundary stress in compound channels and unsteady non-uniform Àows To test the accuracy of the GMS for- Now we relate our results to show how they relate to the computation of boundary resistance 3.0 mula compared with the logarithmic Boards, rectangular in more complicated situations. While doing this we obtain a well-known alternative resistance Masonry, trapezoidal - nearly rectangular formula we obtained from experiment, 2.5 Reuss River, Seedorf formula, which is based on a simpler approximation to our above results. equation (2.11), we consider the re- Boards, rectangular The Chézy-Weisbach Àow formula sults of Strickler (1923, Beilage 4), 2.0 Cement, semi-circular Cement-sand, semi-circ. which have been interpreted as the Boards, semi-circ. Rhine River, St Margrethen Writing shear stress in terms of the result obtained from the Darcy-Weisbach formulation of Àow X Lined tunnel justi¿cation for the exponent 2@3 in (m/s) 1.5 resistance in pipes, the GMS formula, and leading to the = 1X 2> (2.18) “S” in that name. Strickler considered 1.0  8 results from nine very different chan- where  is the Weisbach dimensionless resistance coef¿cient, expressing the relationship between nels. For each the lecturer calculated 0.5 Gauckler-Manning-Strickler velocity and stress. The factor of 1@8 is necessary to agree with the Darcy-Weisbach energy Logarithmic formula, eqn (2.8) the equivalent ns or G, constant for formulation of pipe Àow theory in circular pipes where D@S = Diameter@4. From our simple 0.0 each channel, by least-squares ¿tting 0.0 0.5 1.0 1.5 2.0 force balance we already have equation (2.1): @ = j (D@S ) V. Eliminating @ between of the appropriate Àow formula to the D@S (m) equation (2.18) and this gives the Chézy-Weisbach Àow formula Figure 2.7: Strickler’s results approximated by two Àow formulae p points, with results shown in the ¿g- T 8j D D ure. The Gauckler-Manning-Strickler X = = V = F V> (2.19) D  S S formula gives agreement generally as good as our logarithmic formula obtained from Àuid me- r r chanics experiments, and there seems to be no need to replace it. Using the dimensionless relative where F = 8j@ is the Chézy coef¿cient, named after the French military engineer who ¿rst presented such an open channel Àow formula in 1775. Comparing our GMS formulation equation roughness % = G@(D@S ) to determine resistance seems to have advantages, as set out in the p following section. 25 27

Summary: the Gauckler-Manning-Strickler formula (2.17) we see that it is in the same form, such that 8 In the rest of this course, we sometimes write the Gauckler-Manning formula conventionally as  = > (2.20)  T 1 D 2@3 D 2@3 r X = = sV = n sV> so that our  is clearly related to the resistance coef¿cient , but in the GMS expression,  was a D q S St S μ ¶ μ ¶ function of % = G@(D@S ). 1@6 but if we use the Strickler expression nSt =6=7sj@G we write it in the dimensionless form Generalised resistance coef¿cient which we have called the Gauckler-Manning-Strickler formula: It is often more useful for us to introduce and use the resistance coef¿cient \ such that T D 6=7 G = = where ( )= ,and = (2.17)  1 X  j V  % 1@6 % = D S % D@S \ = = 2> (2.21) r 1@3 8  We no longer have the problem that nSt or q have dif¿cult units (q:L T). • where we do not necessarily consider them constant, but where (%) in general is as given by The characterisation of the resistance has been reduced to that of the dimensionless relative • equation (2.17). In this case, the boundary stress is given by roughness % = G@(D@S ). There are no problems with confusion of Imperial/SI units in various 2@3 1@6 = \X 2= (2.22) similar formulae which group terms differently and contain terms like (D@S ) and G .  To use the formula, we do not have to imagine a value of q or nSt, which have no simple • physical signi¿cance. We do not have to look at pictures of rivers in standard references such Non-uniform and unsteady Àows as Chow (1959, §5-9&10). We do not have to ring a friend to see what they used for a similar We will be considering Àows which are not uniform (vary with position {) and those which are stream 20 km distant some years ago. Instead, we can always use an estimate of G. neither uniform nor steady (vary also with time w). As the length scale of river Àows is much longer Of course, resistance includes also that due to bed forms and vegetation, but expressing it in in space than the cross-sectional dimensions and the time scale of disturbances is much longer than • terms of G is a good basis, with an understandable quantity % = G@(D@S ). that of local turbulence, we will assume that the boundary stress at each place and at each time is given by the local immediate Àow conditions of velocity, in terms of discharge T and area D. 26 28 From equations (2.22) and (2.20) we have 2.4 General situations T ({> w) 2 1 T ({> w) 2 = \X 2 = \ = = (2.23) Presence of different resistance elements  D ({> w) 2(%) D ({> w) μ ¶ μ ¶ In many cases the conditions in the river are more complicated than just a layer of uniform regular Compound cross-sections particles. For example: We consider compound cross-sections such as shown Irregular and variable nature of the bed particle arrangement. in Figure 2.8. Using equation (2.23) we can obtain an 2 3 • expression for the boundary shear force per unit length 1 of channel in each is, generalising equation (2.22) and multiplying by the perimeter of each part: 2 T Sl S = \(% ) l > for l =1> 2>=== Figure 2.8: Compound cross-section l l l D2 l The variability of resistance in real streams is often much greater than has been realised, for the Note that the internal shear forces across faces 1-2 and 1-3 cancel. However, we do not know the arrangement of the bed “grains” or “particles” (even if they are 30 cm boulders) is very impor- individual discharges Tl of the three sections. An approximation would be to neglect interfacial tant, and can change continuously, depending on the Àow history. Nikuradse’s experiments were shear and obtain the discharges for each component from the GMS equation. for sand grains levelled so that their tops were co-planar, and hence most of the particles were The total gravitational component, force per unit length is shielded from the Àow and resistance was small. That is what a bed looks like after a long period 3 of constant Àow, when any individually-projecting grains have been removed, because the force jV Dl> l=1 on them was larger. The bed is said to have been “armoured” – not only is the resistance small, where we have assumed that the slope V is theX same for each part. but individual grains are hard to remove. After a sudden increase of Àow, particles are more likely to have been dislodged, moved, and deposited, leaving a random surface, where those To solve the steady Àow problem then, if we might write Tl = lT,whereT is the total 29 31

Àow.summing the components due to boundary force and equating to the gravitational component most projecting exert larger force on the Àuid and resistance is greater. we obtain Bed forms – ripples, dunes, anti-dunes etc. 3 2 3 • • 2 \(%l)Sll T = jV Dl> l=1 2 l=1 Dl that we could use to calculate theX total Àow T or more complicatedX deductions.

Be very sceptical of the apparently simple formulae for combining nSt or Manning’s q appearing in many books, as exempli¿ed by the list of 17 different compound or composite section formulae in Yen (2002, table 3), or Cowan’s formula (see page 36 of Yen’s paper) q =( l ql) p,where the ql are different contributions from surface roughness, shape and size of channel cross-section, etc., which is irrational nonsense. The proper way to proceed is by a linear sum ofP the forces as we have done here.

Typical bedforms (after Richardson and Simons) The bed-forms which can develop if the bed is mobile will also contribute to variable resistance. Particle movement – if the grains are actually moving, then the force required to move the grains • appears to the water as an additional stress, whether they are moving along the bed, rolling, jumping, or carried suspended in the Àow. Vegetation – trees (standing and/or fallen), grasses, reeds etc • It can be seen that the problem of the current instantaneous resistance in the stream is actually a very dif¿cult and uncertain one ... 30 32 Results for resistance coef¿cients in real rivers Many of the results from each study are for large bed material %84 A 0=1, possibly a reÀection of • the hilly and mountainous nature of New Zealand and Paci¿c North-West of the United States Here we attempt to obtain understanding and a formula for the resistance coef¿cient using results of America (and which applies to Austria ...). from a number of ¿eld measurements. To compare with several experimental works, we use the There is a wide scatter of results. But not all that very wide if we consider that the streams range Chézy-Weisbach formulation. We considered the results of Hicks & Mason (1991), a catalogue • of 558 stream-gaugings from 78 river and canal reaches in New Zealand, of which 55 were sites from large slow-moving rivers with extremely small grains to mountain torrents with 30 cm boulders. Most of the results, unless the grains are moving, fall between \ 0=005 and 0=02. with grading curves for boundary material, so that particle sizes were known. Neither vegetation  nor bed-form resistance can be isolated. Hicks & Mason based their approach on Barnes (1967), There is, as we have seen, slow variation with relative roughness: am increase in % by a factor • who provided values of Manning’s resistance coef¿cient q =1@nSt for a single Àow at each of 50 of 10 leads to an increase in \ of about 2, as we have already seen. separate river sites in the United States of America, of which boundary material details were given The points, we believe, have a tendency to group around three of the curves shown and the rest for 14. We also include those results here. • to be bounded below by the fourth (upper) curve shown. The curves have been drawn using the

From both catalogues we took the values of G84, the boundary particle size for which 84% expression, found by trial and error: of the material was ¿ner, and from the values of D@S , calculated the relative roughness 0=06 + 0=06  \ = > (2.24) % = G @(D@S ), and used the measured values of Chézy’s F to calculate values of  2 84 84 (1=0 0=6  ln %84)   (F = 8j@). Results are shown on the next page. We have plotted them for a parameter with values of  =0> 0=5, 1,and2. The parameter  is an arbitrary one that we use to identify \ = @8, which it will be more convenient for us to use later. p the state of the particles making up the bed. This will now be explained. The ¿rst grouping of points comprises those around the bottom curve. We hypothesise that these • points, having the lowest resistance, are those forming beds where the particles are relatively co-planar such that the bed is armoured. We assigned  =0to this state, and used that in equation (2.24) to plot the curve. The next grouping of points is around the second curve from the bottom, which can be seen • 33 35

100 to substantially coincide with the a curve corresponding to exposed boulders on top of the bed Barnes (1967) Hicks & Mason (1991) — stable bed occupying 0.2 of the surface area. Of course, with a number of these grains thus exposed, the Hicks & Mason (1991) — moving bed resistance is greater. We assigned a value of  =0=5 to this intermediate state. Eq. (2.24) Aberle & Smart (2003, eqn 9) Substituting  =1in equation (2.24) gives the third curve on the ¿gure, passing through Pagliara et al. (2008, eqn 15), V =0, K =0 • what we believe is the third grouping of particles. This is probably the state for the maximum Ditto, but K =0=2 Yen (2003), Eq. (2.25) resistance for a stable bed corresponding to exposed grains occupying something like 50% of

1 Strickler, Eq. (2.17) 103 the surface area. Any more such grains will cause shielding of particles, the bed will start to resemble the co-planar case, and resistance will actually be reduced. Further evidence supporting our assertions is obtained from the expression proposed by Yen  \ = • 8 (2002, eqn 19), who considered results from a number of experimental studies using ¿xed impermeable beds. We used his formula, converted to \ = @8,usedanin¿nite Reynolds number, and converted his equivalent sand roughness %v =2%84. It can be seen that the curve 2 103 passes (left to right) from our curve  =1for small particles, which are unlikely to have the tops levelled so that particles are exposed, to the second curve for larger particles, more likely to be levelled in the laboratory experiments, with  =0. Yen obtained the approximation for :  =2=0 —moving 2 1 1 ns 1=95   =1=0 — fully exposed grains  = log + > (2.25) 4=  10 12= D@S R0=9  =0=5 — irregular μ μ ¶¶  =0=0 — co-planar bed 3 where R is the channel Reynolds number R =(T@S ) @,inwhich is the kinematic viscosity. 103 3 2 1 0 103 103 103 10 The logarithmic formula we obtained above, equation (2.8), leads to, if we use %v =2%84,and Relative roughness %84 •

34 36 as  = 8@ =1@s\, with k, as does the perimeter S(k). So, the right side of the equation varies slowly with k,so 5@3 2 by isolating the k term and taking the 3@5 power of both sides of the equation, we obtain the p 1 11=  2 \ = ln =(6=0 2=5ln(2%84)) > (2.26) equation in a form suitable for direct iteration  %  μ ¶ T 3@5 S 2@5(k) qT 3@5 S 2@5(k) giving results quite similar to those from Yen’s formula. k = = > (2.27) n sV × D(k)@k sV × D(k)@k For points above the third curve almost all experimental points had shear stresses greater than μ St ¶ μ ¶ • the critical one necessary for movement. If particles move, not only do many particles protrude where the ¿rst term on the right is a constant for any particular problem, and the second term above others, increasing the stress, but there is the additional force required to maintain the varies slowly with depth – a primary requirement that the direct iteration scheme be convergent sliding and rolling and jostling of all the particles. Hence, the resistance is greater. And, if there and indeed be quickly convergent. is a need to maintain particles in suspension, that will contribute also to resistance. We have For an initial estimate we suggest making a rough estimate of the approximate width E0 and so, shown the fourth curve as drawn for  =2. making a wide channel approximation, setting D(k)@k E0 and S (k) E0, in the general   Hopefully the ¿gure and approximating curves have given us an idea of the magnitudes and scheme of (2.27) gives 3@5 3@5 variation of the quantities, and maybe even some results for use in practice. T qT k0 = = = (2.28) n E sV E sV μ St 0 ¶ μ 0 ¶ Experience with typical trapezoidal sections shows that the method works well and is quickly convergent.

37 39

2.5 Computation of normal Àow Trapezoidal section

At last we turn to a common practical problem in River Engineering. "Normal Àow" is the name Most canals are excavated to a trapezoidal section, E given to a uniform Àow, and the depth is called the normal depth. If the discharge T, slope V, and this is often used as a convenient approximation resistance coef¿cient n =1@q, and the relationship between area and depth and perimeter and to river cross-sections too. In many of the problems k 1 St S depth are known, the GMS formula becomes a transcendental equation for the normal depth k.To in this course we will consider the case of trapezoidal p Z solve this is a common problem in river engineering sections. Consider the quantities shown in the ¿gure: A numerical method the bottom width is Z , the depth is k, the top width is E, and the batter slope, de¿ned to be the ratio of H:V dimensions is p. Geometrically, E = Z +2pk,areaD = k (Z + pk), wetted Any method for the numerical solution of transcendental equations can be used, such as Newton’s perimeter S = Z +2s1+p2k. method. Here we develop a simple method based on direct iteration, where we develop a trick, giving us rapid convergence. Example 1 Calculate the normal depth in a trapezoidal channel of slope 0.001, q =0=04, bot- tom width Z =10m, with batter slopes p =2, carrying a Àow of 20 m3s 1.WehaveD = In the case of wide channels, (i.e. channels rather wider than they are deep, a common case), the  k (10 + 2 k), S =10+4=472 k.ForE we use Z =10m. Equation (2.28) gives wetted perimeter S does not vary much with depth k. Similarly the width does not vary much with 0 k. Consider the GM formula in the conventional form, written now qT 3@5 0=04 20 3@5 k0 = = × =1=745 m= D5@3 1 D5@3 E sV 10s0=001 T = n sV = sV μ 0 ¶ μ ¶ StS 2@3 qS 2@3 Then, equation (2.27) gives 5@3 3@5 2@5 2@5 we divide both sides by k , and showing functional dependence of D and S on k: qT (10 + 4=472 kq) (10 + 4=472 kq) kq+1 = =6=948 = T (D(k)@k)5@3 1 (D(k)@k)5@3 sV × 10 + 2 k × 10 + 2 k s s μ 0¶ q q 5@3 = nSt V 2@3 = V 2@3 = k S (k) q S (k) With k0 =1=745, k1 =1=629, k2 =1=639, k3 =1=638 m, and the method has converged. The term D(k)@k is approximately the width of the channel, which for many channels varies little 38 40 3. Froude number Near-constancy of Froude number in a stream William Froude (1810-1879, pronounced as in "food") was a naval architect who proposed It is interesting to calculate the Froude number F of a steady uniform Àow given by the Chézy- similarity rules for free-surface Àows. A Froude number is a dimensionless number from a velocity Weisbach formula for discharge: scale X andalengthscaleO, F = X@sjO= In the original de¿nition, of a ship in deep water, the jD3 only length scale was O, the length of the ship. In river engineering it is not obvious what the T = V= r \S length scale is. Might it be the wetted perimeter S , might it be the geometric mean depth D@E, Immediately this gives where D is cross-sectional area and E is surface width? T2E V E F2 = = > In fact, the answer will usually depend on the problem. If we consider the mean total head of a jD3 \ S channel Àow at a section, where X is mean velocity and  is surface elevation: and as E S for wide channels, we see that the square of the Froude number is approximately  X 2 equal to the ratio of bed slope V to resistance coef¿cient \, giving some signi¿cance and physical K =  +  > 2j feeling for \. This means that for a particular reach of river, where slope V is effectively independent of Àow, where E@S also does not vary much with the Àow and \ often does not vary it does not appear explicitly. Neither does it appear in the momentum Àux P at a section much, the Froude number F does not change much with Àow. While a Àood might look more ¯ 2 P = j Dk + X D = dramatic than a more-common low Àow, because it is faster and higher, the Froude number is Here,wearegoingsimplytode¿ne it in terms¡ of a vertical¢ length scale, the depth scale D@E: roughly the same for both. X 2 T2E F2 = = > (3.1) jD@E jD3 where T = XD is discharge. For many years the lecturer was more speci¿c, using terms of kinetic and potential energy, now all he says is that F2 is a measure of dynamic effects relative to gravitational, the latter measured by mean depth. 41 43

Even if we were to consider rather more complicated problems such as the unsteady propagation of 4. The effect of obstructions on streams – an approximate waves and Àoods, and to non-dimensionalise the equations, we would ¿nd that the Froude number F itself never appears in the equations, but always as F2 or F2, depending on whether energy or method momentum considerations are being used. Structures such as weirs can almost completely block Flows which are fast and shallow have large Froude numbers, and those which are slow and deep a river, but there are also other types of obstacles have small Froude numbers. Generally F2 is an expression of the wave-making ability of a Àow, that are only a partial blockage, such as the piers and in conversation we usually use “high/ low Froude number” as an expression of how fast a Àow of a bridge, blocks on the bed, Iowa vanes, etc. or 1 is. For example, consider a river or canal which is 2mdeep Àowing at 0=5ms (makesomeeffort possibly more importantly, the effects of trees placed to imagine it - we can well believe that it would be able to Àow with little surface disturbance!). in rivers (”Large Woody Debris”), used in their We have environmental rehabilitation. It might be important X 0=5 to know what the forces on the obstacles are, or, F = =0=11 and F2 =0=012 > sjG  s10 2 more importantly for us, what effects the obstacles × and we can imagine that the wavemaking effects are small. Now consider Àow in a street gutter River Traun, Bad Ischl, Oberösterreich have on the river. Here we set up the problem in 1 after rain. The velocity might also be 0=5ms , while the depth might be as little as 2cm.The conventional open channel theoretical terms. Then, Froude number is however, we obtain an analytical solution by making X 0=5 the approximation that the effects of the obstacle are F = =1=1 and F2 =1=2 > sjG  s10 0=02 small. This gives us more insight into the nature and and we can easily imagine it to have many× waves and disturbances on it due to irregularities in the importance of the problem. gutter.

42 44 The physical problem and its idealisation turbulence in the inertial momentum Àux by writing the integral on the right of equation (4.1)

Surface if no obstacle: slowly-varying Àow 2 2 2 Surface along axis and sides of obstacle x gD = x¯ + x0 gD= (4.3) Mean of surface elevation across channel ZD ZD ³ ´ { Usually we do not know the nature of the turbulence structure, or even the actual velocity distribution across the Àow, so that we approximate this in a simple sense such that we write for the integral in space of the time mean of the squared velocities: T 2 T2 2 2 2 ¯ 2 x gD = x¯ + x0 gD X D =  D =  = (4.4) 1 2  D D Z Z μ ¶ (a) The physical problem, longitudinal section showing backwater { D D ³ ´ at obstacle decaying upstream to zero The coef¿cient  is called a Boussinesq coef¿cient, after the French engineer who introduced it to allow for the spatial variation of velocity. Allowing for the effects of time variation, { turbulence, has been a recent addition.  has typical values of 1=05 to something like 1=5 or more in channels of irregular cross-section. Almost all textbooks introduce this quantity for open 1 2 channel Àow (without turbulence) but then assume it is equal to 1. In this course we consider it important and will include it. 12 In equation (4.1), collecting contributions (4.2) and (4.4), we have the expression for the momentum (b) The idealised problem, uniform channel with no friction or slope Àux at a section Figure 4.1: A typical physical problem of Àow past a bridge pier, and its idealisation for hydraulic purposes T2 P =  jDk¯ +  = (4.5) D μ ¶ 45 47

Momentum Àux in a channel Momentum conservation The momentum Àux across a section is de¿ned to be the sum of the pressure force, plus the mass Consider the momentum conservation equation if a force S isappliedinanegativedirectiontoa rate of transport multiplied by the velocity. For a vertical section, the mass rate of transport is Àow between two sections 1 and 2: x dD, so the momentum Àux is T2 T2 S =  jDk¯ +   jDk¯ +  > (4.6) D  D P = s + x2 dD= μ ¶1 μ ¶2 Usually one wants to calculate the effect of the obstacle on water levels. The effects of drag can be Z D ¡ ¢ estimated by knowing the area of the object measured transverse to the Àow, d, the drag coef¿cient Substituting the hydrostatic pressure distribution, s = j ( }),where is the free surface F ,andx, the mean Àuid speed past the object:  D elevation, we obtain 1 S = F x2d> (4.7) 2 D P =  j ( })+x2 gD= (4.1)  and so, substituting into equation (4.6) gives, after dividing by density, Z D ¡ ¢ 2 2 The integral ( }) gD is simply the ¿rst moment of area about a transverse horizontal axis 1 2 ¯ T ¯ T • D  FDx d = jDk +  jDk +  = (4.8) at the surface, we can write it as 2 D  D R μ ¶1 μ ¶2 ¯ We consider the velocity x on the obstacle as being proportional to the upstream velocity, such that D( }) gD = Dk (4.2)  we write where k¯ is the depth of the centroid of the section below the surface. R T 2 The Àuid inertia contribution x2 gD: although it has not been written explicitly, it is x2 = K > (4.9) D D • understood that equation (4.1) is evaluated in a time mean sense. In equation (1.2) we saw that μ 1¶ where K is a coef¿cient which recognises that the velocity which impinges on the object is if a Àow is turbulent, then x2 =¯R x2 + x 2, such that the time mean of the square of the velocity 0 generally not equal to the mean velocity in the Àow. For a small object near the bed, K could be is greater than the square of the mean velocity. In this way, we should include the effects of quite small; for an object near the surface it will be slightly greater than 1; for objects of a vertical 46 48 scale that of the whole depth, K 1. Equation (4.8) becomes approximation for the dimensionless drop across the obstacle {@(D @E ),whereD @E is the  2 2 2 2 1 T2 T2 T2 mean downstream depth: K F d = jDk¯ +  jDk¯ +  (4.10) D 2 { 1 K F F2 2 D1 D 1  D 2  2 D 2 d2 μ ¶ μ ¶ = 2 = (4.11) A typical problem is where the downstream water level is given (sub-critical Àow, so that the D2@E2 1 F D2  2 control is downstream), and we want to know by how much the water level will be raised upstream This explicit approximate solution has revealed the important quantities of the problem to us and 2 2 3 if an obstacle is installed. As both D1 and k1 are functions of k1, so that we would need to know in how they affect the result: downstream Froude number F2 = T E2@jD2 and the relative blockage 2 detail the geometry of the stream, and then to solve the transcendental equation for k1.However, area d2@D2. For subcritical Àow F2 ? 1 the denominator in (2.27) is positive, and so is {,so 2 by linearising the problem, solving it approximately, we obtain a simple explicit solution that tells that the surface drops from 1 to 2, as we expect. If the Àow is supercritical, F2 A 1,we¿nd { 2 us rather more. negative, and the surface rises between 1 and 2. If the Àow is near critical F2 1, the change in depth will be large, which is made explicit, and the theory will not be valid.  E

e We could immediately estimate how important this is. We see that, for small Froude number { 2 2 2 1 F2 1, such that 1 F2 1, the relative change of depth is equal to 2 times K 1 (for a D1 ¿   2  body extending the whole depth), times FD 1 for cylinders etc=, multiplied by F2, usually small, D2 k¯2 k¯1  multiplied by the blockage ratio d2@D2, which is also probably small. So, the relative result is usually small. However, the absolute value might still be ¿nite compared with resistance losses, as   d1 1 2 will be seen below. d2 Another bene¿t of the approximate analytical solution is that it shows that such an obstacle forms a control in the channel, so that the ¿nite sudden change in surface elevation { is a function of 2 Figure 4.2: Cross-section showing dimensions for water levels at 1 and 2 T ,orT a function of s{, in a manner analogous to a weir. In numerical river models it should ideally be included as an internal boundary condition between different reaches as if it were a type of ¿xed control. 49 51

Consider the stream cross-section shown in Fig. 4.2, with a small change in water level The mathematical step of linearising has revealed much to us about the nature of the problem that 1 = 2 + {. We now use geometry to obtain approximate expressions for quantities at 1 in terms the original momentum equation did not. of those at 2. It is easily shown that Example 2 It is proposed to build a bridge, where the bridge piers occupy about 10% of the "wetted D = D + E { + R ({)2 > and Dk¯ = Dk¯ + D { + R ({)2 > 1 2 2 1 2 2 area" of a river with Froude number 0=5 (which is quite large). How much effect will this have on ³ ´ ¡ ¢ ¡ ¢ 2 ³ ´ the river level upstream? and similarly we write for the blockage area d1 = d2 + e2 { + R ({) ,wheree2 is the surface As the bridge piers occupy all the depth, we have K =1. A typical drag coef¿cient is FD 1.We width of the obstacle (which for a submerged obstacle would be zero).³ We´ have actually been using  Taylor series expansions, but the physical interpretations seem simpler than the mathematical! will use  =1(this is an estimate!). So we ¿nd, using equation (2.27): { 1 K F F2 d The momentum equation (4.10) gives us = 2 D 2 2 D @E 2 D 1 T2 T2 T2 2 2 1 F2 2 K F d = jD { +   =  2 D 2 2 2 1 0=5 2 D1 D2 + E2 {  jD2 1 1 0=1  2 × × × 1 0=52 × Now we use a power series expansion in { to simplify the term in the denominator: =0=017,  1 1 1 1 1 = = (1 + E2 {@D2) (1 E2 {@D2) > about 2% of the mean depth. This seems small, but if the river were 2mdeep, there is a 4cmdrop D2 + E2 { D2 (1 + E2 {@D2) D2  D2  4 across the bridge. If the slope of the river were V =10 , this would correspond to the surface neglecting terms like ({)2 (see equation A-1). The momentum equation becomes level change in a length of 400 m, which can hardly be neglected. 2 2 1 T T E2 K FD d2 {D2 1  = 2 jD2   jD3 1 μ 2 ¶ 2 2 3 As D1 = D2 + R ({) we replace D1 by D2 and introduce F2 = T E2@jD2, the square of the Froude number of the downstream Àow. The equation is easily solved to give an explicit 50 52 5. Reservoir routing Numerical solution of the differential equation by Euler’s method Consider the problem shown in ¿gure 5.1, Euler’s method is the simplest (but least-accurate) of all methods, being of ¿rst-order accuracy L(w) Surface Area D + {D D where a generally unsteady inÀow rate L(w) only. For river engineering purposes it is usually quite good enough. However there is a good  enters a reservoir or a storage tank, and we method for making it more accurate, which we will use. Euler’s method is to approximate the Surface Area D have to calculate what the outÀow rate T(w) derivative in a differential equation at a time step l by a forward difference expression in terms of a is, as a function of time w. The action of the time step {, here applying it to equation (5.1): } =  + { reservoir is usually to store water, and to d l+1 l L(wl) T(l>wl) release it more slowly, so that the outÀow  =  > T((w)>w) dw l  { D(l) } =  is delayed and the maximum value is less ¯ giving the scheme to calculate the¯ value of  at wl+1 as than the maximum inÀow. Some reservoirs, ¯ ¯ L(w ) T( >w) Figure 5.1: Reservoir or tank, showing surface level varying notably in urban areas, are installed just  =  + { l l l + R {2 > (5.2) with inÀow, determining the rate of outÀow l+1 l  for this purpose, and are called detention D(l) reservoirs or storages. The procedure of solving the problem is also called Level-pool Routing. whereweusethenotationl for the solution at time step l.Wehaveshownthattheerrorofthis¡ ¢ approximation is proportional to {2. It is necessary to take small enough { that this is small. InÀow L The process is shown in ¿gure 5.2. When a Àood comes down the river, inÀow increases, the water level rises in the reservoir until at Accurate results with simple methods – Richardson extrapolation the point O when the outÀow over the spillway now balances the OutÀow T inÀow. At this point, outÀowandsurfaceelevationinthereservoir We introduce a clever device for obtaining more accurate solutions from Euler’s method and others. have a maximum. After this, the inÀow might reduce quickly, but Consider the numerical value of any part of a computational solution for some physical quantity ! it still takes some time for the extra volume of water to leave the obtained using a time or space step {, such that we write !({). Let the computational scheme be reservoir. of known qth order such that the global error of the scheme at any point or time is proportional to Time Figure 5.2: 53 55

It is simple and obvious to write down the relation- {q,thenif!(0) is the exact solution, we can write the expression in terms of the error at order q: ship stating that the rate of surface rise d@dw is !({)=!(0) + e{q + === > (5.3) equal to the net rate of volume increase divided by surface area: where !(0) is the solution for a vanishingly small time step, so that it should be exact. The e is an q+1 d L(w) T(> w) unknown coef¿cient; the neglected terms vary like { . If we have two numerical simulations or =  > (5.1) dw D() approximations with two different {1 and {2 giving numerical values !1 = !({1) and !2 = !({2) then we write (5.3) for each: where  is the free surface elevation, and D() is q the surface area, possibly given from planimetric !1 = !(0) + e{1 + === > q information from contour maps, and T(> w) is the !2 = !(0) + e{2 + === = Detention reservoir in a public park volume rate of outÀow, which is usually a simple These are two linear equations in the two unknowns !(0) and e. Eliminating e, which is not in Melbourne, Australia function of the surface elevation , from a weir or important, between the two equations and neglecting the terms omitted, we can solve for !(0),an 1@2 3@2 gate formula, usually involving terms like ( }outlet) and/or ( }crest) ,where}outlet is the   approximation to the exact solution: elevation of the pipe or tailrace outlet to atmosphere and }crest is the elevation of the spillway crest. ! uq! There might be extra dependence on time w if the outÀow device is opened or closed. This is a !(0) = 2  1 + R {q+1> {q+1 > (5.4) 1 uq 1 2 differential equation for the surface elevation itself. The procedure of solving it is called Level-pool  where u = { @{ . The errors are now proportional¡ to step size to¢ the power q +1,sothatwe Routing. 2 1 have gained a higher-order scheme without having to implement any more sophisticated numerical The traditional method of solving the problem, described in almost all books on hydrology, is to methods, just with a simple numerical calculation. This procedure, where q is known, is called use an unnecessarily complicated method called the “Modi¿ed Puls” method of routing, which Richardson extrapolation to the limit. solves a transcendental equation for a single unknown quantity, the volume in the reservoir, at 1. For simple Euler time-stepping solutions of ordinary differential equations, q =1,andifwe each time step. It is simpler and more fundamental to treat the problem as a differential equation (Fenton 1992). :-) 54 56 perform two simulations, one with a time step { and then one with {@2,wehave point. Next, applying Richardson extrapolation, equation (5.5), gave the results shown by the solid points. They almost coincide with the accurate solution, and cross the inÀow hydrograph with an !(w> 0) = 2 !(w> {@2) !(w> {)+R {2 > (5.5)  apparent horizontal gradient, as required, whereas the less-accurate results do not. Overall, it seems where the numerical solution at time w has been shown as a function¡ ¢ of the step. This is very that the simplest Euler method can be used, but is better together with Richardson extrapolation. In simply implemented. fact, there was nothing in this example that required large time steps – a simpler approach might 2. For the evaluation of an integral by the trapezoidal rule, q =2. have been just to take rather smaller steps. 3 1 3 1 The role of the detention reservoir in reducing the maximum Àow from 20 m s to 14=7ms is Example 3 Consider a small detention reservoir, square in plan, with dimensions 100m by 100m, clear. If one wanted a larger reduction, it would require a larger spillway. It is possible in practice with water level at the crest of a sharp-crested weir of length of e =4m, where the outÀow over that this problem might have been solved in an inverse sense, to determine the spillway length for a the sharp-crested weir can be taken to be given maximum outÀow. T ()=0=6sje3@2> (5.6) 2 where j =9=8ms . The surrounding land has a slope (V:H) of about 1:2, so that the length of a reservoir side is 100 + 2 2 ,where is the surface elevation relative to the weir crest, and × × D() = (100 + 4)2 . The inÀow hydrograph is: w 5 1 w@Wmax L(w)=Tmin +(Tmax Tmin) h  > (5.7)  W μ max ¶ where the event starts at w =0with Tmin and has a maximum Tmax at w = Wmax. This general form of inÀow hydrograph mimics a typical storm, with a sudden rise and slower fall, and will be used in other places in this course. In the present example we consider a typical sudden local storm event, 57 59

3 1 with Tmin =1ms at Wmax = 1800 s. 6. The one-dimensional equations of river hydraulics

In ow These are the fundamental equations that are used to describe the propagation of Àoods and 20 Out ow — accurate disturbances in rivers. They are called the long wave equations,theshallow water equations,or Euler — step 200s Euler — step 100s the Saint-Venant equations, and are mass and momentum conservation equations for water. Richardson extrapn 15 The equations are a pair of partial differential equations in the independent variables { (distance along the stream) and w (time). A typical Àood routing problem is for large extra values of discharge Discharge T to be introduced at the upstream initial point, and then for a number of time steps, to solve the 3 1 m s 10 equations along the channel to obtain the progress of the Àood at each time. ¡ ¢ We will also consider a mass conservation equation for soil. In their steady form, the equations describe how water level and velocities vary along a stream, and what effects boundary changes 5 such as sand removal might have on Àooding. We make the traditional approximation that all rivers are straight. Later we will see that it is quite 0 0 1000 2000 3000 4000 5000 6000 accurate, even for meandering streams. Time (sec) The model is one-dimensional. We do not consider details of motion in the plane across the stream Figure 5.3: Computational results for the routing of a sudden storm through a small detention reservoir – all quantities are averaged across it. This does not mean that we assume they are constant. The problem was solved with an accurate 4th-order Runge-Kutta scheme, and the results are shown This approach requires surprisingly few approximations – the model is a good simple model of as a solid blue line on ¿gure 5.3, to provide a basis for comparison. Next, Euler’s method (equation complicated reality. 5.2) was used with 30 steps of 200 s, with results that are barely acceptable. Halving the time step It is easier to use cartesian co-ordinates, for which we use { the horizontal distance along the to 100 s and taking 60 steps gave the slightly better results shown. It seems, as expected from stream, | the horizontal transverse co-ordinate, and } the vertical, relative to some arbitrary origin. knowledge of the behaviour of the global error of the Euler method, that it has been halved at each 58 60 6.1 Mass conservation of water and soil Upstream Volume

l{{ Consider Figure 6.1, showing an elemental slice The mass conservation equation (6.1) suggests the introduction of a function Y ({> w) which is the of channel of length {{ with two stationary volume upstream of point { at time w, such that for the channel Àow vertical faces across the Àow. It includes two CY CY { {{ = D and = l ({0) d{0 T= (6.3) T different control volumes. The free surface and C{ Cw  Z the interface between them may move. The The derivative of volume with respect to distance { gives the area, as shown, while the time rate of D surface shown by solid lines contains water and D + {D T + {T change of volume upstream is given by the rate at which the volume is increasing due to inÀow, possibly suspended soil grains. The surface minus the rate at which volume is passing the point and therefore no longer upstream. Substituting Tb shown by dotted lines contains the soil moving for D and T into equation (6.1): } Db as bed load and extends down into the soil such { Db + {Db Tb + {Tb C CY C CY C that there is no motion at its far boundaries. + + l ({0) d{0 = l> | Cw C{ C{  Cw C{ Each is modelled separately. We assume that the μ ¶ μ ¶ Z density of the Àuid (water plus suspended soil which is identically satis¿ed. By introducing Y we automatically satisfy one of the two { particles) is constant, so that we can consider conservation equations and reduce the number of unknowns from two (D and T) to one (Y ). volume conservation. Sometimes this can be very helpful. Figure 6.1: Elemental length of channel showing control volumes On the upstream vertical face at any instant, In the case of the bed load, a similar quantity Yb({> w) can be introduced such that the mass there is a volume Àux (rate of volume Àow) T, conservation equation (6.2) is identically satis¿ed. and that on the downstream face is T + {T,sothat Use of surface elevation instead of cross-sectional area CT Net volume Àow rate of Àuid leaving across vertical faces = {T = {{ + terms in ({{)2 . C{ We usually work in terms of water surface elevation (“stage”) , which is easily measurable and If rainfall, seepage, or tributaries contribute an inÀow volume rate l per unit length of stream, the which is practically more important. We make a signi¿cant assumption here, but one that is usually 61 63 volume Àow rate of this other Àuid entering the control volume is l {{. Ifthesumofthetwo accurate: the water surface is horizontal across the stream. Now, if the surface changes by an contributions is not zero, then volume of Àuid is changing inside the elemental slice, so that the amount  in an increment of time w, then the area changes by an amount D = E,whereE water level will change in time. The rate of change with time w of Àuid volume is CD@Cw {{. is the width of the stream surface. Taking the usual limit of small variations in calculus, we obtain For volume to be conserved (mass, but we assume the water is incompressible) this is equal× to CD@Cw = EC@Cw, and the mass conservation equation can be written the net rate of Àuid entering the control volume, dividing by {{ and taking the limit as {{ 0 C CT $ E + = l= (6.4) gives Cw C{ CD CT + = l= (6.1) The discharge T could be written as T = XD,whereX is the mean streamwise velocity over a Cw C{ section, and substituted into this. However, the discharge is more practical and fundamental than This is the mass conservation equation. Remarkably for hydraulics, it is almost exact. The only the velocity, and that will not be done here. approximation has been that the channel is straight. It is also linear in the two dependent variables D and T. 6.2 Momentum conservation equation for channel Àow

The composite bulk density b of the bed load composed of larger soil particles and is assumed to The equation be constant. The bed has a cross-sectional area D , the bulk volumetric Àow rate is T , and there is b b The conservation of momentum principle is now applied to the mixture of water and suspended an inÀow of mass rate p per unit length, possibly due to deposition or erosion. Mass conservation l solids in the main channel for a moving and deformable control volume (White 2003, §§3.2 & is calculated following the same reasoning as for the channel, giving Exner’s1 equation: 3.4). The {-component is CD CT p b + b = l= (6.2) d Cw C{  x dY + x ur nˆ dV = S{> (6.5) b dw · The volume transport rate used here is the bulk Àow rate; it is related to the volume Àow rate of CVZ CSZ solid matter Ts used in transport formulae, by Ts = Tb (1 *),where* is the porosity. Unsteady term Fluid inertia term  where u is the Àuidvelocitywith{-component x,dY is an element of volume, ur is the velocity | {z } | {z } 1 https://de.wikipedia.org/wiki/Felix_Maria_von_Exner-Ewarten – Austrian – Director of the Zentral Anstalt vector of the Àuid relative to the local element of the control surface, which is possibly moving für Meteorologie und Geodynamik 62 64 itself, nˆ is a unit vector with direction normal to and directed outwards, and dV is an elemental will consider only the {-component of the momentum equation, which have chosen to be area of the surface. The quantity u nˆ is the component of relative velocity normal to the surface at horizontal, as gravity only has a component in the } direction, there will be no contribution r  any point. It is this velocity that is responsible· for the transport of any quantity across the surface, from gravity to our equation! The manner in which gravity acts is to cause pressure gradients momentum here. S{ is the force exerted on the Àuid in the control volume by both body forces, in the Àuid, giving rise to the following term, due to pressure variations around the control which act on all Àuid particles, and surface forces which act only on the control surface. surface. Hydraulic approximations b. Pressure forces: these act normally to the control surface. The direction of the pressure force on the Àuid at the control surface is nˆ,wherenˆ is the outward-directed normal; its local 1. Unsteady term magnitude is s dV,wheres is the pressure and dV an elemental area of the control surface. The element of volume is dY = {{ dD, and the term contribution can be written Hence, the total pressure force is the integral CS snˆ dV. Thisisdif¿cult to evaluate d CT for arbitrary control surfaces, as the pressure and the non-constant unit vector have to be  {{ x dD =  {{ > (6.6) dw Cw integrated over all the faces. A much simpler derivationR is obtained if the term is evaluated ZD using Gauss’ Divergence Theorem: where the integral D x dD has a simple and practical signi¿cance – it is just the discharge T, so that the contribution of the term can be written simply as shown, but where again it has been snˆ dV = s dY> R   u necessary to use the partial differentiation symbol, as T is a function of { as well. No additional CSZ CVZ approximation has been made in obtaining this term. It can be seen that the discharge T plays a where s =(Cs@C{>Cs@C|>Cs@C}), the vector gradient of pressure. This has turned a simple role in the momentum of the Àow. complicatedu surface integral into a volume integral of a simpler quantity. 2. Fluid inertia term We only need the { component Cs@C{dY , the volume integral of the streamwise  CV The second term on the left of equation (6.5) is x ur nˆ dV, has its most important pressure gradient. The hydraulic approximation now has a problem, because we have not CS · attempted to calculate the detailedR pressure distribution throughout the Àow. However, in contributions from the stationary vertical faces perpendicular to the main Àow. R most places in most channel Àows the length of disturbances is much greater than the depth, a. Top and bottom, possibly moving surfaces: we have chosen our control surface to coincide so that streamlines in the Àow are only very gently sloping and gently curved, and the 65 67

with these boundaries so that no Àuid crosses them, ur nˆ =0and there is no contribution. pressure in the Àuid is accurately given by the equivalent hydrostatic pressure, that due to a · stationary column of water of the same depth. Hence we write for a point of elevation }, our b. Stationary vertical faces: on the upstream face, ur nˆ = x, giving the contribution to the term of  x2 dD. The downstream face at {·+ {{has a contribution of a similar equation (1.6) gives  D nature, but positive, and where all quantities have increased over the distance {{. The net s = j Depth of water above point = j( })> R 2 ×  contribution, the difference between the two, after neglecting terms like ({{) , can be written where  is the elevation of the free surface above that point. The quantity that we need is the C  {{ x2 dD= horizontal pressure gradient Cs@C{ = j C@C{, and so the streamwise pressure gradient is C{ entirely due to the slope of the free surface. The contribution is ZD Cs C C In equation (4.4), much earlier, we approximated the integral over the cross section and dY  {{j dD  {{jD > (6.8) with a mean in time, in terms of a Boussinesq coef¿cient  such that the contribution to the  C{  C{  C{ ZD ZD equation is then simply but approximately. where any variation with | has been ignored, as the surface elevation usually varies little C T2 across the channel, and so C@C{ is constant over the section and has been able to be taken  {{  = (6.7) C{ D outside the integral, which is then simply evaluated. μ ¶ It is useful to retain the , unlike many presentations that implicitly assume it to be 1.0, as it c. Resistance due to shear: there is little that we can say that is exact about the shear forces. is a signal and reminder to us that we have introduced an approximation. We have already considered resistance in some detail, however, and in equation (2.22) we we c. Lateral momentum contributions: If there is also Àuid entering or leaving from rainfall, have 2 2 tributaries, or seepage, there are contributions over the other faces. Their contribution to T ({> w) 1 T ({> w) = \X 2 = \ = > momentum exchange is small and uncertain and we will ignore them.  D ({> w) 2(%) D ({> w) μ ¶ μ ¶ where we could use a value of \ from the ¿gure on page 34 or from the formulae given 3. Contributions to force S{ a. Body force: for the straight channel considered, the only body force acting is gravity; we therein, or we could use our result for the Gauckler-Manning-Strickler formula where (%)=6=7@%1@6= The value of is the mean around the solid boundary, so to obtain the force 66 68 we multiply by the wetted perimeter S and length of the element {{ and instead of T2 we is given by the hydrostatic pressure corresponding to the depth of water above each point. write T T , to allow for possible negative T in estuaries, as resistance always opposes the 3. Effects of curvature of the stream course are ignored.  | | motion: 4. In the momentum Àux term the effects of both non-uniformity of velocity over a section and T T Total horizontal shear force on control surface =  {{ \ | |S= (6.9) turbulent Àuctuations are approximated by a momentum or Boussinesq coef¿cient. D2  5. Surface elevation  across the stream is constant. Collection of terms and discussion Relating surface slope C@C{ and CD@C{ Now all contributions from the hydraulic approximations to terms in equation (6.5) are collected, E In the momentum equation (6.10) when using equations (6.6), (6.7), (6.8), and (6.9), and bringing all derivatives to the left and others to expanded, the dependent variables are At { the right, all divided by {{, gives the momentum equation: { At { + {{ discharge T and a mixture of derivatives CT C T2 C T T of area CD@C{ and surface elevation +  + jD = \S | |= (6.10) } D Cw C{ D C{  D2 C@C{. We must relate the two, and now μ ¶ It is convenient to generalise the resistance term so as to be able to incorporate Gauckler-Manning- {] consider the bottom geometry in greater Strickler resistance as well as situations where a Rating Curve is known from river measurements, | detail, although in practice the precise giving a relationship between measured discharge, supposed steady and uniform, and local details of the bed are often not known. cross-sectional area, Tr(D). We write the equation as Figure 6.2: Two channel cross-sections separated by {{ This will help us know when to make CT C T2 C approximations. +  + jD = lT T > (6.11) Cw C{ D C{  | | The cross-section of a river in Figure 6.2 shows how ambiguous and possibly non-unique the μ ¶ concept of the “bottom” of the stream may be. In a distance {{ the surface elevation may change where the coef¿cient l, of dimensions L 3, is a function of resistance coef¿cient, cross-sectional  by an amount { as shown, so that the contribution to the change in cross-section area {D is E {,where{ is usually negative as the surface drops downstream. The change in the bed is 69 × 71 area, and wetted perimeter. {], which in general varies across the section, with contribution to {D of {]g|,thearea  E jDV@T˜ 2(D)> in terms of rated discharge T (D); between the solid and dotted lines on the ¿gure corresponding to the bed at the two locations. The r r R l = \S@D2> Chézy-Weisbach, where \ = @8=j@F2; (6.12) minus sign is because, if the bed drops away and {] is negative, as usual, the contribution to area ; 2 4@3 7@3 4@3 2 7@3 increase is positive. Combining the two terms, ? q jS @D = jS @nStD > Gauckler-Manning-Strickler= {D = E { {] d| (6.14) =  Example 4 Verify the use of the three resistance forms for steady uniform Àow, on a uniform slope ZE V˜ = V. For the second contribution, the integral of the change in bed elevation across the stream, we introduce the symbol V˜ for the mean downstream bed slope across the section such that In this case, the Àow is steady so the ¿rst term in equation (6.11) is zero, and uniform so that 1 C] the second is zero. The surface slope C@C{ = V,andasT is positive, T T = T2 and the V˜ = d|> (6.15) momentum equation (6.11) gives jDV = lT2,sothat | | E C{   ZE jDV 2 where the negative sign has been introduced such that in the usual case when ] decreases with {, jDVTr = Tr> in terms of rated discharge Tr(D); jDV this mean downstream bed slope at a section is positive. In an important problem where bed details T = = ; qjD3V j D (6.13) \S = D \ S V> Chézy-Weisbach; might be known, this can be evaluated. In the usual case where bed topography is poorly known, a r l A ?A q jDV 7@3 1 D 2@3 reasonable local approximation or assumption is made. Using equations (6.14) and (6.15) we can 2 4@3 D q= D sV> Gauckler-Manning-Strickler= q jS q S write A q =A ¡ ¢ {D = E { + EV˜ {{> At this stage the non-trivial assumptions in the derivation are stated, roughly in decreasing order of where in a distance {{ the mean bed level across the channel then changes by V˜ {{ under the importance (they are actually not very restrictive at all!): water. In the rare case where the sides of the stream are vertical diverging or converging × walls, an 1. Resistance to Àow is modelled empirically. The Navier-Stokes equations are not being used. 2. All surface variation is suf¿ciently long and of small slope that the pressure throughout the Àow 70 72 extra term would have to be included. Taking the usual calculus limit, we obtain 6.4 Examples of Àood propagation CD C = E +V˜ > (6.16) C{ C{ In ow μ ¶ 10 which might have been able to have been written down without the mathematical details. nSt =67, F0 =0=46 Long wave eqns

8 nSt =20, F0 =0=15 Long wave eqns

6 2 1 T@E m s ¡ ¢ 4

2

0 0 1224364860 Time (h) As an example we consider an in¿nitely-wide (no side friction) channel with a channel slope V =0=0005 and length 50 km. Two different boundary resistances were considered, Strickler nSt =67for a smooth boundary to give a large Froude number and nSt =20for a natural boundary.

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6.3 Forms of the governing equations 6.5 Hierarchy of one-dimensional open channel theories and approximations We use equation (6.16) to eliminate ¿rst C@C{ and then CD@C{ to give two alternative forms of Real stream Boussinesq approximations, the momentum equation governing Àows and long waves in waterways. In both cases, we restate non-hydrostatic, can Assume pressure hydrostatic describe transition between Two-dimensional equations, sub- and super-critical Àow the corresponding mass conservation equation, using (6.1) and (6.4), to give the pairs of equations: possibly including moment of momentum, to include secondary Àows, not yet established Formulation 1 – Long wave equations in terms of area D and discharge T Vertically averaged Two-dimensional equations, no Eliminating C@C{ gives the equations in terms of D and T: secondary Àows Assume stream curvature small CD CT Assume stream straight, no cross-stream variation + = l> (6.17a) One-dimensional long wave One-dimensional long wave equations in Cw C{ equations for curved streams (D> T) (eqns 6.17) or (>T) (eqns 6.18) 2 CT C T jDCD Characteristic formulation, §6.12, +  + = jDV˜ lT T = (6.17b) Misleading outmoded results Finite-difference-based Cw C{ D E C{  | | numerical methods §6.6, Characteristic-based Preissmann Box Scheme μ ¶ numerical models Formulation 2 – Long wave equations in terms of stage  and discharge T Assume small disturbances Assume disturbances very long

Linearised Telegrapher equation, §6.8 Slow change routing equation §6.9, simple reveals nature of propagation of equation & computational method Now eliminating CD@C{, but retaining D in all coef¿cients, as it can be calculated in terms of : disturbances C 1 CT l Both assumptions: small and very long disturbances + = > (6.18a) Linearised, and slow-change/slow-Àow, Cw E C{ E Advection-diffusion equation, §6.10, 2 2 reveals nature of most disturbances CT TCT T E C T E Assume diffusion small +2 + jD  =  V˜ lT T = (6.18b) Cw D C{  D2 C{ D2  | | Interchange of time & space μ ¶ differentiation, Muskingum routing §6.11 These equations are the basis of computational hydraulics and Àood routing. There is much Assume diffusion zero The very long wave equation commercial software written to solve them. They are actually quite simple in the form here! 74 76 6.6 Numerical solution of the long wave equations – FTQS scheme 6.7 Initial and boundary conditions w Initial conditions wq+1 Upstream { Downstream inÀow Usually there is some initial Àow in the channel which is constant if there is no inÀow, wq control T = T0(w) {p 1 {p {p+1 T = I() T({> w0)=T0. The next step is to determine the initial distribution of surface elevation .The speci¿ed  w1  conventional method is to solve the Gradually-varied Àow equation, using the equations and methods described in §7, as well as the downstream boundary condition, which is about to be w0 { described. A simpler method is to use the unsteady equations and computation scheme that will be p =0 Initial conditions at w = w0, T({> w0) and ({> w0) speci¿ed p = P used later anyway – simply start with an approximate solution for ({> w0) (a straight line?) and let Figure 6.3: ({> w) axes showing computational grid, initial and boundary conditions, and three computational modules the unsteady dynamics take over, allowing disturbances to propagate downstream and out of the We use a scheme where time derivatives are approximated using forward differences, and where computational domain until the solution is steady. Then, for example, the main computation can {-derivatives are approximated using quadratic approximation, ¿tting a quadratic to three points, be started. giving the Forward-Time-Quadratic-Space scheme. The {-derivatives are

Ci 3i0 +4i1 i2 =   + R () > (6.19a) C{ 2 ¯0 Ci ¯ ip+1 ip 1 ¯ =   + R () > for p =1>===>P 1 > (6.19b) C{ ¯ 2  ¯p Ci ¯ iP 2 4iP 1 +3iP ¯ =    + R () = (6.19c) C{ ¯ 2 ¯P ¯ ¯ ¯ 77 79

Using the obvious forward difference expressions for the time derivatives, the scheme applied to Boundary conditions the (D> T) formulation, equations (6.17), becomes Upstream Dp>q+1 Dp>q CT  = l > (6.20a) {  C{ It is usually the upstream boundary condition that drives the whole model, where a Àood or wave ¯p>q ¯ 2 enters, via the speci¿cation of the time variation of T = T ({ >w) at the boundary. The surface Tp>q+1 Tp>q C ¯ T jDCD ˜ 0  = ¯ + jDV lT T = (6.20b) elevation there is obtained as part of the computations. A common model inÀow hydrograph is: { C{ D  E C{  | | p>q μ ¶ ¯ 5 ¯ w Both expressions are easily re-arranged to give explicit formulae for the terms in red, the values of 1 w@Wmax ¯ T ({0>w)=Tmin +(Tmax Tmin) h  > D and T at point p at the next time level q +1,eachintermsofthreevaluesof¯ D and three values  Wmax μ ¶ of T at the computational points at the previous time wq, using one of the equations (6.19). where the event starts at w =0with Tmin and has a maximum Tmax at w = Wmax. Liggett & Cunge (1975) claimed that the above scheme, the simplest and most obvious, was We have two variables, however, T and either D or . To obtain this we just use the mass unconditionally unstable. This had some important implications, for it meant that the world was conservation FTQS expression (6.20a) to obtain the updated value of D or  at p =0at wq+1.The forced into using complicated schemes such as the Preissmann Box scheme, which form the basis equation of course applies up to and including the boundary point. of all commercial software. The lecturer (Fenton 2014) has discovered that their analysis is wrong, and that the scheme has a quite acceptable stability limitation, and it opens up the possibility for simpler computations of Àoods and Àows in open channels. The Preissmann Box Scheme allows much larger time steps, but it is very complicated to apply.

78 80 Downstream boundary - known stage-discharge relationship 6.8 Nature of the equations and solutions Where there is a downstream control structure such as a spillway, weir, gate, or Àume, the • The Telegraph equation as a model for long wave propagation stage-discharge relationship T ({P >w)=I ( ({P>w)) must be known. For example, a weir might have a Àow formula such as We recall the long wave equations in terms of area, equations (6.17): 3@2 CD CT T =0=6sje ( }c) + = l>  Cw C{ 2 2 where e is the crest length and }c is the elevation of the crest. CT C T jDCD T +  + = jDV˜ 1 > We assume that the T = I() relationship is not affected by unsteadiness and non-uniformity, Cw C{ D E C{  T2(D) μ ¶ μ r ¶ • which probably holds for relatively short control structures mentioned where we have now written the resistance term in terms of the function of area Tr(D) which is A potential dif¿culty – we have one equation too many: we have the FTQS ¿nite difference the Rating Curve relationship, or that given by, say, the Gauckler-Manning-Strickler formula. For • formulae based on mass conservation for  ({P >wq+1) and momentum conservation for steady uniform Àow such that all derivatives on the left are zero, the equation becomes T = Tr(D), T ({P >wq+1) and the relation between T and  which we require. We also recall the relationships (6.3) for the upstream volume: However, a sudden change in section where a typical spillway, weir, gate, or Àume is placed CY CY { • = D and = l ({0) d{0 T= (6.21) actually violates a fundamental assumption of the long wave momentum equation, that variation C{ Cw  Z in the channel is long. We can easily ignore that equation near such a sudden change Substituting these into the mass conservation equation, as we did in §6.1.1, we ¿nd that it is Fortunately, the mass conservation equation, is still valid near a sudden change – it requires only identically satis¿ed (we expect volume to satisfy a volume conservation equation). The momentum • the assumption that water surface is horizontal across the channel. conservation equation becomes 2 2 2 2 2 The procedure is: obtain the updated value  ({P >wq+1) from the FTQS ¿nite difference formula C Y T C Y T jD C Y CY@Cw • +2 +  + jDV˜ 1 =0 for the mass conservation equation (6.20a), using values of T at {P 2, {P 1, {P and wq and Cw2 D C{Cw D2  E(D) C{2   T (D)   μ ¶ Ã μ r ¶ ! then use the stage-discharge relationship to calculate T ({P>wq+1)=I ( ({P >wq+1)) where symbols T and D have been retained in coef¿cients of second derivatives. 81 83

Open downstream boundary The momentum equation has become a second-order partial differential equation in terms of the • single variable Y . A common boundary is where the computational domain is arti¿cially truncated at some point • in the stream. This is sometimes called Normal Depth boundary and standard practice is that the And it is unusable in this ugly form. It is more useful in theoretical works and where • computational domain be arti¿cially extended and this boundary condition be used far enough approximations can be made, as we now do. downstream from the study area that it does not affect the results there. We linearise the equation by considering relatively small disturbances about a uniform Àow • The lecturer prefers a different approach, and this is simply to treat the boundary as if it were with area D0 and discharge T0. Substituting the series • just any other part of the river (which it is!) and to use both long wave equations to update Y = D0{ T0w + %y> D = D0 + %y{>T= T0 %yw> and Tr(D)=T0 + T0 %y{> both  and T there, calculating the necessary derivatives C@C{ and CT@C{ from upstream   where %y is a small quantity, a deviation of upstream volume from that of uniform Àow, ¿nite difference formulae and simply treating the end point as if it were an ordinary point in yw = Cy@Cw, y{ = Cy@C{,andT0 =dTr@dD 0. the stream and using both FTQS formulae for  ({P>wq+1) and T ({P >wq+1) there, with the | three-point leftwards approximations for the last point {P in terms of values at {P 2, {P 1, Performing power series operations, the second order derivatives can simply be written down with   and {P. constant coef¿cients. The gravity and resistance terms become, making use of the power series q This works very well in practice. expansion to ¿rst order, (1 + %d) =1+%qd + ===: • 2 2 T0 %yw 1 %yw@T0 j (D0 + %y{) V0 1  = jD0V0 1  ×  T + T %y ×  1+T @T %y à μ 0 0 {¶ ! à μ 0 0 {¶ ! = jD0V0 (1 (1 2%yw@T0)(1 2T0 @T0 %y{)) ×    0 = jD0V0 (1 (1 2%yw@T0 2T0@T0 %y{)) 2jD V× Cy  Cy  = % 0 0 + T = T Cw 0C{ 0 μ ¶ 82 84 We obtain the linearised momentum equation as the Telegraph equation: Very long waves – the longest Àood waves 2 2 2 Cy Cy C y C y 2 2 2 C y 2 2 0 + f0 + +2X0 (F  X ) =0= (6.22) For disturbances with a long period, such that C @Cw 0C@Cw, “very long waves”, the last Cw C{ Cw2 C{Cw  0  0 C{2 • ¿ μ ¶ three terms in the equation can be neglected, and it becomes the advection equation 0 – resistance parameter / inverse time scale: this is actually an important channel parameter, Cy Cy • determining the nature of wave behaviour and computational solution properties + f =0, (Very long wave equation) Cw 0C{ 2 2jD0V0 jV0 C T Solution: y = i1 ({ f0w) > (6.23) 0 = =2 = jDV  T X CT T2 0 0 μ r ¶¯0 where i1 (=) is an arbitrary function given by the upstream conditions. To show this consider a ¯ It is the derivative with respect to T of the resistance term in the¯ momentum equation. We moving variable [ = { f0w,andy = i1([).Bythechain rule for partial differentiation, could argue by a rough electrical analogy that the resistance term in¯ the momentum equation is  Cy Ci1([) gi 1([)C[ gi 1([) equivalent to potential difference or voltage, while discharge T is equivalent to current. As the = = = f0 > and Cw Cw g[ Cw  g[ derivative of voltage with respect to current gives electrical resistance, 0 can be thought of as a Cy Ci ([) gi ([)C[ gi ([) = 1 = 1 =1 1 > resistance parameter in our nonlinear case. C{ C{ g[ C{ × g[ f0 – wave speed: This will be shown to be the speed of very long period waves, which means and the equation is satis¿ed for any i1([), whatever the upstream conditions determine. • for us the propagation speed of Àood waves: This solution is a wave propagating downstream at speed f0 with no change or diffusion. • dTr f = = The equation has been known as the “kinematic wave equation” and f0 the “kinematic wave 0 dD • ¯0 speed”, because the approximation has previously been believed to be such that dynamic terms ¯ 5@3 2@3 2 For the Gauckler-Manning-Strickler equation, Tr¯ = nStD @S sV, for a wide stream, of order F in the momentum equation have been neglected. 5 ¯ ignoring change of S with D,thisgivesf0 X0, so that a good estimate of the speed of Here we have shown that the only approximation has been that the wave period is long. No  3 propagation of a Àood wave is to multiply the stream velocity by 5@3. This velocity is important, • approximation has been made by neglecting dynamical terms. A better name is the Very Long and will be studied more practically later. Wave Equation,VLWE. 85 87

X0 = T0@D0 – mean Àuid velocity: used for simplicity. Not-so-long waves – in the shorter limit of waves from the long wave equations • F – the speed of not-so-long waves: 2 2 0 In the other limit, for disturbances which are shorter, such that C @Cw 0C@Cw, for which we • • À 2 2 use the term “not-so-long” waves, the Telegraph equation becomes F0 = jD0@E0 +   X0 >  2 2 2 q C y C y 2 2 2 C y In most textbooks this is written, not unreasonably,¡ implicitly¢ with  =1such that F0 = +2X0 (F0  X0 ) =0> Cw2 C{Cw   C{2 jD @E , which is usually said to be the “celerity” or “long wave speed” or “dynamic wave 0 0 which is a second-order wave equation with solutions speed”. Below it will be shown that it is actually the speed of waves only in the limit of p y = i21 ({ (X0 + F0) w)+i22 ({ (X0 F0) w) shorter waves, but still long enough that the hydrostatic approximation holds. We call these    “not-so-long” waves. They occur when waves are due to rapid gate movements. This velocity is where i21 (=) and i22 (=) are arbitrary functions determined by boundary conditions both less-important than is generally believed. upstream and downstream. We now obtain some simple solutions to the Telegraph equation in two limits. In this case the solutions are waves propagating upstream and downstream at velocities of • X0 F0, such that in the usual terminology F0 is the “long wave speed”, and the waves travel ± relative to an advection velocity X0, where the presence of  is slightly surprising.

We have shown here that F0 is the speed of waves that are actually not so long, apparently • paradoxically – they are long enough that the pressure distribution in the Àuid is still hydrostatic, but they are short in terms of time scales given by the resistance characteristics.

86 88 Intermediate period waves It is interesting that just using the rating expression T = Tr(D) is already an approximation to the momentum equation. In the general case, solutions of the long wave equations show wave propagation characteristics, • velocity and rate of decay, that depend on the period of the waves, so that the waves are actually This approximate simple momentum equation is surprisingly accurate, and can be used for long – diffusive – different period components decay at different rates, and wave propagation problems in streams. It is possible to substitute it into the mass conservation equation to give a single partial differential equation in area D ({> w). However, that is complicated, – dispersive – different components travel at different speeds and inÀow boundary conditions are often expressed in terms of a discharge hydrograph, when the One can obtain solutions for the propagation behaviour in terms of wave period, but the area D formulation is not so convenient. • operations are complicated, and they are not included here. A more general approach in terms of Upstream Volume can be developed. Substituting for The widespread belief, printed in all textbooks, is wrong, that all waves obeying the long wave D = CY@C{ and T = CY@Cw gives the single equation in the single dependent variable Y : • equations travel at a speed F jD@E sj Depth. The behaviour is very much more  2 complicated. There is no such thing as “thelong wave× speed”. CY 1 C Y + Tr (CY@C{) 1 =0> (6.25) p Cw  VE(CY@C{) C{2 Solving the long wave equations numerically overcomes all such problems, but it is nice to know s what physical processes are at work. where both breadth E and Tr have been written as functions of area D = CY@C{.Wecall this the Slow change routing equation. It is a single equation in a single unknown. The only approximation relative to the long wave equations has been that the variation with time is slow, such that jVW@X & 10. Boundary conditions involving discharge T or stage  can be incorporated using equations (6.21) and the geometrical relationship between D and surface elevation  at a point. The equation can be used for simulations, for which it is necessary to use the quadratic approximation to the second derivative, similar to those for the ¿rst derivative in equation (6.19)

89 91

6.9 Slow change routing equation on page 77: 2 C i ip 1 2ip + ip+1 The Telegraph equation (equation 6.22 on page 85) is =   > C{2 2 2 2 2 2 ¯p Cy Cy C y C y 2 2 C y 2 C y ¯ 0 + f0 + +2X0 +  X F =0= and as the second derivative of a quadratic function is constant, this is the value used also at p 1 Cw C{ Cw2 C{Cw 0 C{2  0 C{2 ¯  μ ¶ and p +1at boundaries if necessary.¯ For most Àood routing problems the slow change routing Previously for very long waves, we included just the ¿rst 0 terms. We now make a rather better equation gives results as good as the long wave equations. For short waves it is less accurate. The approximation, including the last term, ignoring just the terms shown light blue. The corresponding only approximation relative to the long wave equations has been that the variation with time is full momentum conservation equation is slow, such as for Àood waves. 2 2 CT C T jDCD ˜ T 6.10 Advection-diffusion model equation +  + = jDV 1 2 = Cw C{ D E C{  Tr μ ¶ μ ¶ In fact, the slow change routing equation is more use to us here for purposes of understanding – to It can be shown that neglecting those ¿rst two terms of the momentum equation, the time derivative show us the nature of solutions and how waves in rivers behave. and the Àuid momentum term, one is making, surprisingly, not a low-Froude number “kinematic” approximation as has been believed, but a Very Long Wave or Slow Change approximation. The It is a fully-nonlinear (no assumption of smallness) advection-diffusion equation, which is made criterion is in terms of the dimensionless time scale 0W where 0 =2jV@X0,inwhichX0 is the clearer if we write Tr = XrD,whereXr is the mean Àow velocity as given by Gauckler-Manning- mean Àow velocity, and W is the time scale.For the approximation to be accurate, waves should be Strickler etc.: suf¿ciently long that 0W & 20, where here W is wave period. The equation can be re-arranged CY CY 1 C2Y to give the momentum equation in the form of a simple expression for T for the period limitation + Xr (CY@C{) 1 =0> (6.26) Cw C{s  VE(CY@C{) C{2 shown: so that in this form with little approximation, the advective term with the single derivative CY@C{ 1 CD is multiplied by a term containing a diffusive correction with 2 2. T(D)= Tr(D) 1 > for jVW@X0 10 = (6.24) C Y@C{ ×  E(D)V C{ & Steady, uniforms

| {z } 90 92 Advection term Diffusion term To the lowest level of approximation, when waves are so long that diffusion plays no role, equation Now we consider the effects of the “diffusion term”, with the second derivative. (6.26) becomes the nonlinear advection equation In physics, the process of diffusion occurs because of a continuous process of random particle CY CY movements, where any irregularities in concentration ! of a substance are smoothed out. In a + Xr(Y{) =0= (6.27) Cw C{ stationary medium, the governing diffusion equation is This generally nonlinear equation has some interesting properties. Super¿cially it seems that C! C2! C2! C2! volume Y is advected at a velocity of X , the mean velocity of Àow in the stream, as one might =  + + > (Diffusion equation) r Cw C{2 C|2 C}2 expect. However nonlinearity of the equation causes us a surprise. We consider a general unsteady μ ¶ where  is the coef¿cient of diffusion. The signi¿cance of the equation is that any regions of Àow superimposed on a steady uniform Àow, of area D0 and discharge T0 = X0D0, such that, as previously to obtain the Telegraph equation, we write Y = D { X D w + y ({> w),wherey ({> w) "curvature" (where there is not a linear variation of !) will be smoothed out. For example, near 0 0 0 2 2 is the unsteady or non-uniform contribution. Substituting into equation (6.27) a local maximum in concentration, even in just one dimension, C !@C{ is negative, and this means that C!@Cw there is negative, and the concentration is reduced. The reverse applies near a Cy Cy Cy X0D0 + + Xr D0 + D0 + =0> minimum. Quantities that show diffusive behaviour are temperature, electric charge, and pollution  Cw C{ × C{ μ ¶ μ ¶ concentration. and now assuming that the unsteady/non-uniform terms Cy@Cw and Cy@C{ are small compared Linearising the slow change routing equation, now including the with the underlying Àow, and taking a Taylor expansion of X about D , we obtain the advection r 0 second derivative term, gives the Advection-diffusion equation: equation 2 Cy Cy T0 C y Cy Cy Cy gXr Cy + f =0> (Advection-diffusion equation) + f = + X + D =0. (6.28) 0 2 Cw 0C{ Cw 0 0 gD C{ Cw C{  2E0V C{ μ ¯0¶ We have obtained, rather more simply this time, the very long¯ wave equation (Very long wave and we see that the diffusion coef¿cient is given by T0@2E0V, ¯ where E0 is the width of the undisturbed stream. Typical solutions equation). It is now clearer that, even if the underlying Àow is¯ carried at a velocity X0,anyvariation of the equation are shown in the ¿gure. This is a good simple approximate model of Àood 93 95

in the Àow is advected at the very long wave speed f0 given by the Kleitz-Seddon formula: propagation. It tells us most importantly, how fast the Àood wave moves. gXr gTr As a ¿rst estimate, we might assume that diffusion is unimportant, and that the Àood peak might f (D)=X + D = = (6.29) 0 0 0 gD gD be the same downstream as it was upstream (as implied by the very long wave advection equation). ¯0 ¯0 ¯ ¯ For practical problems, however, the problem then arises as how to estimate the importance of This nonlinear mathematical artefact is not so obvious,¯ physically!¯ Evaluating f0 for some cases, we ¿nd ¯ ¯ diffusion. We can go back to the momentum equation (6.24) and consider the term which leads to diffusion dTr@dD 0 > General expression 1@2 3 | 1 1 CD f0 = ; 2X0 1 3D0S00@S0 > Chézy-Weisbach > 1 = A 5  2  E(D)V C{ ?A X0 1 D0S 0@S0 > Gauckler-Manning-Strickler μ ¶ 3 ¡  5 0 ¢ This is not a particularly helpful criterion, as we do not know what the {-derivative is until we where S =dS@dD . solve a problem. However, we can use our knowledge that, to ¿rst approximation,theÀood wave 00 0 A ¡ ¢ | = propagates unchanged, satisfying the very long wave equation Example 5 Estimate the effect of side resistance on Àood wave speed for a river of bottom width CD CD + f =0= 20 m, side slopes 2:1 (H:V) and a depth of 2m. Cw 0 C{ Using our relationships E = Z +2pk, D = k (Z + pk),andS = Z +2s1+p2k we have The term containing diffusion, using a power series expansion to ¿rst order becomes dS@dk =2s1+p2 and dD@dk = E 1 CD 1 CD 1 1 =1+1 2 2 2 2D0 dS 2D0 dS@dk 2D0 dS@dk 2 k (Z + pk) 2s1+p  E(D)V C{ f0E(D)V Cw = = = 1 C 5 S0 dD 5 S0 dD@gk 5 S0 E 5Z +2s1+p2kZ +2pk 1 ¯0 ¯0 ¯0 =1+2 > ¯ 2 2(10+2¯ 2) 2s1+2¯2 f0V Cw ¯ = ¯× ¯ = 15% ¯ 510 + 2 2s¯ 1+22 10 + 2 2¯ 2  × × × 94 96 where  is the surface elevation, such that CD@Cw = EC@Cw.Wehavetheresult Summary of theories, names, and ranges of application 1 C Long wave equations Relative importance of diffusion = = (6.30) @k 10 2f0V Cw  FTQS Numerical method This has an interesting simple physical signi¿cance: we can consider it to be Telegraph equation Relative importance of diffusion = Slow change routing equation Vertical water surface velocity @ Horizontal water surface velocity Muskingum equations 1 = 2 Vertical fall of bed @ Horizontal distance Very long wave equation Wave speed Wave speed Wave speed is a function of period At the beginning of any problem one does not know C@Cw. However, it is not dif¿cult to estimate sj Depth 5@3 Water velocity  ×  × it to give the relative importance of diffusion. Historical records should give us an approximate “Not-so-long” waves Very long waves idea of the maximum (Peak) Àood level P to be expected, as well as the time at which it occurs, or Fast gate movements Floods which value is given for a particular event. In this case we have 1 10100 1000 10000 1   0W Relative importance of diffusion = P  0> (6.31) 2f0V wP w0  and this is enough for an estimate. Our expressions might also be useful to give us a general idea of when diffusion is important. As f0 is proportional to X0 and that is proportional to sV, whether Gauckler-Manning-Strickler or Chézy-Weisbach resistance formula are used, we obtain 3@2 the important result that the relative importance of diffusion is proportional to V ,whichisa very strong result showing that diffusion is more important for small slopes.

97 99

Two examples, showing smaller and greater diffusion effects 6.11 Muskingum methods

The ¿gure shows results from two computations, for a 50 km length of river, ¿rst with a very The advection-diffusion equation, re-written with the symbol G0 for the coef¿cient of diffusion, smooth boundary and steeper slope so that diffusion is not so large, the second for a rougher G0 = T@2E0V,is boundary and smaller slope. Cy Cy C2y + f0 G0 =0= (Advection-diffusion equation) Cw C{  C{2 10 In ow It is a good simple approximate model of Àood propagation – not as good as the fully nonlinear Long wave eqns, FTQS Small diusion Slow change eqn, FTQS slow change routing equation. Both have, however, a ¿nite stability criterion – strangely, the Slope V =0=00025 numerical simulation with the second derivative diffusion term makes the computation less stable! 8 Smooth channel, nSt =70 However, numerical solution is not a problem – one simply takes smaller steps until it works. In WP =12h the last 40 years, however, there have been a large number of papers published using Muskingum Larger diusion 6 methods, named after a river in the USA where such a method was ¿rst applied. They mimic 2 1 Slope V =0=0001 t m s Rough channel, nSt =20 the advection-diffusion equation, are supposed to be simple and plausibly seem so, and have ¡ ¢ WP =12h been widely used. People have obtained the methods, sometimes from a simple reservoir routing 4 approach, sometimes from the long wave equations, using long, complicated and arbitrary methods. The problem is to obtain a single ¿nite difference equation in a single variable. (Using upstream q 2 volume Y solves that problem rather better!). A typical Muskingum scheme is written, where fp q q q q q is the very long wave speed gTr@gD p,andGp is the coef¿cient of diffusivity Gp = Tp@2EpV, andsoon: | 0 01224364860 Time (h)

98 100 6.12 The method of characteristics q q q q q q q q w (  {fp +2Gp@fp) Tp +(  + {fp+1 2Gp+1@fp+1) Tp+1   q+1 q+1 q+1 q+1  q+1 q+1 q+1 q+1 +  {fp 2Gp @fp Tp +  + {fp+1 +2Gp+1@fp+1 Tp+1 =0=   ({ >w ) This can be re-arranged to give an explicit expression for Tq+1 , the top right point shown in the p q+1 ¡ ¢ ¡ p+1 ¢ wq+1 X + F X F blue computational module in Figure 6.4 in terms of the known two values of the discharge at time  1 1 q q q+1 { q, Tp and Tp+1 and the known value at the previous space point Tp .

wq w q+1 q+1 + ip ip+1 {p 1 { {p { {p+1 { (q +1){  FTQS Figure 6.5: ({> w) axes, showing computational module with characteristics { { Muskingum & Box schemes This method is described in many books. The lecturer believes that it is something of an accident i q i q i q of history, and that the deductions that emerge from it are misleading and have caused several q{ p 1 p p+1  important misunderstandings about the nature of wave propagation in open channels.   Each of the pairs of long wave equations (6.17) and (6.18), which are partial differential equations, (p 1) p (p +1) {  can be expressed as four ordinary differential equations. Two of the differential equations are for paths for {(w), a path known as a characteristic: Figure 6.4: Computational stencils g{ q = X F> (6.32) However, if one performs a Consistency Analysis, and writes the point values Tp etc as bi- gw ± dimensional Taylor series, one ¿nds that the differential equation that the Muskingum formula 101 103 actually satis¿es is where X = T@D is the mean Àuid velocity in the waterway at that section and the velocity F is 2 CT CT G0 C T + f + =0> jD 2 Cw 0 C{ f C{Cw F = + X 2   > 0 E  and not the desired Advection-diffusion equation r often incorrectly described as the “long wave speed”.¡ It is, as¢ equation (6.32) shows, the speed 2 CT CT C T of the characteristics relative to the Àowing water. The two contributions F correspond to + f0 G0 =0= Cw C{  C{2 downstream and upstream propagation of information. Two characteristics± that meet at a point One can use the ¿rst two terms in the Muskingum equation to write CT@Cw f0CT@C{ and are shown on Figure 6.5. The “downstream” or “+” characteristic has a velocity at any point of  substitute this into the mixed derivative C2T@C{Cw to give the advection-diffusion equation, but X + F. In the usual case where X is positive, both parts are positive and the term is large. that is accurate only for small diffusion. As shown on the diagram, the “upstream” or “-” characteristic has a velocity X F,whichis  Muskingum methods work surprisingly well for small-diffusion problems, but in general, they usually negative and smaller in magnitude than the other. Not surprisingly, upstream-propagating solve the wrong equation, are numerically diffusive, and are to be avoided. disturbances travel more slowly. The characteristics are curved, as all quantities determining them are not constant, but functions of the variable D, E,andT. The other two differential equations for  and T can be established from the long wave equations: T g gT T2E T T E  F + =  V˜ \S | |> (6.33)  D ± gw gw D2  D2 μ ¶ On each of the two characteristics given by the two alternatives of equation (6.32), each of these two equations holds, taking the corresponding plus or minus signs in each case. To advance the solution numerically means that the four differential equations (6.32) and (6.33) have to be solved over time, usually using a ¿nite time step {. Figure 6.5 shows the nature of the process on a plot of { against w. 102 104 The usual computational problem is, for a time wq+1 = wq + {, and for each of the discrete points 6.14 Results + {p, to determine the values of { and { at which the characteristics cross the previous time level wq. From the information about  and T at each of the computational points at that previous time Evolution of Àoodwave–largediffusioncase + + level, the corresponding values of  , , T ,andT are calculated and then used as initial values in the two differential equations (6.33) which are then solved numerically to give the updated values  ({p>wq+1) and T ({p>wq+1), and so on for all the points at wq+1. 10 In textbooks and research papers, characteristics seem wrongly to be believed to have an almost supernatural property that the partial differential equations do not. An advantage of characteristics 8 has been believed to be that numerical schemes are relatively stable. The lecturer is not convinced Depth 6 that they are any more stable then ¿nite difference approximations to the original partial differential (m) 4 equations, but this remains to be proved conclusively. 2 In fact, the use of characteristics has led to a widespread misconception in hydraulics where F is 500 understood to be the speed of propagation of all waves. It is not – it is the speed of characteristics. 40 If surface elevation were constant on a characteristic there would be some justi¿cation in using the { (km) 30 term”wavespeed”forthequantityF, as disturbances travelling at that speed could be observed. 20 60 48 54 However as equation (6.33) holds, in general neither  (surface elevation – the quantity that we 10 36 42 24 30 see), nor T, is constant on the characteristics and one does not have observable disturbances, 0 18 w (h) 6 12 something that we would call a wave, travelling at F relative to the water. While F may be the 0 speed of propagation of information in the waterway relative to the water, it cannot properly be Figure 6.6: termed the wave speed as it would usually be understood. In this course we have already examined at length the real nature of the propagation speed of waves. 105 107

6.13 Implicit methods – the Preissmann Box scheme Comparison of different methods The most popular commercial numerical method for solving the long wave equations in time are Implicit Box (Preissmann) models, where the derivatives are replaced by ¿nite-difference 10 Small diusion example, p.98 equivalents based on the rectangular blue module in Figure 6.4 on page 101: In ow Ci 1 8 LongwaveeqnsFTQS—OpenBC q+1 q+1 q q Slow change routing FTQS — Open BC (p> q)  ip+1 ip +(1 )(ip+1 ip) > C{  Characteristics — Uniform BC     2 1 6 t m s Preissmann Implicit — Uniform BC Ci 1 q+1 q q+1 q (p> q) £ ¡i i ¢ + i i > ¤ Muskingum Cw  2{ p+1  p+1 p  p ¡ ¢ 4 1 i¯(p> q) £¡i q+1 + i q+1 ¢+(1¡ )(i q ¢¤+ i q ) >  2 p+1 p  p+1 p 2 where  is a coef¿cient that determines£ ¡ how much¢ weight is attached to values¤ at time q +1 0 (unknown, shown red) and how much to those at q (known, shown blue). Now, in the long 10 Finite diusion example, p.98 wave equations (6.17) or (6.18) we use these expressions for all derivatives and also the averaged quantities i¯ for those that occur algebraically. Considering all the modules at a certain time level, 8 we have a set of 2P simultaneous complicated nonlinear algebraic equations in the values of T 2 1 6 and  at all points along the channel. The method is very complicated, but it is robust and stable, t m s 1 ¡ ¢ 4 and large time steps can be taken. It is neutrally stable if  = 2. In practice, one uses a larger value, such as  =0=6, and the scheme is stable because it is computationally-diffusive. Several 2 well-known commercial programs are available. For human purposes, it is simpler and better to use an explicit ¿nite difference FTQS scheme. 0 0122436486072 Time (h) 106 108 Conclusions 7.1 The gradually-varied Àow equation (GVFE) Small diffusion case: Use of area D and application to streams of unknown bathymetry All methods performed well. Muskingum was accurate; the incorrect uniform Àow downstream For steady Àow where T is constant so that CT@Cw and CT@C{ are zero, the long wave momentum • boundary condition did not matter (all motion is like simple advection – there is little diffusion equation (6.17b) on page 74 in terms of cross-sectional area D, gives one version of the GVFE in for downstream effects to be felt upstream). terms of area D: Finite diffusion case: dD V˜ T2@N2 V˜ T2S 4@3@n2 D10@3 = E  = E  St = (7.1) Muskingum showed far too much diffusion when that was important. Such methods have been d{ 1 F2 1 T2E@jD3 •  known to be problematical for small slopes, but nobody has called them out. In the resistance term we are using the conveyance N, which is a function of section properties and The uniform Àow downstream boundary condition: the Preissmann Implicit Box scheme and the Strickler coef¿cient nSt • the Method of Characteristics agreed quite well with each other, but there were ¿nite differences D5@3 N = n > (7.2) with those of the open downstream boundary condition. StS 2@3 ˜ Both the FTQS ¿nite difference schemes (solving the long wave equations and the slow such that for uniform Àow, V = V = constant, Tr = NsV. • change routing equation) agreed closely with each other using the more correct open boundary The ordinary differential equation (7.1) is valid also for non-prismatic channels. The mean bed condition. They are the simplest methods and the best. One has to take relatively small time slope at a section V˜, can be variable but is, usually poorly known and is often just estimated, like steps (30 s used in the examples, compared with 900 s for the implicit and Muskingum methods), the other parameters of the problem;  might be something like 1=1.Thecoef¿cient nSt is also but their simplicity means that computational time is short. often poorly known. One can devise an example with a more rapidly-rising Àood wave where the slow change routing • equation no longer agrees so well. It is simple, and is in terms of a single variable, so that we used it to show the nature of approximations. In general, however, solving the long wave equations themselves using our explicit FTQS scheme is the best of all. 109 111

7. Steady Àow In the differential equation there are strongly-varying functions of the dependent variable itself, D3 and possibly D10@3, plus the usually slowly-varying functions E(D) and S(D). This suggests that Vegetation using the GVFE in terms of D has an important advantage: one needs few details of the under-water topography. It is not necessary to know the precise details of the underwater bathymetry other than Control structure / bridge those weakly-varying functions E(D) and S(D). The obvious approximation could be made that they are constant and equal; river width often does not vary much. Sand mining To start numerical solution, one would need to know the area at a control where surface elevation A common task in river engineering is to calculate the free surface elevation along a steadily might be known. The solution in terms of area might be enough, to give an idea of how far • Àowing stream. upstream the effects of a structure or channel changes extend. It is surprising that we can do so much with so little information. However, if one needed a value of surface elevation  at a certain Simply the solution of a ¿rst-order differential equation – often obscured in writings. • value of {, one would then need cross-sectional details there to go from the computed D to . Flow is usually sub-critical, so the control / boundary condition is at the downstream end and • one computes upstream. Customary use of a quantity k called the “water depth” Alternative approach suggested here, using cross-sectional area as the dependent variable, The long wave momentum equation (6.18b) in terms of surface elevation ,forT constant so that • requiring little knowledge of the details of the underwater topography. CT@Cw and CT@C{ are zero gives another version of the GVFE: Traditional textbook methods are unsatisfactory: the “Standard Step” method is unnecessarily • complicated and the “Direct Step” method is incorrect. d V˜ F2 T2@N2 =  T2@N2 for F2 small, the common case Application of simple explicit numerical methods is described. d{ 1 F2  •  If D is not used, for non-prismatic streams all methods require much data. Often that is not ¡ ¢ • available. An approximate linearised model of Àow in a river is made. This gives us insight into The tradition is not to use , but instead a depth-like quantity k =  ]0,where]0 is the elevation  the nature of the problem, as well as simple approximate answers. 110 112 ({> w) 7.2 Traditional textbook methods – some old, complicated, and wrong The“Standard”stepmethod k({> w) k({> w) } The almost trivial energy derivation, ignoring non-prismatic effects, is that the rate of change of } ]0({) – Reference axis total head K is given by the empirical expression for the energy gradient ]({> |) ]0({) Bed pro¿les ]({> |) 2 dK 2 2 T { | = T @N ({> k) where K = ]0({)+k +  = d{  2jD2({> k) Figure 7.1: Complicated reality The computational approximation scheme is of a longitudinal axis, almost always the supposed bed of the channel. The GVFE becomes K (k ) K (k ) 1 1 ˜ 2 2 2 l+1 l+1 l l 1 2 dk V0 +  V V0 F T @N  = 2T 2 + 2 =   > {l+1 {l  N ({l>kl) N ({l+1> kl+1) d{ 2  μ ¶ ¡ 1 ¢F The method advocated by Chow (1959) in a pre-computer  • where V0 = d]0@d{, the slope of the reference axis, positive downwards. We almost never know era and still suggested by textbooks. ˜ ˜ the details of V so here we assume that V = V0, which we now write as V,giving K(k) and N({> k) are both complicated geometrical dk V T2@N2 • functions of k, the unknown k is deep inside left and =  l+1 d{ 1 F2 right sides.  where in general both N and F are functions of both { and k, while in a prismatic channel, Requires numerical solution of a transcendental equation at functions just of k. • each time step. J. Fenton, Australia 1966; H. Honsowitz, Austria, 1970? solving transcendental equations 113 115

Because of our use of k, we pretend that we know the bed in great detail, or, that our channel looks The “Direct” step method – distance calculated from depth like this: Applied by taking steps in the water depth and calculating the corresponding step in {. • It has some advantages: iterative methods are not necessary (“Direct”). • Practical disadvantages are: Normal depth k0 • k({) – It is applicable only to prismatic sections k 1 – Results are not obtained at speci¿ed points in { – As uniform Àow is approached the steps become in¿nitely large

Figure 7.2: Our simpler model – AND, it is wrong, as we now show

Consider the “speci¿c head”, the head relative to the local channel bottom, denoted here by K0: 2 This shows a typical subcritical Àow retarded by a structure, showing the free surface disturbance T K0(k)=K(k) ] = k +  = decaying upstream, and if the channel is prismatic, to constant normal depth.  2jD2(k) The differential equation becomes, after inverting each side d{ 1 = 2 2= dK0(k) V T @N  A mistake and a correction The differential equation is now approximated, the left side by a ¿nite difference expression • ({l {l+1) @ (K0>l K0>l+1).   114 116 For the right side the numerical method as set out in textbooks is to take the mean of just the Euler’s method is the simplest but least accurate – yet it might be appropriate for open channel • denominator at beginning and end points, and so to write • problems where quantities may only be known approximately K K { = { + 0>l+1 0>l One can use simple modi¿cations such as Heun’s method to gain better accuracy, or use l+1 l 1 2 2  2 2 • Vl T @N + Vl+1 T @N Richardson extrapolation, or even more simply, just take smaller steps  2  l  l+1 where the red shows the quantity that is a supposed mean value. For greater accuracy one can use the Trapezoidal method, simply repeating the second Heun ¡ ¢ • While this is a plausible approximation, it is not mathematically consistent. What should step several times, setting kl+1 = kl+1 each time • be done is to use the mean value at beginning and end points of the whole right side of the Often these two methods are not presented in hydraulics textbooks as alternatives, yet they are differential equation, to give a trapezoidal approximation of the right side, which leads to • simple and Àexible, and reveal the nature of what we are doing 1 1 1 The step  can be varied at will, to suit possible irregularly spaced cross-sectional data {l+1 = {l +(K0>l+1 K0>l) 2 2 + 2 2 = •  2 Vl T @N Vl+1 T @N 2 2 μ  l  l+1¶ In many situations, where F 1, we can ignore the F term in the denominators, giving a • notationally simpler scheme ¿

117 119

7.3 Standard simple numerical methods for differential equations Comparison of schemes

We write the differential equation as 3 1 Example 6 A Àow of 11=33 m s passes down a trapezoidal channel of gradient V =0=0016,bed dk V({) T2@N2 ({> k) width 6=10 m and channel side slopes K : Y =2,  =  =1=1,andnSt =40.At{ =0the Àow = i ({> k)=  2 d{ 1 F ({> k) is backed up to a depth of 1=524 m. Compute the backwater curve for 1000 m in 10 steps and then  The two simplest numerical methods are: 20, then perform Richardson extrapolation for a more accurate estimate. Euler Heun

kl+1 i ({l>kl) kl+1 2=5 Runge-Kutta 4th order i {l+1>kl+1 = Mean kl+1 Euler  =100m ¡ ¢ = Euler  =50m Gradient i ({ >k) l l Richardson 2=0 kl kl  (m)

{l  {l+1 {l  {l+1 1=5 2 kl+1 kl + i({l>kl)+R  k kl + i({l>kl) >  l+1   3 ¡ ¢ kl+1 kl + i ({l>kl)+i {l+1>kl+1 + R   2 ¡ ¡ ¢¢ ¡ ¢ 1000 800 600 400 200 0    {(m)  

118 120 Convergence of numerical schemes Similarly we also let the possible non-constant slope be

V = V0 + %V1({)+==== Euler 2 10 Convergence 1  Euler, half step size In a real stream varying along its length, both N and F are functions of { and k. We write the Heun series: Direct step 2 N = N0 + %N1({)+%k1({)Nk0 + R % > 3 2 Corrected direct step 10 Convergence  Mean  Euler & Richardson where N is a change caused by a change in the channel properties in {, whether the resistance Standard step 1 ¡ ¢ elevation Trapezoidal error (m) coef¿cient or the cross-section, and Nk0 =dN@dk 0 expresses the change of conveyance with water depth. We also write | 4 10 2 2 F = F0 + R (%)+===> in which we will ¿nd that terms in % are not necessary. 2 5 Multiplying through by 1 F , setting gk0@g{ to zero for uniform Àow and neglecting terms in 10 20 50 100 %2:  Computational step  (m) 2 dk1({) T N1({) Nk0 Figure 7.3: Comparison of accuracy - logarithmic scales 2 % 1 F0 = V0 + %V1({) 2 1 2% 2%k1({) =  d{  N0  N0  N0 Using Euler, then applying Richardson extrapolation, gave the third most accurate of all the μ ¶ ¡ 0 ¢ • methods, more than enough for practical purposes At zeroeth order % we obtain 2 2 The most accurate were the Standard step method and the Trapezoidal method V0 T @N0 > •  There is something wrong with the conventional Direct step method as we have suggested, an expression of whichever Àow formula is being used, and is identically satis¿ed. • while the corrected scheme is highly accurate 121 123

7.4 A simple model of steady Àow in a river At %1, we obtain the linear differential equation Often the precise details of a stream are not known, and it is quite legitimate to make dk1 k1 = !({) • approximations d{  where  is a constant: These might give us more insight and understanding of the problem • Now a model is made where the GVFE is linearised and a general solution obtained dN@dk 0 • 2 | > General expression; Simple deductions as to the length of backwater effects can be made ; N0 V0Nk0@N0 V0 E dS@dk • =2 = A 0 0 (7.3) One can calculate an approximate solution for a whole stream if the variation in the resistance  2 2 A 3 | > Chézy-Weisbach; • 1 F0 1 F0 × A D0  S0 coef¿cient and geometry are known or can be estimated   ?A 10 E0 4 dS@dk 0 There is more of a balance between what we know (usually little) and the (un)sophistication of 3 3 | > Gauckler-Manning; • A D0  S0 the model A andtheforcingtermontherightis A =A The GVFE is ( ) 2 ( ) 2 2 V0 V1 { N1 { dk V T @N ({> k) !({)= 2 + > (7.4) =  1 F V0 N0 d{ 1 F2({> k)  0 μ ¶  showing the effects of fractional changes in slope and conveyance N. We linearise the problem (similar to obtaining the Telegraph equation) and consider small Solving the differential equation perturbations about an underlying uniform Àow of slope V0 and depth k0, such that we write

k = k0 + %k1({)+===> The differential equation is in integrating factor form, and can be solved by multiplying both sides by h { and writing the result where % is a small quantity expressing the magnitude of deviations from uniform.  d h {k = h {! ({) > d{  1  122 ¡ ¢ 124 which can be integrated to give General solution for channel { { {0 k1 = h h ! ({0) g{0 + Constant > Here we neglect any boundary conditions and consider just the solution due to the forcing function μZ ¶ ! due to changes in the channel: where {0 is a dummy variable. Returning to physical variables, k = k0 + %k1 gives the solution 4 ({ {0) { k = k0 h  ! ({0) g{0 (7.6) { ({ {0)  k = k0 + Kh + h  ! ({0) g{0 Z{ Z This is a simple result: at any point { in subcritical Àow, any disturbance is due to the integrated The part of the solution Kh{ is that obtained by Samuels (1989), giving the solution for backwater effects of the disturbance function ! for all downstream points, from { to , weighted according level in a uniform channel by evaluating the constant of integration using a downstream boundary to the exponential decay function. 4 condition k = K at { =0. The solution shows how the surface decays upstream at a rate h{,as{ Effect decaying upstream exp({) becomes increasingly negative, because  is positive,  For a wide channel, the terms in gS@gk in the formulae for  are unimportant (and are often not Local effect • well known), so that D0@E0 k0, the channel depth, and for small Froude number this gives  V0  3 > (7.5) Local change  k0 showing that the rate of exponential decay is small for gently sloping and deep streams and greatest for steep and shallow ones. Consider the position { upstream for the effect of a blockage to diminish by a factor of 1@2. • 1@2

125 127

Then exp {1@2 =1@2,or Example 7 The effect on a river of a ¿nite length of greater resistance ¡ ¢ ln 1@2 ln 1@2 k0 k0 {1@2 = 0=2   3 V0  V0 Consider, as an example, a case where over a ¿nite length O of river, the carrying capacity is reduced 4 So for a gently-sloping river V0 =10 and 2mdeep, the effect of any backwater decreases by by the conveyance N decreasing by a relative amount N1@N0 = , such as by local deposition  2 1@2 in a distance of 4km. To diminish to 1@16,say,thedistanceis16 km.Forasteeperriver, of material, between { =0and { = O, and constant in that interval. Assume F0 negligible and the say V0 =0=0016 from the example simulated above, where k0 1m, the “half-length” is about river wide. 150 m. This is roughly in agreement with the computed results in Example 6 above. If the approximate exponential decay solution were shown on that ¿gure, it would not agree The forcing function form equation (7.4) is: • closely with the computed results, because the checked-up disturbance is as large as 50% of 0> if { 6 0; the depth, when the linear solution is not all that accurate. The beauty of Samuels’ result is in !({)= V0> if 0 6 { 6 O; its ability to give a quick estimate and an appreciation of the quantities that affect the length of ;  ? 0> if { > O. backwater. For { downstream, { > O, !({)=0,and=k = k0, which is correct in this sub-critical Àow, there are no downstream effects. For { in the section where the changes occur, 0 6 { 6 O, the solution is O V  ({ {0) 0 ({ O) k = k0 + V0 h  g{0 = k0 + 1 h  = {   Z ³ ´ For { upstream, { 6 0, where there is no extra resistance, O V  { {0 0 { O k = k0 + V0h h g{0 = k0 + h 1 h =   126 Z0 128 ¡ ¢ These solutions are all shown in the ¿gure 8.2 The Problem Undisturbed with an arbitrary vertical scale such that Constricted reach the slope is exaggerated. The calculations Almost universally the routine measurement of the state of a river is that of the stage, the surface Upstream ow were performed for V =0=0005, k = elevation at a gauging station. While that is an important quantity in determining the danger If constriction continued 0 0 1m, and with a constricted length of of Àooding, another important quantity is the actual volume Àow rate past the gauging station. O = 1000 m, with a 10% increase in Accurate knowledge of this instantaneous discharge - and its time integral, the total volume of Àow resistance there, such that  =0=1.Using - is crucial to many hydrologic investigations and to practical operations of a river and its chief environmental and commercial resource, its water. Examples include decisions on the allocation of these ¿gures, and with  =3V0@k0,the computed backwater at the beginning of water resources, the design of reservoirs and their associated spillways, the calibration of models, the constriction calculated according to the and the interaction with other computational components of a network. 2O O 0 O   formula was 2=6cm. Stage is usually simply measured. Measuring the Àow rate, the discharge, is rather more dif¿cult. In the reach of increased resistance the surface is raised, as one expects and shows an exponential Almost universally, occasionally (once per month, or more likely, once per year) it is obtained by measuring the velocity ¿eld in detail and integrating it with respect to area. At the same time, the approach to the changed depth V0@ if O . $4 water level is measured. This gives a pair of values (l>Tl) whichobtainedonthatday.Overa The abrupt changes of gradient violate our physical assumptions of the long wave equations, but long period, a ¿nite number of such data pairs are obtained using this laborious method. A curve they give us a clear picture of what happens, possibly obvious in retrospect, but hopefully of that approximates those points is calculated, to give a function Tr(),aRating Curve. assistance. Separately, the actual stage can be measured easily and monitored almost continuously at any We have obtained an approximate solution to the problem, with little input data necessary. time, and automatically transmitted and recorded at intervals of one hour or one day, to give a Stage Hydrograph, a discrete representation of q = (wq)>q=0> 1>===. To get the corresponding Discharge Hydrograph, each value of q is considered and from the rating curve the corresponding Tq = Tr (q) >q=0> 1>===are calculated. Values of q and Tq are published and made available. 129 131

8. Hydrometry - measurement and analysis 8.3 Routine measurement of water levels 8.1 De¿nitions Most water level gauging stations are equipped with a sensor or gauge plus a In English, the traditional word used to describe the measurement of water levels and Àow recorder. In many cases the water level is volumes is “Hydrography”. That is ambiguous, for that word is also used for the measurement measured in a stilling well, thus eliminating of water depths for navigation purposes, has been so used since the great navigators of the strong oscillations. eighteenth century. Organisations with names like “National Hydrographic Service” are usually Staff gauge: only concerned with the mapping of an area of sea and surrounding coastal detail. This is the simplest type, with a graduated Here we follow Boiten (2000) and Morgenschweis (2010) who provide a refreshingly modern gauge plate ¿xedtoastablestructuresuch approach to the topic, calling it “Hydrometry”, the “measurement of water”. In these notes, a as a pile, bridge pier, or a wall. Where the practitioner will be called a hydrometrician, but the term hydrograph will be retained for a record, range of water levels exceeds the capacity either digital or graphical, of the variation of water level or Àow rate with time. of a single gauge, additional ones may Two modern documents from the World Meteorological Organization provide more background. be placed on the line of the cross section Stilling well & level recorder (Morgenschweis 2010) Experimental techniques are described in WMO Manual 1 (2010), and methods of analysis in normal to the plane of Àow. WMO Manual 2 (2010). It is remarkable, however, that a ¿eld so important has received little Float gauge: bene¿t from hydraulics research. A Àoat inside a stilling well, connected to the river by an inlet pipe, is moved up and down by the water level. Fluctuations caused by short waves are almost eliminated. The movement of the Àoat is transmitted by a wire passing over a Àoat wheel, which records the motion, leading down to a counterweight.

130 132 Pressure transducers: required is the area integral of the velocity, that is T = xgD. If we express this as a double integral we can write Water level is measured as hydrostatic pressure and transformed into an electrical signal via a R semi-conductor sensor. These are best suited for measuring water levels in open water (the effect ](|)+k(|) of short waves dies out almost completely within half a wavelength down into the water). They T = xg}g|> (8.1) should compensate for changes in the atmospheric pressure, and if air-vented cables cannot be ZE ]Z(|) provided air pressure must be measured separately. so that we must ¿rst integrate the velocity from the bed } = ](|) to the surface } = ](|)+k(|), Peak level indicators: where k is the local depth. Then we have to integrate these contributions across the channel, for There are some indicators of the maximum level reached by a Àood, such as values of the transverse co-ordinate | over the breadth E. arrays of bottles which tip and ¿ll when the water reaches them, or a staff Calculationofmeanvelocityinthevertical coated with soluble paint. The ¿rst step is to compute the integral of velocity with depth, which hydrometricians think of Bubble gauge: as calculating the mean velocity over the depth. Consider the law for turbulent Àow over a rough This is based on measurement of the pressure which is needed to produce bed: x } ] bubbles through an underwater outlet. These are used at sites where it would x =  ln  > (8.2) be dif¿cult to install a Àoat-operated recorder or pressure transducer. From a pressurised gas  }0 cylinder or small compressor gas is led along a tube to some point under the water (which will where x is the shear velocity,  =0=4, ln() is the natural logarithm to the base h, } is the elevation  remain so for all water levels) and small bubbles constantly Àow out through the ori¿ce. The above the bed, and }0 is the elevation at which the velocity is zero. (It is a mathematical artifact pressure in the measuring tube corresponds to that in the water above the ori¿ce. Wind waves that below this point the velocity is actually negative and indeed in¿nite when } =0– this does not should not affect this. usually matter in practice). If we integrate equation (8.2) over the depth k we obtain the expression Ultrasonic sensor: These are used for continuous non-contact level measurements in open channels. The sensor points 133 135 vertically down towards the water and emits ultrasonic pulses at a certain frequency. The inaudible for the mean velocity: sound waves are reÀected by the water surface and received by the sensor. The round trip time is ](|)+k(|) measured electronically and appears as an output signal proportional to the level. A temperature 1 x k x¯ = xg} =  ln 1 = (8.3) probe compensates for variations in the speed of sound in air. They are accurate but susceptible to k  }0  Z μ ¶ wind waves. ](|) Now it is assumed that two velocity readings are made, obtaining x1 at }1 and x2 at }2.Thisgives 8.4 Occasional measurement of discharge enough information to obtain the two quantities x @ and }0. Substituting the values for point 1  Most methods of measuring the rate of volume Àow past a into equation (8.2) gives us one equation and the values for point 2 gives us another equation. Both point are single measurement methods which are not designed can be solved to give the simple formula for the mean velocity in terms of the readings at the two for routine operation. Below, some will be described that are points: x1 (ln(}2@k)+1) x2 (ln(}1@k)+1) methods of continuous measurements. x¯ =  = (8.4) ln (}2@}1) Velocity area method (“current meter method”) As it is probably more convenient to measure and record depths rather than elevations above the The area of cross-section is determined from soundings, and bottom, let k1 = k }1 and k2 = k }2 be the depths of the two points, when equation (8.4)   Àow velocities are measured using propeller current meters, becomes x1 (ln(1 k2@k)+1) x2 (ln(1 k1@k)+1) Traditional manner of taking current electromagnetic sensors, or Àoats. The mean Àow velocity is x¯ =    = (8.5) meter readings. In deeper water a boat is deduced from points distributed systematically over the river ln ((k k2) @ (k k1))   used. cross-section. In fact, what this usually means is that two or This expression gives the freedom to take the velocity readings at any two points. This would more velocity measurements are made on each of a number of simplify streamgauging operations, for it means that the hydrometrician, after measuring the depth vertical lines, and any one of several empirical expressions used to calculate the mean velocity on k, does not have to calculate the values of 0=2k and 0=8k and then set the meter at those points, as each vertical, the lot then being integrated across the channel. is done in current practice. Instead, the meter can be set at any two points, within reason, the depth and the velocity simply recorded for each, and equation (8.5) applied. This could be done either in Calculating the discharge requires integrating the velocity data over the whole channel - what is 134 136 situ or later when the results are being processed. This has the potential to speed up hydrographic l =1, then the Mean-Section Method gives measurements. 1 T0 = e0x1k1> If the hydrometrician were to use the traditional two points, then setting k1 =0=2k and k2 =0=8k 4 in equation (8.5) gives the result while the Trapezoidal rule gives 1 x¯ =0=4396 x0=2k +0=5604 x0=8k 0=44x0=2k +0=56x0=8k > (8.6) T0 = e0x1k1>  2 whereas the conventional hydrographic expression is which is correct, and we see that the Mean-Section Method computes only half of the actual 1 contribution. The same happens at the other side. Contributions at these edges are not large, and in x = (x0=2k + x0=8k) > (8.7) 2 the middle of the channel the formula is not so much in error, but in principle the Mean-Section that is, the mean of the readings at 0.2 of the depth and 0.8 of the depth. The nominally more Method is wrong and should not be used. Rather, the Trapezoidal rule should be used, which is accurate expression is just as simple as the traditional expression in a computer age, yet is based 3 1 just as easily implemented. In a gauging in which the lecturer participated, a Àow of 19=60 m s on an exact analytical integration of the equation for a turbulent boundary layer. was calculated using the Mean-Section Method. Using the Trapezoidal rule, the Àow calculated 3 1 The formula (8.6) has been tested by taking a set of gauging was 19=92 m s , a difference of 1.6%. Although the difference was not great, practitioners should results. A canal had a maximum depth of 2.6m and was be discouraged from using a formula which is wrong. 28m wide, and a number of verticals were used. The An alternative global “spectral”approach with least-squares ¿tting conventional formula (8.7), the mean of the two velocities, was accurate to within 2% of equation (8.6) over the whole It is strange that only very local methods are used in determining the vertical velocity distribution. Figure 8.1: Cross-section of stream, show- range of the readings, with a mean difference of 1%. That Here we consider a signi¿cant generalisation, where we consider velocity distributions given by a ing velocity measurement points error was always an overestimate. The more accurate more general law, assuming an additional linear and an additional quadratic term in the velocity formula (8.5) is hardly more complicated than the traditional one, and it should in general be pro¿le: preferred. Although the gain in accuracy was slight in this example, in principle it is desirable } ](|) 2 x(|> })=d0(|)ln  + d1(|) } + d2(|) } > (8.8) to use an expression which makes no numerical approximations to that which it is purporting to }0 137 139

evaluate. This does not necessarily mean that either (8.7) or (8.6) gives an accurate integration of but where the coef¿cients d0(|), d1(|),andd2(|) are actually polynomials in the transverse the velocities which were encountered in the ¿eld. In fact, one complication is where, as often co-ordinate. The whole expression is a global function, that approximates the velocity over the happens in practice, the velocity distribution near the surface actually bends back such that the whole section. If the polynomials in | each have M terms, then the total number of unknown maximum velocity is below the surface. coef¿cients is 3M. Consider a number of Àow measurements Xq for q =1to Q, where we presume Integration of the mean velocities across the channel that the corresponding bed elevation is also measured, however that is not essential to the method. We compute the total sum of the errors squared, using equation (8.8): The problem now is to integrate the readings for mean velocity at each station across the width Q of the channel. Here traditional standards commit an error – often the Mean-Section method is % = (x (| >} ) X )2 > used. In this the mean velocity between two verticals is calculated and then this multiplied by the q q q q=1  area between them, so that, given two verticals l and l +1separated by e the expression for the X l and we use package software to ¿nd the coef¿cients such that the total error % is minimised. Or, as contributiontodischargeisassumedtobe one says, the function is “¿tted to the data”. This method does not require points to be in vertical 1 Tl = el (kl + kl+1)(xl + xl+1) = lines, although it is often convenient to measure points like that, as well as the corresponding bed 4 level. An example of the results is given in ¿gure 8.2, where more than two points were used This is not correct. From equation (8.1), the task is actually to integrate across the channel the on each vertical line. It can be seen that results are good - and we see an import feature that the quantity which is the mean velocity times the depth. For that the simplest expression is the conventional method ignores – almost everywhere there is a velocity maximum in the vertical. Trapezoidal rule: 1 T = e (x k +x k ) l 2 l l+1 l+1 l l To examine where the Mean-Section Method is worst, we consider the case at one side of the channel, where the area is a triangle. We let the water’s edge be l =0and the ¿rst internal point be Figure 8.2: Cross-section of canal with velocity pro¿les and data points plotted transversely, showing ¿t by global function

138 140 Dilution methods Electromagnetic methods

In channels where cross-sectional areas are dif¿cult to determine (e.g. steep mountain streams) or Coil for producing magnetic ¿eld The motion of water Àowing in an open channel cuts a where Àow velocities are too high to be measured by current meters dilution or tracer methods can Signal probes vertical magnetic ¿eld which is generated using a large be used, where continuity of the tracer material is used with steady Àow. The rate of input of tracer coil buried beneath the river bed, through which an electric is measured, and downstream, after total mixing, the concentration is measured. The discharge in current is driven. An electromotive force is induced in the the stream immediately follows. water and measured by signal probes at each side of the Ultrasonic Àow measurement channel. This very small voltage is directly proportional to the average velocity of Àow in the cross-section. This Figure 8.4: Electromagnetic installation, In this case, sound generators are placed along the side of showing coil and signal probes is particularly suited to measurement of efÀuent, water in a channel and beamed so as to cross it diagonally. They are treatment works, and in power stations, where the channel reÀected on the other side, and the total time of travel of is rectangular and made of concrete; as well as in situations where there is much weed growth, the sound waves are measured. From that it is possible to or high sediment concentrations, unstable bed conditions, backwater effects, or reverse Àow. This calculate the mean water velocity along the channel – the has the advantage that it is an integrating method, however in the end recourse has to be made to sound “samples” the water velocity at all points. Then, to empirical relationships between the measured electrical quantities and the Àow. get the total discharge it is necessary to integrate the mean Figure 8.3: Array of four ultrasonic beams velocity of the paths in the vertical. This, unfortunately, is in a channel where the story ends unhappily. The performance of the trade and scienti¿c literature has been poor. Several trade brochures advocate the routine use of a single beam, or maybe two, suggesting that that is adequate (see, for example, Boiten 2000, p141). In fact, with high-quality data for the mean velocity at two or three levels, there is no reason not to use accurate integration formulae. However, practice in this area has been quite poor, as 141 143 trade brochures that the author has seen use the inaccurate Mean-Section Method for integrating 8.5 Rating curves – the analysis and use of stage and discharge measurements vertically over only three or four data points, when its errors would be rather larger than when it is used for many verticals across a channel, as described previously. The lecturer found that no-one The state of the art – the power function wanted to know of his discovery. The generation of rating curves from data is a problem that is of crucial importance, but has had Acoustic-Doppler Current Pro¿ling (ADCP) methods little research attention and is done very badly all around the world. Almost universally it is believed that they must follow a power function In these, a beam of sound of a known frequency is transmitted into the Àuid, often from a boat.  T = F (k k0) > (8.9) When the sound strikes moving particles or regions of density difference moving at a certain speed,  the sound is reÀected back and received by a sensor mounted beside the transmitter. According where k is water surface elevation (stage), F is a constant, k0 is a constant elevation reference to the Doppler effect, the difference in frequency between the transmitted and received waves is level, for zero Àow, and  is a constant with a typical value in the range 1=5 to 2=5. a direct measurement of velocity. In practice there are many particles in the Àuid and the greater the area of Àow moving at a particular velocity, the greater the number of reÀections with that frequency shift. Potentially this method is very accurate, as it purports to be able to obtain the velocity over quite small regions and integrate them up. However, this method does not measure in the top 15% of the depth or near the boundaries, and the assumption that it is possible to extract detailed velocity pro¿le data from a signal seems to be optimistic. The lecturer remains unconvinced that this method is as accurate as is claimed. US Geological Survey - ¿tting three straight lines to data segments If we take logarithms of both sides of equation (8.9),

log T =logF +  log (k k0) > (8.10)  then on a ¿gure plotting log T against log (k k0), a straight line is obtained. Often in practice,  142 144 the data is divided and different straight-line approximations are used, as in the ¿gure. Distant control: There may be something such as the larger river downstream shown as a distant control in the ¿gure. In our work on Steady Flow, we saw that backwater inÀuence can extend A problem is that k0, the nominal zero Àow point, is not initially known and has to be found, which is actually a dif¿cult nonlinear problem. A larger problem is that the equation is only a rough for a long way. approximation, but because of its ability occasionally to describe roughly almost all of a rating In practice, the natures of the controls are unknown. curve, it has acquired an almost-sacred status, and far too much attention has been devoted to it Data approximation rather than addressing the problems of how to approximate rating data generally and accurately. Much modelling and computer software follow its dictates. It really has been believed to be a The global representation of T by a polynomial has been in the background for some time: “law”. P 2 P p The reason that the power law has been believed to be so powerful (sorry) is that hydrologists T = d0 + d1k + d2k + ===+ dP k = dpk > (8.11) p=0 believe the hydraulic formula – and it does describe the discharge of a sharp-crested in¿nitely-wide X weir in water of in¿nite depth and the steady uniform Àow in an in¿nitely-wide channel. We know where d0, d1, ===, dP are coef¿cients. Standard linear least-squares methods can be used to enough hydraulics to know that not all rivers satisfy those conditions ... and we treat the problem determine the coef¿cients, but it has never really succeeded. as one, not of hydraulics, but of data approximation, because the hydraulics are complicated. Fenton & Keller (2001, §6.3.2) suggested writing the polynomial for T raised to the power , speci¿ed apriori: P  2 P p T = d0 + d1k + d2k + ===+ dPk = dpk > (8.12) p=0 X 2 1@ which is a simple generalisation of the power function to T = d0 + d1k + d2k + === .A 1 value of  = 2 was recommended as that was the mean value in the hydraulic discharge formulae for a sequence of weir and channel cross-sections that modelled local¡ and channel control.¢

145 147

The hydraulics of a gauging station The use of such a fractional value has two effects: 1. For small Àows, k small, the data usually is such that 1 Flood  2 T = d0 + d1k> (8.13) High Àow 3 Low Àow 4 that is 1@  5 T =(d0 + d1k) = F (k k0) > Channel control Distant 1  Gauging Local control so it looks like the simple power law! In fact,  = 2 is a good approximation. In that low-Àow station control limit the polynomial just has to simulate nearly-linear variation, which it can easily do. Larger body Channel control 1@2 of water 2. For large Àows, the use of T means that the magnitude of the dependent variable to be 4 3 1 approximated is much smaller, so that, instead of a range of say, T =1to 10 m s ,a Figure 8.5: Section of river showing different controls at different water levels with implications for the stage dis- 2 charge relationship at the gauging station shown numerical range 1 to 10 has to be approximated. Local control: Just downstream of the gauging station is often some sort of ¿xed control which The lecturer (Fenton 2015) has shown it is better to generalise equation (8.12) by considering may be some local topography such as a rock ledge which means that for relatively small Àows the approximating function to be made up, not of monomials kp, but more general Chebyshev there is a de¿nite relationship between the head over the control and the discharge, similar to a functions weir. This will control the Àow for small Àows. Channel control: For larger Àows the effect of the ¿xed control is to ”drown out”, to become P  unimportant, and where the control is due to resistance in the channel. T = dp Wp (|) = (8.14) p=0 Overbank control: For larger Àows when the river breaks out of the main channel and spreads X onto the surrounding Àoodplain, the control is also due to resistance, but where the geometry of With these modi¿cations, global approximation is more stable and accurate. channel and nature of the resistance is different. 146 148 Problems with global interpolation and approximation An example

p The simplest set of basis functions are the monomials s ({)={ . They are not very good, as 2=0 16 p (a) Small ows — natural axes (b) All ows — natural axes 14 (c) Semi-logarithmic axes 14 they all look rather like each other for large { and for p =2or greater. Individual basis functions 1=8 12 12 sp({) should look different from each other so that irregular variation can be described ef¿ciently. 10 1=6 10 8 (a) Interval [100> 110] (b) Interval [0> 10] k (m) 8 k (m) 1=4 6 {@100 {@10 6 2 2 1=2 4 ({@100) ({@10) Polynomial series: degree 10 4 3 3 ({@100) ({@10) Splines: 5 intervals 2 2 1=0 Spline knots, Km by hand 0 010200 1000 2000 3000 4000 1101001000 3 1 3 1 3 1 T (m s3 ) T (m s3 ) log T,(T in m s3 ) Figure 8.6: Noxubee River near Geiger, AL, USA, USGS Station 02448500, 1970s The example has quite a striking ideal form showing local, channel and overbank control. 100{ 110 010{ • (c) Interval [ 1> +1] Variation at the low Àow end can be rapid, and can have a vertical gradient and high curvature.  {p, p =0>===>4 • p The discharge may extend over 3 or 4 orders of magnitude (factor of 1000 or 10000) { , p =5>===>10 • There can be rapid variation between these different regimes (d) Chebyshev Wp({) on [ 1> +1] •  There are two curves that have been ¿tted by least-squares methods, one using Chebyshev • polynomials, the other using local spline approximation. Both have worked well.

10 110 1  

149 151

Least-squares approximation Possible problems

The coef¿cients dp can be obtained by least-squares software, minimising the sum of the weighted There are several problems associated with the use of a Rating Curve: squares of the errors of the approximation over Q data points, Discharge is rarely measured during a Àood, and the quality of data at the high Àow end of the • Q P 2 curve might be quite poor.  %2 = zq dp Wp (|q) Tq > There are a number of factors which might cause the rating curve not to give the actual q=1 Ãp=0  ! • X X discharge, some of which will vary with time. Factors affecting the rating curve include: where the |q are obtained from the kq by scaling the range of all stage measurements [kmin>kmax]. – The channel changing as a result of modi¿cation due to dredging, bridge construction, or The zq are the weights for each rating point, giving the freedom to weight some points more if one vegetation growth. wanted the rating curve to approximate them more closely, or they could be set to be a decaying – Sediment transport - where the bed is in motion, which can have an effect over a single Àood function of the age of the data point, so that the effects of changes with time could be examined. event, because the effective bed roughness can change during the event. As a Àood increases, Or, a less-trusted data point could be given a smaller weight. Often, however, all the zq will be 1. any bed forms present will tend to become larger and increase the effective roughness, so that friction is greater after the Àood peak than before, so that the corresponding discharge for a given stage height will be less after the peak. – Backwater effects - changes in the conditions downstream such as the construction of a dam or Àooding in the next waterway. – Unsteadiness - in general the discharge will change rapidly during a Àood, and the slope of the water surface will be different from that for a constant stage, depending on whether the discharge is increasing or decreasing, also contributing to a Àood event appearing as a loop on a stage-discharge diagram. – Variable channel storage - where the stream overÀows onto Àood plains during high discharges, giving rise to different slopes and to unsteadiness effects. 150 152 Modelling rating curve changes with time The effects of bed roughness and bed changes on rating curves in alluvial The importance of each data point can be weighted according to their age, so that the oldest streams points have the smallest contributions to the least squares error, and the most recent gaugings can If there is a rapidly-changing Àow event such as a Àood, roughness and hence resistance might also be rationally incorporated to give the most recent rating curve. In fact, the rating curve can be change relatively quickly, and the relationship between stage and discharge changes with time such constructed for any day, now or in the past. that if we were able to measure and plot it throughout the unsteady event we would obtain a looped We use a smooth function, decaying into the past, to keep all the points to some extent in curve with two discharges for each stage, and vice versa, before and after the Àow maximum. This determining the shape of the curve, the exponential weight factor z ( )=exp(  ),where is is usually described as a looped rating curve. The lecturer has usually been sceptical of that term,  a decay constant. Writing 1@2 for the “half-life”, the age at which the weight decays by a factor of viewing it more as a looped Àow trajectory on rating axes. 1 2, then the expression becomes Let us consider the mechanism by which changes in resistance cause the Àood trajectories to (w0 wq)@ 1@2 be looped, by considering a hypothetical and idealised situation. We do not know how much 1  z (w0 wq)= > bed-forms and how much individual grains are responsible for most resistance.  2 μ ¶ The ¿gure on the next page is plotted with rating curve axes, stage versus discharge. The rating where w0 is the date for which the rating curve is required, and wq is the date when point q curves which would apply if the resistance were a particular value are shown, for a Àat bed with was established. If wq Aw0 a value of zero is used. This was applied to 31 years of data from USGS Station 02448500 on the Noxubee River near Geiger, AL, USA, shown in Fig. 8.7. The co-planar grains, and for various increasing resistances. approximating spline method with 6 hand-allocated intervals was used, with a T1@2 ¿t. It can be In the top left corner is a stage-time graph with two Àood events, and another not yet completed. seen how the rating curve, and presumably the bed, has moved down over the 31 years. The points labelled O, A, ..., G are also shown on the Àow trajectory, showing the actual relationship between stage and discharge at each time.

153 155

O: Àow is low, over a Àat co-planar bed Stage Large bedforms Medium bedforms after a period of steady Àow. 12 C Small bedforms D G Flat bed A: the Àow increases. The Àow is not OA BCDEFG 1990-01-01 Time enough to change the nature of the bed, 1995-01-01 Stage and the Àood trajectory follows the 10 2000-01-01 B 2005-01-01 Àat-bed rating curve up to here. 2010-01-01 E A 2015-01-01 F B: the bed is no longer stable, grains Flood trajectory 8 Rating curves – for constant roughness move and bed forms develop. Accord- O ingly, the resistance is greater and the k (m) Discharge stage increases. 6 C: Àood peak has arrived, resistance continues to increase, a little later the stage is a maximum. D: resistance and bedforms have continued to grow until here, although Àow is decreasing. 4 E: the Àow has decreased much more quickly than the bed can adjust, and the point is close to the instantaneous rating curve corresponding to the greatest resistance. F: over the intervening time, Àow has been small and almost constant, however the time has 2 1 10 100 500 3 1 been enough to reduce the bed-forms and pack the bed grains to some extent. Now another log T,(T in m s ) Figure 8.7: Calculation of rating curves on speci¿c days using weights that are a function of measurement age — Àood starts to arrive, and this time, instead of following the Àat-bed curve, it already starts from Noxubee River near Geiger, AL, USA, USGS Station 02448500, from 1984-10-02 to 2015-05-11 a ¿nite resistance. G: after this, the history of the stage will still depend on the history of the Àow and the characteristics of the rate of change of the bed. 154 156 The rating trajectory and rating envelope — generalisations of the single curve 16 The ephemeral nature of the resistance means that in highly mobile bed streams it is not possible • to calculate accurately the Àow at any later time. It is not known what Àows and bed changes 14 will occur in the future up until the moment the curve is required to give a Àow from a routine stage measurement. 12 The view of the rating curve here is that it is an approximate curve passing through a more-or- k (m) 10 • less scattered cloud of points, where at least some of that scatter is due to Àuctuations in the preceding Àows and instantaneous state of the bed when each point was determined. 8 If a long period of time is considered, with many Àow events of different magnitudes, the Data points and time trajectory • Rating curve Àow trajectory will consist of a number of different paths and loops, the whole adding up to a 6 Computed envelope complicated web occupying a limited region. 0 5 000 10 000 15 000 20 000 3 1 Usually, however, one does not measure rating points very often, and instead of a continuous Àow T (m s ) • trajectory following an identi¿able path, one just sees a discrete number of apparently-random points, occupying a more-or-less limited region. The ¿gure shows an example for three years (1995–1997) of gaugings from Station 41 on the Red River, Viet Nam. The Àow trajectory has several large loops, barely visible in the ¿gure. Four The more stable the bed, the less will be the scatter. • passes of the halving procedure for each of the upper and lower envelopes were applied, starting Generally the points will fall in a band between a lower boundary, corresponding to smaller with 217 data points, at the end there were about 217@24 15 for each envelope. It can be seen • resistance, with an armoured bed, and an upper boundary corresponding to greater resistance that the method worked well.  with individual grains protruding and possibly bed-forms prominent when, for a given Àow, the water will be deeper and stage higher.

157 159

This leads to an extension of the idea of a single rating curve: that in a stream of variable bed 9. Sediment transport • conditions, one can never know the situation when rating data is actually to be used to predict a Àow, and so it might be helpful also to compute a rating envelope, to provide expressions for Most rivers and canals have beds of soil, of a more-or-less erodible nature, which is composed curves approximating both upper and lower bounds. of particles. Water Àowing in such streams has the ability to remove and carry the particles, or to deposit them, hence changing the bed topography. This is of great importance, for example Suggested procedure: • in predicting the scouring around and potential collapse of bridges, weirs, channel banks etc., – Calculate the approximation to all the points, the rating curve estimating the rate of siltation of reservoirs, predicting the possible form changes of rivers with a – Then delete those points which lie below it. threat to aquatic life etc.. – Approximate the remaining points, and repeat as many times as necessary, to give the upper 9.1 Sediment stability envelope. – Repeat, successively deleting all points above each curve. Dimensional analysis – As approximately half the data points are lost with each pass, the number of passes is The grains forming the boundary of an alluvial stream can be brought into motion if the Àuid limited. In practice what one would be doing is approximating the 1/8 or 1/16, say, of all forces acting on a sediment particle are greater than the resisting forces. This is usually expressed data points, those which lie furthest from the approximation to all the points. in terms of mean shear stress on the bed. If the shear stress acting at a point on the Àow boundary is greater than a certain critical value cr then grains will be transported. The variables which dominate the problem of the removal and transport of particles are: 3 2  Density of water ML j Gravitational acceleration LT 3 v Density of solid particles ML k Depth of Àow L 2 1 1 2  Kinematic viscosity of water L T Shear stress of water on bed ML T G Diameter of grain L

158 160 As we have 7 such quantities and 3 fundamental dimensions involved, there are 4 dimensionless English translation: numbers which can characterise the problem. In fact, it is convenient to replace j by j0 = j (v@ 1), the apparent submerged gravitational acceleration of the particles, and to replace by the shear velocity x = @.  Convenient dimensionless variables,p partly found from physical considerations, which occur are x2 X = = , Dimensionless stress or the Shields parameter, j G ( ) jG 0 v  roughly the ratio of the shear force on a particle to its submerged weight x G R =  , Grain Reynolds number, roughly the ratio of Àuid in-   ertia forces to viscous forces on the grain  J = v, Speci¿cgravityof the bed material, often J 2=65  (quartz)  G Shields diagram from Henderson (1966, p413). Ratio of grain size to water depth k

161 163

9.2 Incipient motion A better representation in terms of grain size The Shields diagram A problem with this is that the Àuid velocity (in the form of shear velocity x ) occurs in both abcissa R and ordinate X. It is better to form another dimensionless quantity from which x has In the 1930s Albert Shields conducted a number of experiments in Berlin and found that there was been eliminated, the dimensionless grain size (see p7 of Yalin & Ferreira da Silva 2001):  a band of demarcation between motion and no motion of bed particles, called incipient motion. He 2 1@3 1@3 represented this band on a ¿gure of X versus R . This is the best-known contribution in the study R j0  G =  = G = of sediment stability and movement.  X 2 μ ¶ μ ¶ 2 Here we consider what G means. If we take the common value of J =2=65,plusj =9=8ms , 6 2 1  4  =10 m s ,(for20C), then we obtain G 2=5 10 G with G in units of metres. For a range of particle sizes we have   × × G 0=1 1 10 100 1000 G (cm) 0=0004 0=004 0=04 0=44 This means that instead of the Shields plot (R > X) we can plot (G > X),andasG G,itis effectively a plot of (G> X), so that to determine whether a particular bed of given particle 2 size G is stable, we can just read Xcr off the ¿gure or calculate it using the formulae that we will present below. There are a number of different results and features of the ¿gure: The number of experimental points and an outlined region of experimental points show how • variable the results can be. There might be different de¿nitions of what “incipient” motion is – e.g. how many particles have to move to be deemed unstable? Shields approach was well-judged, showing a ¿nite band of results. 162 164 Fenton & Abbott (1977) followed Bagnold’s suggestion and examined the effect of protrusion • of particles into the stream. Although they did not obtain de¿nitive results, they were able to recommend that for large particles the value of Xcr was more like 0=01 than 0=045,whichseems 0=4 Neill (1967) Fenton & Abbott (1977, table 3) to be an important difference, the factor of 1@4 requiring a Àuid velocity for entrainment into the Expts in air, Miller et al (1977) Bu!ngton & Montgomery (1997) Àow of randomly-placed particles to be about half that of the sheltered case. 0=2 Envelope from Paphitis (2001, g. 4(b)) Yalin & Ferreira da Silva (2001, p8) present a mathematical approximation to the Shields results Beheshti & Ataie-Ashtiani (2008, g. 1) • Yalin’s approximation, eqn (9.1) for incipient motion, a single curve giving the critical value Xcr: 0=1 Eqn (9.2) 0=392 0=015 G2 0=068 G X =0=13 G h +0=045 1 h  = (9.1) Beds at in laboratories cr     Dimensionless 0=06   shear stress Eqn (9.2), s@G =0 Above the curve, for larger values of X (and hence larger¡ velocities or¢ smaller and lighter X 0=04 grains), particles will be entrained into the Àow. For large particles the critical shear stress approaches a constant value of Xcr =0=045.Thearti¿cial dip in the curve occurs at about 0=02 G =10, corresponding to a grain size of 0=4mm,a¿ne sand. A simpler and more general expression can be obtained which includes relative protrusion s@G 0=01 • as a parameter, and which seems to describe all the features of the ¿gure well: Eqn (9.2), s@G =1 All beds random on the 2 scale of the particles Beds random in nature 0=13 + (0=035 0=025 s@G)G Xcr = 0=5  2  > (9.2) 0=1 0=0004 cm 1 0=004 cm 10 0=04 cm 100 0=4 cm 1000 4cm G + G  Grain size: Dimensionless G and dimensional G     where  is a parameter whose value determines the rate of approach to the large G behaviour Figure 9.1: Stability diagram, dimensionless stress X plotted against grain size, showing experimental results for Xcr  and approximating functions. when Xcr 0=035 0=025 s@G. The curves in the ¿gure on page 166 were plotted with  =0=01.$ One might use s@G =0for levelled structures such as Block Ramps, in natural sediments it is expected that s@G 1 would be more applicable.  165 167

Bagnold (personal communication) suggested that Criterion in terms of stream parameters • there was probably no Àuid mechanical reason for Now we consider some simple relations to relate this to physical quantities. The shear velocity x the apparent dip in the data on the Shields diagram  itself, rather that there is an arti¿cial geometric effect is a very convenient quantity indeed. If we consider the steady uniform Àow in a channel, then the due to scale, and suggested that the Shields diagram component of gravity force down the channel on a slice of length {{ is jD {{V,whereV is slope. However the shear force resisting the gravity force is S {{. Equating the two we has been widely misinterpreted. For experiments × × with small particles, while the overall bed may have obtain D been Àattened, individual small grains may sit on = j V= top of others and may project into the Àow, so that S In this work it is sensible only to consider the wide channel case, such that D@S = k, the depth, the assemblage is random on a small scale. For giving large particles (gravel, boulders, etc.)innature, they too are free to project into the Àow, however = jk V> in the experiments which determined the Shields or in terms of the shear velocity: diagram, the bed was made Àat by levelling the tops x = = jk V> of the large particles. Hence, there is an arti¿cial   r scale effect. Grains in nature are free to project into andsointermsofthedimensionlessstress: p the Àow above their immediate neighbours, so that x2 jkV Vk experimental results for large grains with Àattened X =  = = = (9.3) j G j G (J 1) G beds are not applicable to natural situations. 0 0 

166 168 9.3 Laboratory experiments and dimensional similitude There are a number of transport formulae. Moveable-bed models are much more dif¿cult to design and operate than ¿xed-bed models, where Meyer-Peter and Mueller (1948): This is considered to be the ¿rst reliable empirical bed load the major requirement is just that the bed roughness correspond to the full scale. transport formula. They performed Àume experiments with uniform particles and with particle mixtures. Based on data analysis, a relatively simple formula was obtained, which is frequently In experiments with sediment transport, as in other areas of Àuid mechanics, it is desirable to have used: the same dimensionless numbers governing both experimental and full-scale situations. In this case tsb 1=5 =8(X Xcr) > we would like the dimensionless particle size AND the dimensionless shear stress each to have the 3 j0G  same values in both model and full scale. Using the subscript m for model and with no subscript where tsb is the volume rate of solids per unit width of stream transported as bed load. for the full scale situation, we then should have p 1@3 1@3 Einstein (1950) introduced statistical methods to represent the turbulent behaviour of the Àow. He jm0 j0 gave a detailed but complicated statistical description of the particle motion in which the exchange G m = G such that Gm = G =   2 2 probability of a particle is related to the hydrodynamic lift force and particle weight. μ m¶ μ ¶ If water is used, as it is most commonly, viscosity m =  – although note the variation with Bagnold (1966) introduced an energy concept and related the sediment transport rate to the work temperature: done by the Àuid. Temperature C 0 10 20 30 t = x G (X X ) > 2 1 6 sb b cr water (ms 10 ) 1.81.31.00.80  × where  is a function of G ,andxb is the Àow velocity in the vicinity of the bed. In the case of As gravitational acceleration j is a constant, a rough turbulent Àow,   0=5. Yalin & Ferreira da Silva (2001, p9) state that this formula is 1@3 1@3  Gm (Jm 1) = G (J 1) = preferred, as it is simple, as accurate as any, and reÀects the meaning of the bed-load rate.   If we use the same bed material, Jm = J, the implication is that we should use Gm = G,which Engelund and Hansen (1967) presented a simple and reliable formula for the total load transport might be very dif¿cult. in rivers.

169 171

We also require the same dimensionless shear stress, so from equation (9.3): 9.5 Bedforms V k Vk m m = = Two-dimensional (Jm 1) Gm (J 1) G   It is dif¿cult to be able to satisfy all the dimensionless numbers, and often loose-bed models can In most practical situations, sediments behave as non-cohesive materials, and the Àuid Àow can do little other than provide qualitative information (where does deposition occur?) rather than distort the bed into various shapes. The interaction process is complex. At low velocities the quantitative information (how rapidly does deposition occur?). bed does not move. With increasing Àow velocity the inception of movement occurs. The basic 9.4 Transport of sediment bed forms encountered are ripples (usually of heights less than 0=1m), dunes, Àat bed, standing waves,andantidunes. At high velocities chutes and step-pools may form. Typical bed forms are Consider the concept of a relative tractive force at a point, @ cr. If this is slightly greater than 1, summarised in Figure 9.2 below. only the grains forming the uppermost layer of the Àow boundary can be detached and transported. Bed forms for steady Àow over a sand bed can be classi¿ed into If @ is greater than 1, but less than a certain amount, then grains are transported by deterministic cr Lower transport regime with Àat bed, ribbons and ridges, ripples, dunes and bars, jumps in the neighbourhood of the bed (“saltation”). These modes of grain transport are referred • to as bed-load. If the ratio @ is large, then grains will be entrained into the Àow and will be Transitional regime with washed-out dunes and sand waves, cr • carried downstream by turbulence. This transport mechanism is known as suspended-load.The Upper transport regime with Àat mobile bed and sand waves (anti-dunes). • total transport rate is the sum of the two. The simultaneous motion of the transporting Àuid and the When the bed form crest is perpendicular to the main Àow direction, the bed forms are called transported sediment is a form of two-phase Àow. transverse bed forms, such as ripples, dunes and anti-dunes. Ripples have a length scale much A formula for the criterion of the initiation of suspension is from Van Rijn: smaller than the water depth, whereas dunes have a length scale much larger than the water depth. 0=3 Ripples and dunes travel downstream by erosion at the upstream face (stoss-side) and deposition Xcr,susp = +0=1(1 exp ( 0=05G )) , 1+G    at the downstream face (lee-side). Antidunes travel upstream by lee-side erosion and stoss-side  deposition. Bed forms with their crest parallel to the Àow are called longitudinal bed forms such as andinthelimitoflargeG , Xcr,susp 0=1.  $ ribbons and ridges. 170 172 References Aberle, J. & Smart, G. M. (2003), The inÀuence of roughness structure on Àow resistance on steep slopes, Journal of Hydraulic Research 41, 259–269. Barnes, H. H. (1967), Roughness characteristics of natural channels, Water-Supply Paper 1849, U.S. Geological Survey, Washington. Beheshti, A. A. & Ataie-Ashtiani, B. (2008), Analysis of threshold and incipient conditions for sediment movement, Coastal Engineering 55(5), 423–430. http://linkinghub. elsevier.com/retrieve/pii/S0378383908000021 Boiten, W. (2000), Hydrometry,Balkema. Buf¿ngton, J. M. & Montgomery, D. R. (1997), A systematic analysis of eight decades of incipient motion studies, with special reference to gravel-bedded rivers, Water Resources Research Figure 9.2: Bedforms and the Froude numbers at which they occur (after Richardson and Simons) 33(8), 1993–2029. In the literature, various bed-form classi¿cation methods for sand beds are presented. The types Chow, V. T. (1959), Open-Channel Hydraulics, McGraw-Hill, New York. of bed forms are described in terms of basic parameters (Froude number, suspension parameter, particle mobility parameter; dimensionless particle diameter). Fenton, J. D. (1992), Reservoir routing, Hydrological Sciences Journal 37, 233–246. Fenton, J. D. (2011), A note on the role of Area/Perimeter in calculating open channel Àow resistance, Technical report, Alternative Hydraulics Paper 4. Fenton, J. D. (2014), Long waves in open channels - their nature, equations, approximations, and numerical simulation, in Proceedings of the 19th IAHR-APD Congress, September 2014,Ha Noi, Viet Nam. 173 175

Three-dimensional structures Fenton, J. D. (2015), Generating stream rating information from data, Technical Report 8, Alternative Hydraulics. The largest bed forms in the lower regime are sand bars Fenton, J. D. & Abbott, J. E. (1977), Initial movement of grains on a stream bed: the effect of (such as alternate bars, side bars, point bars, braid bars relative protrusion, Proc. Roy. Soc. London Ser. A 352, 523–537. and transverse bars), which usually are generated in areas with relatively large transverse Àow components (bends, Fenton, J. D. & Keller, R. J. (2001), The calculation of streamÀow from measurements of conÀuences, expansions). Alternate bars are features stage, Technical Report 01/6, Cooperative Research Centre for Catchment Hydrology, with their crests near alternate banks of the river. Braid Melbourne. http://www.ewater.org.au/archive/crcch/archive/pubs/ bars actually are alluvial "islands" which separate the pdfs/technical200106.pdf anabranches of braided streams. Numerous bars can be Hamming, R. W. (1973), Numerical Methods for Scientists and Engineers, second edn, McGraw- observed distributed over the cross-sections. These bars Hill. have a marked streamwise elongation. Transverse bars are Henderson, F. M. (1966), Open Channel Flow, Macmillan, New York. Point bar at a river meander: the Cirque de la Madeleine diagonal shoals of triangular-shaped plan along the bed. Hicks, D. M. & Mason, P. D. (1991), Roughness Characteristics of New Zealand Rivers,DSIR in the Gorges de l’Ardèche, France. Photograph by Jean- One side may be attached to the channel bank. These type Christophe Benoist. Marine and Freshwater, Wellington. of bars generally are generated in steep slope channels with a large width-depth ratio. The Àow over transverse bars is sinuous (wavy) in plan. Side bars Keulegan, G. H. (1938), Laws of turbulent Àow in open channels, J. Res. Nat. Bureau Standards are bars connected to river banks in a meandering channel. There is no Àow over the bar. The 21, 707–741. planform is roughly triangular. Special examples of side bars are point bars and scroll bars. Liggett, J. A. & Cunge, J. A. (1975), Numerical methods of solution of the unsteady Àow equations, in K. Mahmood & V. Yevjevich, eds, Unsteady Flow in Open Channels,Vol.1, Water Resources Publications, Fort Collins, chapter 4. Miller, M. C., McCave, I. N. & Komar, P. D. (1977), Threshold of sediment motion under unidirectional currents, Sedimentology 24(4), 507–527. http://doi.wiley.com/10. 174 176 1111/j.1365-3091.1977.tb00136.x Appendix A. A general tool for simplifying problems and results Morgenschweis, G. (2010), Hydrometrie – Theorie und Praxis der DurchÀussmessung in offenen Gerinnen, Springer. – series and Taylor expansions Neill, C. (1967), Mean velocity criterion for scour of coarse, uniform bed material, in Proc. 12th It is worthwhile here suggesting a method that can often be usefully applied to give simpler results Congress IAHR, Vol. 3, Fort Collins, Colorado, pp. 46–54. that make problems and results simpler and help our understanding. Sometimes results from Pagliara, S., Das, R. & Carnacina, I. (2008), Flow resistance in large-scale roughness condition, hydraulic analyses are complicated, or the importance of results are not clear. If one can assume Canadian Journal of Civil Engineering 35(11), 1285–1293. that a parameter of the problem is small, then one can use power series expansions, often to give a simpler result with more understanding. Usually only one term in the approximation need be Paphitis, D. (2001), Sediment movement under unidirectional Àows: an assessment of empir- taken, when we say that we linearise. ical threshold curves, Coastal Engineering 43(3-4), 227–245. http://linkinghub. elsevier.com/retrieve/pii/S0378383901000151 Example 1 – Power series Samuels, P. G. (1989), Backwater lengths in rivers, Proc. Inst. Civ. Engrs, Part 2 87, 571–582. A power series expansion is where we write Strickler, A. (1923), Beiträge zur Frage der Geschwindigkeitsformel und der Rauhigkeitszahlen q (q 1) (1 + %)q =1+q% +  %2 + R %3 ,(A-1) für Ströme, Kanäle und geschlossene Leitungen, Mitteilungen 16, Amt für Wasserwirtschaft, 2! Bern. where we have introduced the “Big O” notation to show the size of¡ terms¢ we have neglected. White, F. M. (2003), Fluid Mechanics, ¿fth edn, McGraw-Hill, New York. Another common example is that of the logarithmic function WMO Manual 1 (2010), Manual on Stream Gauging Volume I – Fieldwork, Technical Report 1 2 3 1044, World Meteorological Organization. ln (1 + %)=loge (1 + %)=% % + R % >  2 WMO Manual 2 (2010), Manual on Stream Gauging Volume II – Computation of Discharge, another is the exponential function ¡ ¢ Technical Report WMO 1044, World Meteorological Organization, Geneva. 1 exp (%)=1+% + %2 + R %3 > Yalin, M. S. & Ferreira da Silva, A. M. (2001), Fluvial Processes, IAHR, Delft. 2! 177 179 ¡ ¢

Yang, S.-Q., Tan, S.-K. & Lim, S.-Y. (2004), Velocity distribution and dip-phenomenon in smooth Example 2 – Taylor series uniform open-channel Àows, Journal of Hydraulic Engineering 130(12), 1179–1186. Here we write the value of a quantity at { +  in terms of that at {, and its derivatives. It can be Yen, B. C. (2002), Open channel Àow resistance, Journal of Hydraulic Engineering 128(1), 20–39. written as the shift operator Hi({)=i({ + ), and can be evaluated as the exponential of the derivative operator G =d@d{ using the power series for exponential d 1 d2 di 1 d2i i({+)=hG = 1+ + 2 + === i({)=i({)+ ({)+ 2 ({)+==== (A-2) d{ 2! d{2 d{ 2! d{2 μ ¶ Combining expressions Often there will be more than one series in a compound expression. Usually these can be arranged so that they appear as a product (multiplication) of two or more series. All such combinations are treated as one might expect, multiplying through and retaining just the terms we need. For example 2 (d1 + d2%)(e1 + e2%)=d1e1 + % (d2e1 + d1e2)+R % = (A-3) ¡ ¢

178 180