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Open channel hydraulics for engineers. Chapter 7 unsteady flow

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This chapter introduces issues concerning unsteady flow, i.e. flow situations in which hydraulic conditions change with time. Many flow phenomena of great importance to the engineer are unsteady in character, and cannot be reduced to steady flow by changing the viewpoint of the observer. The equations of motion are formulated and the method of characteristics is introduced as main part of this chapter. The concept of positive and negative waves and formation of surges are described. Finally, some solutions to unsteady flow equations are introduced in their mathematical concepts.......

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Nội dung Text: Open channel hydraulics for engineers. Chapter 7 unsteady flow

  1.  OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Chapter UNSTEADY FLOW _________________________________________________________________________ 7.1. Introduction 7.2. The equations of motion 7.3. Solutions to the unsteady-flow equations 7.4. Positive and negative waves; Surge formation _________________________________________________________________________ Summary This chapter introduces issues concerning unsteady flow, i.e. flow situations in which hydraulic conditions change with time. Many flow phenomena of great importance to the engineer are unsteady in character, and cannot be reduced to steady flow by changing the viewpoint of the observer. The equations of motion are formulated and the method of characteristics is introduced as main part of this chapter. The concept of positive and negative waves and formation of surges are described. Finally, some solutions to unsteady flow equations are introduced in their mathematical concepts. Key words Unsteady flow; method of characteristics; positive and negative waves; surge; numerical solution. _________________________________________________________________________ 7.1. INTRODUCTION In unsteady flow in an open channel, velocities and depths change with time at any fixed spatial position. Open channel flow in a natural channel almost always is unsteady, although it often is analyzed in a quasi-steady state, e.g. for channel design or floodplain mapping. Unsteady flow in open channels by nature is non-uniform as well as unsteady because of the free surface. Mathematically, this means that the two dependent flow variables (e.g. velocity and depth or discharge and depth) are functions of both distance along the channel and time for one-dimensional applications. Problem formulation requires two partial differential equations representing the continuity and the momentum principle in the two unknown dependent variables. The full differential forms of the two governing equations are called the Saint-Venant equations or the dynamic wave equations. Only through rather severe simplifications of the governing equations analytical solutions are available for unsteady flow. This situation has led to the extensive development of appropriate numerical techniques for the solution of the governing equations. A complete theory of unsteady flow is therefore required, and will be developed in this chapter. The equations of motion are not soluble in the most general case, but we shall see that explicit solutions are possible in certain cases which are physically very simple but are real enough to be of engineering importance. For the less simple cases, approximations and numerical methods can be developed which yield solutions of satisfactory accuracy. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 130
  2. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 7.2. THE EQUATIONS OF MOTION 7.2.1. Derivation of Saint-Venant equations Although the governing equations of conservation of mass and momentum can be derived in a number of ways, we apply a control volume of small but finite length, x, that is reduced to zero length in the limit to obtain the final differential equation. The derivations make use of the following assumptions (Yevjevich, 1975; Chaudhry, 1993): 1. The shallow-water approximations apply, so that vertical accelerations are negligible, resulting in a vertical pressure distribution that is hydrostatic; and the depth, h, is small compared to the wavelength, so that the wave celerity c = (gh)½. 2. The channel bottom slope is small, so that cos2 in the hydrostatic pressure force formulation is approximately unity, and sin  tan = io, the channel bed slope, where  is the angle of the channel bed relative to the horizontal. 3. The channel bed is stable, so that the bed elevations do not change with time. 4. The flow can be presented as one-dimensional with a) a horizontal water surface across any cross section such that transverse velocities are negligible, and b) an average boundary shear stress that can be applied to the whole cross-section. 5. The frictional bed resistance is the same in unsteady flow as in steady flow, so that the Manning or Chezy equations can be used to evaluate the mean boundary shear stress. Additional simplifying assumptions made subsequently may be true in only certain instances. The momentum flux correction factor, , for example, will not be assumed to be unity at first, because it can be significant in river overbank flows. 7.2.2. The equations of motion We proceed to obtain equations describing unsteady open channel flow. The terms used are defined in the usual way, and are illustrated in Fig.7.1. V2 2g h B b h h+h A H x P z Datum Fig.7.1. Definition sketch for the equations of motion Consider the channel section shown in Fig. 7.1; assuming that the slopes are small and the pressure distribution hydrostatic, the pressure difference along any horizontal line drawn longitudinally through the element has a magnitude of gh, where h is defined as the amount by which the water surface rises from the upstream to the downstream face of the element. The total horizontal hydrostatic thrust on the element, taken positive in the ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 131
  3. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- downstream direction, is therefore equal to - gbh, if h/h is small. The summation of this force over the whole cross-section clearly gives the result - gAh, where A is the cross-sectional area. The acting shear force is equal to oPx, where P is the wetted perimeter of the section and o the mean longitudinal shear stress acting over this perimeter. The two forces are not quite parallel, but it is consistent with our assumption of small slopes to regard the two forces as parallel. The net force in the direction of flow is therefore equal to: - gAh - oPx (7-1) We now consider the state of uniform flow, in which the channel slope and the cross section, as well as the flow depth and the mean velocity, remain constant as we move downstream. In this state there is no acceleration, and the net force on any element is zero. Hence from Eq. (7-1): ôo = ñgRio (7-2) where R = A/P is termed the hydraulic mean radius and io is the bed slope. io = -dz/dx (in the limit), which is equal to the water surface slope – dh/dx (in the limit) in the case of uniform flow. Note that we define these slopes so as to get positive numbers when the surface concerned is dropping in the downstream direction. Consider now the more general case in which the flow is non-uniform; the velocity may therefore be changing in the downstream direction. The force given by Eq. (7-1) is no longer zero, since the flow is accelerating. We consider steady flow, in which the only acceleration is convective, and equal to: V V x The force given by Eq. (7-1) applies to a mass Ax; therefore the equation of motion becomes: V -ñgAh - ô o PÄx = ñAV x x  dh V dV  d  V2  i.e. in the limit ôo = - ñgR     - ñgR  h +   dx g dx  dx  2g  ô o = ñgRi f (7-3) where if = - dH/dx, the slope of the total energy-head line, and may be termed the “energy- head slope” or “friction slope”. We see therefore that for any state of steady flow the shear stress o can be written as: ô o =ñgRi (7-4) dH We know that, when the flow is steady, the gradient, , of the total energy-head line is dx V2 equal in magnitude and opposite in sign to the “friction slope” if  2 . Indeed this CR statement was taken as the definition of if; however, in the present context we have to recognize the two independent definitions: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 132
  4. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ôo V2 if   2 (7-5) ñgR C R H   V2  and  z h   (7-6) x x  2g  introducing partial derivative operators, because the quantities involved may now vary with time as well as with x. To allow for variation with time, we need only to reproduce, with appropriate extensions, the argument leading up to Eq. (7-3). The acceleration term VdV/dx in that argument must now be replaced by the more general expression: V V ax  V  dV (7-7) dt x t V V where ax is the fluid acceleration in the x direction of flow; is the local and V is t t the convective acceleration, respectively. The equation of motion therefore becomes:  V V   Ah   o Px   Ax  V   (7-8)  x t    (z  h) V V 1 V  i.e. in the limit  o   R     (7-9)  x g x g t   H 1 V  so that  o   R    (7-10)  x g t  from Eq. (7-6). Substituting from Eq. (7-5), we now have: H 1 V V 2   0 (7-11) x g t C 2 R and this equation may be rewritten ie + ia + if = 0 (7-12) naming the three terms of Eq. (7-11) the energy-head slope, the acceleration slope and the friction slope, respectively. A more radical restatement of Eq. (7-11) may be made by using Eq. (7-6), and recalling that the bed slope io is equal to -z/x. We have, from Eq. (7-6): H z h V V    x x x g x ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 133
  5. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- H  h V V   io   x  x g x H 1 V   if (7-13) x g t from Eq. (7-11). Hence, Eq. (7-11) can be written: h V V 1 V V 2 if  i o     (7-14) x g x g t C 2 R steady uniform flow steady non-uniform flow unsteady non-uniform flow this equation being applicable as indicated. This arrangement shows clearly how non- uniformity and unsteadiness introduce extra terms into the dynamic equation. Like the steady-flow equations of which they are an extension, Eqs. (7-11) and (7-14) are true only when the pressure distribution is hydrostatic, i.e., when the vertical components of acceleration are negligible. The equation of continuity for unsteady flow can be derived by considering a cross section of the channel with a very short length x, as shown in Fig.7.2. 1 2 Q1 Q2 h +z x datum Fig. 7.2. Definition sketch for the equation of continuity In Section 1.1.3, Chapter 1, the equation of continuity is written in the form: Q1 = Q2 = constant But in this case, the discharges at the two ends are not necessarily the same, but will differ by the amount: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 134
  6. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Q Q 2  Q1  x x and this term gives the rate at which the volume within the region considered is decreasing. The partial derivative is necessary, because Q may be changing with time as well as the distance x along the channel. Now if h + z is the height of the water surface above the datum plane, then the volume of water between sections 1 and 2 is increasing at the rate: h z B x (Note that  0) t t where B is the water-surface width. The two terms derived must therefore be equal in magnitude but opposite in sign, i.e. Q h B 0 (7-15) x t When the channel is rectangular in section, the substitution Q = Bq leads to: q h  0 (7-16) x t Q  (AV) An alternative form of Eq. (7-15) may be written by expanding the term  , x x leading to: V A h A V B 0 (7-17) x x t the three terms of which are known as the prism-storage, wedge-storage, and rate-of-rise term, respectively. The significance of this terminology will become apparent in the treatment of flood routing problems. 7. 3. SOLUTIONS TO THE UNSTEADY-FLOW EQUATIONS 7.3.1. Characteristic differential equations The treatment of the method of characteristics dates back from the nineteenth century. A practical recent account is due to Stoker (1957). It has been further developed by many other authors, most notably by Lai (1965), McLaughlin et al. (1966), Amein (1967), Liggett (1967, 1968), Evangelisti (1969) and Strelkoff (1970). Following Courant and Friedriechs (1954) and Lai (1965), one converts the two partial differential equations of Saint-Venant into a set of four ordinary differential equations, which are called the “characteristic differential equations”. The unsteady flow equations of conservation of momentum, energy and mass were first developed by Saint-Venant (1871). Keulegan (1942), Liggett (1967, 1975), Ktrelkoff (1969) and Yen (1973), among others, made them -under several forms- suitable for the solution of particular problems. General expressions for the continuity equation and the momentum equation are introduced by Sergio Montes (1997) as: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 135
  7. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- A Q Continuity:   ql (7-18) t x V V h V  g  g  io  i f   l (V  v l ) q Momentum: (7-19) t x x A where q l is the lateral discharge Ql per unit length. The lateral inflow Ql may be due to seepage through the bed of the channel, precipitation over the free surface or, in the case of a side-overflow spillway, due to the weir discharge into the channel. Similarly, vl is the lateral inflow velocity involved in the lateral inflow Ql. In Eq. (7-19), the term V V h V  g  g  i o  if  is another form of Eq. (7-14). t x x To do this, for the case of a prismatic channel, the continuity equation (7-18) is multiplied by an unspecified coefficient  and added to the momentum equation (7-19), having set the lateral inflow velocity vl = 0: V V h  h A V h  V  g  g  i o  i f   l V     V  l q q (7-20) t x x A  t B x x  B  h A V   h q l  Applying Eq. (7-20):      V   , where B is the channel width at the  t B x   x B  free surface. Eq. (7-20) results in: V  A  V  h  g  h  q  V       V     l V  g  i o  if    l q t    x  A (7-21)  B  x  t  B In order to give some physical meaning to this algebraic manipulation, it may be remarked that the terms involving the differentials of V and h on the left hand side of Eq. (7-21) have a form closely resembling a perfect differential with respect to time: dV V  dx  V dh h  dx  h    and    dt t  dt  x dt t  dt  x The terms within the square brackets on the left hand side of Eq. (7-21) could be reduced dx to perfect differentials, if the ratio could be identified with the quantities dt  A  g  V   B  and  V+   . That is:  V  V dx A g     dt B  This double identity allows one to give a more definite expression to , as this parameter must now satisfy the condition: 2  g  =  g B B (7-22) A A gA One recognizes that the term corresponds to the velocity of translation of an B infinitesimal wave on still water, c, so that Eq. (7-21) can be written as: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 136
  8. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  V V  B  h h    V  c c    V  c    g  io  if   l  V  c  q  (7-23)  t x  A  t x  A To reduce the number of dependent variables the depth h may be replaced by the “stage variable”  as a measure of water level in the channel (Escoffier, 1962), defined by:    dA   c dh   g dh A h h c B B (7-24) o A o A o A Following from its definition,  has the dimension of a velocity. For any given cross- section  and c are only functions of the depth h and, according to Sergio Montes (1998), their relation /c equals 2 for a rectangular section and /c = 4 for a triangular section, with values comprised between these limits for trapezoidal and parabolic sections. It is possible to replace the derivatives of the depth by similar expressions in terms of : h A   . (7-25a) t gB t h A  and  . (7-25b) x gB x The momentum equation is transformed into:  V V        V  c    V  c   g io  if    V  c ql  (7-26)  t x   t x  A It is useful now to conceive of a fictitious observer moving in the x,t-plane with the velocity V  c, which is the absolute velocity in the infinitesimal wave. The slope of the trajectory of the observer dx/dt equals the speed V  c. As V and  are functions of x and t, the substantial derivatives (i.e. derivatives following the fluid motion) are: DV V V dx D   dx   and   (7-27) Dt t x dt Dt t x dt It follows from Eqs. (7-26) and (7-27) that the observer will perceive the rates of change of V and  as:  V     g(i o  i f )  l  V  c  D q (7-28) Dt A which together with the differential equation for the path of the observer:  Vc dx (7-29) dt represent the “characteristic system of equations” that replaces the partial differential equations of the unsteady-flow motion. The paths described by the observer and defined through Eq. (7-29) are called the “characteristic directions” or simply the “characteristics” of the system. There is a forward characteristic C1 defined by: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 137
  9. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  Vc dx dt and a backward characteristic C2 defined by:  Vc dx dt Both are shown in Fig. 7.3. backward characteristic forward characteristic t P  Vc dx  Vc dx dt dt dt dx x1, 0 V1, 1 x2, 0 V2, 2 x P1 P2 Fig. 7.3. Forward and backward characteristics The integration of the system of equations is now conducted along a very special path: along the forward and backward characteristics as defined above. 7.3.2. Initial condition A solution of the original system of Eqs. (7-18) and (7-19) requires that at some initial time, say t = 0, the values of the (new set of) dependent variables V,  are specified for all values of x. The same conditions are needed for the solution of the characteristic equations. Suppose then that x, V and  (or V, h and c) are known at the points P1 and P2, both at t = 0 , and that forward and backward characteristics C1 and C2 are drawn from these points. One may seek the values of V and  at the point of intersection, P, of these characteristic paths (Fig. 7.3). To accomplish this, one can integrate Eqs. (7-28) and (7-29), the “characteristic equations”, along the characteristic trajectories, and obtain: along the forward characteristic:   Vp   p  V1  1   g  i o  i f   l  V  c  dt tp q (7-30) 0   A x p  x1    V  c  dt tp (7-31) 0 along the backward characteristic: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 138
  10. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------   Vp   p  V2   2   g  io  i f   l  V  c  dt tp q (7-32) 0   A x p  x 2    V  c  dt tp (7-33) 0 The set of Eqs. (7-30) to (7-33) is a set of four equations in four unknowns: Vp, p, xp, tp. Accepting for the moment that a solution is possible, it is seen that the values of the dependent variables are determined at the points of intersection of the characteristics emanating from the points with previously determined values of Vp and p. It is furthermore possible to progressively extend the solution from the initially known conditions at t = 0, through a network of characteristics as shown in Fig. 7.4, until the region of interest in the x,t-plane is covered by an appropriate number of points. It is clear that the intersections of the characteristic lines do not occur at regular intervals of x or t, at the nodes of a uniform grid in the x,t-plane. Fig. 7.5. shows such a uniform grid superimposed on the network of characteristics. The conditions at grid point M, for example, may be deducted by bi-variate interpolation from the known values at points P, Q and S. S1 t R1 R2 Q1 Q2 Q3 x P1 P2 P3 P4 Fig. 7.4. Network of characteristics t x M Q2 t Q1 S1 S3 S2 x P1 P2 P3 Fig. 7.5. Characteristics do not generally intersect at grid-points M ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 139
  11. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 7.3.3. The simple-wave problem A simple wave is defined to be a wave for which io = if = 0, with an initial condition of constant depth and velocity and with the water extending to infinity in at least one direction. Though neglecting gravity and friction forces may not be very realistic, the simple-wave assumption is useful for illustrating the solution of an unsteady flow problem in the characteristic plane. We have specified that the undisturbed flow is also uniform: this property defines the “simple-wave” problem. It may be thought that the conditions set out above are so special as to be of limited practical interest, but this is not so. There are many practical situations in which the flow changes so quickly that the acceleration terms in Eq. (7-14) are large compared with io and if, which may therefore be neglected – to a good first approximation anyway. An example is the release of water from a lock into a navigation canal; in this case the initial surge (which is of most interest to the engineer) can be accurately treated by neglecting slope and resistance, whose effect becomes appreciable only after the wave has traveled some distance. Moreover, quite apart from immediate applications of the simple- wave problem, its study will, as we shall see, disclose principles whose interest and usefulness extend far beyond the simple wave problem alone. We now consider Eq. (7-26) in detail, for a simple case, assuming that the lateral inflow, ql, is neglected. Since io = if = 0, Eq. (7-26) can be written as:  V V       t   V  c  x    t   V  c  x   0 (7-26a)      For a rectangular cross-section we have seen that  2 , so that: c  c  c 2 and 2 t t x x Substitution in Eq. (7-26a) yields:  V V   c c   t   V  c  x   2  t   V  c  x   0 (7-26b)     from which it follows:   (V  2c)  (V  2c)   V  c   0  Dt  V  2c  D   t x  and   (V  2c)  (V  2c)    V  c   0  Dt  V  2c  D   t x  D being the total-derivative operator, representing the rate of change from the viewpoint Dt of an observer moving with the velocity V+c and V-c, respectively. The total derivatives in these above equations are zero; this means that to observers moving with velocity (V  c), the quantities (V  2c) appear to remain constant. The paths of these observers can be traced on the x,t-plane, as in Fig. 7.6, giving rise to two families of lines, called ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 140
  12. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- characteristics. Along each member of the first group, which we shall designate the C1- family, the inverse slope of the line is (V + c), and (V + 2c) is a constant; similarly, along each member of the second group (the C2-family) the inverse slope is (V – c), and (V – 2c) is a constant. C1 family t B A C2 G E D C2 F 1 Vo + co zone of quiet 0 x Fig. 7.6. Characteristic curves on the x,t-plane The two families of curves are therefore contours of (V + 2c) and (V - 2c). We shall see that in the simple-wave problem the members of the C1-family are also contours of (V + c) and are therefore straight lines, as in Fig. 7.6. To establish this result it is first necessary to prove the following introductory theorem: THEOREM: If io = if = 0, and if any one curve of the C1- or C2-family of characteristics is a straight line, then so are all other members of the same family. To prove the theorem we consider the two C1-lines AB, DE, in Fig. 7.6. DE is a straight line, and we are to prove that AB (which may be any other member of the family) is also straight. Both (V + c) and (V + 2c) are constant along DE, so that their difference c must be constant, and hence V also. It follows that cD = cE, VD = VE; also we can write, from the C2-characteristics AD and BE: VA – 2cA = VD – 2cD VB – 2cB = VE – 2cE (7-34) whence VA – 2cA = VB – 2cB (7-35) And since AB is a C1-characteristic, we have VA + 2cA = VB + 2cB (7-36) ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 141
  13. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Eqs. (7-35) and (7-36) can be satisfied only if VA = VB, cA = cB, i.e. if AB is a straight line. The theorem is therefore proved for the C1-family of characteristics; a similar proof can readily be obtained for the C2-family. We now formulate the simple-wave problem in detail. We consider a channel in which the flow is initially uniform, i.e.. V and c are constant and equal to Vo and co, respectively. A disturbance is now introduced at the origin of x, at the left-hand end of the channel; this disturbance takes the form of a prescribed variation of V (and/or c) with the time t. We postpone for the present the question of whether V(t) and c(t) may be prescribed independently, or whether the one will be dependent on the other. Also, we assume that the disturbance propagates into the undisturbed region with a velocity co relative to the undisturbed fluid, i.e. with a velocity (Vo + co). This implies that the disturbance sets up a wave front small enough to have a velocity co, i.e. we assume for the present that an abrupt wave front of finite height will not form. It follows that we can draw a straight line OF, of constant inverse slope (Vo + co), dividing the undisturbed flow, or “zone of quiet”, from the disturbed region above OF. This line will also be a C1-characteristic – the first of that family – and since it is straight, so are all the other members of the family, as in Fig. 7.6. However, the C2-characteristics are not straight lines. If we could now calculate the values of V and c appropriate to every C1- characteristic, we could obtain V and c at every point on the x,t-plane, and we should have the complete solution to the problem. This calculation is, in fact, easily carried out, given the prescribed values of V = V(t) and/or c = c(t) along t-axis. Consider any point G on this axis, of ordinate t; the C1- characteristic issuing from this point will have an inverse slope equal to:  V(t)  c(t) dx (7-37) dt We can examine the interdependence of V(t) and c(t) by drawing a C2-characteristic (shown dotted) from G to OF; whatever the form of this line may be, it indicates the result: V(t) – 2c(t) = Vo – 2co (7-38) and this equation tells us that V(t) and c(t) are not independent; only one or the other need be prescribed (indeed only one of them can be prescribed), as a description of the disturbance at the origin of x. From Eqs. (7-37) and (7-38) it follows that the inverse slope of the C1-characteristic issuing from G can be expressed in either of the two forms:  V(t)  Vo  c o dx 3 1 (7-39a) dt 2 2  3c(t)  Vo  2c o dx (7-39b) dt and from these equations V and c can readily be obtained at any point in the x,t-plane. The argument leading up to Eq. (7-39) is undoubtedly circuitous, but the end result, in the form of Eq. (7-39), presents an extremely simple treatment of problems which are physically quite real, and indeed of some practical importance. Consider, for example, uniform flow in a river discharging into a large lake or estuary. Initially the water level in the estuary is the same as the water level at the river mouth; then under tidal action the ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 142
  14. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- estuary level begins to fall. It is clearly of some interest to ask how long it will be before the water level falls by a certain amount at some specified distance upstream from the mouth. Eq. (7-39) can give the solution. 7.3.4. Numerical solution of the characteristic differential equations Although it is possible to apply the simple-wave solution to a number of cases, a more comprehensive utilization of the method of characteristics demands that the slope, friction and lateral flow terms of the complete characteristics equations, Eqs. (7-28) and (7- 29), be retained. There are three currently used methods which are based on the characteristics equations: i). The first involves the solution of the characteristics equations by finite differences, and was advocated by Stoker in his book “Water waves” in 1957. ii). The second is a modification of the general method of integrating the characteristics equations along the characteristics by restricting the range of integration to a specific time-interval t, and a fixed space-interval x. The method is attributed to Hartree (1958). iii). The third involves the construction of a general network of characteristics in which the values of the dependent variables, say V and h, are calculated at non- regular points with time and space intervals which may vary at different locations in the x,t-plane. Conceptual schemes for methods (i) and (ii) are shown in Fig. 7.7, Fig. 7.8 and Fig. 7.9. Due to the limitation of time and knowledge requirements, it is impossible to go further into details for each method. Students may have a chance to meet this subject again when climbing up to the Master’s training programme in Water Resources Engineering. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 143
  15. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- t t + t P (j+1)t t L M R t jt x x (i –1)x ix (i +1)x Fig. 7.7. Finite-difference grid for numerical solution by Stoker’s method forward characteristic backward characteristic t t Q t + t P C_ C+ C_ N S t L M R x x x=0 x-x x x+x Fig. 7.8. Determination of conditions at points P and Q from those at the previous time step t t t + t P Q Q C+ C_ C+ N t L S M R L S M x x x-x x x+x x=L Fig. 7.9. Method of characteristics with specified time intervals ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 144
  16. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 7.4. POSITIVE AND NEGATIVE WAVES; SURGE FORMATION In the simple-wave problem discussed in Section 7.3.3 the disturbance introduced at one end of the channel may be positive, i.e. such as to increase the depth, or negative, reducing the depth. An important difference between the two consequent types of wave becomes apparent on considering Eq. (7-39b). If the disturbance is negative, c(t) deceases as t increases, so the inverse slope dx/dt given by Eq. (7-39b) diminishes as we move up to the t-axis. It follows that the slopes of the C1-characteristics increase as t increases, i.e. that the characteristics diverge outwards from the t-axis, as shown in Fig. 7.10. From this it follows that negative waves are dispersive, i.e. that sections having a given difference in depth move further apart as the wave moves outwards from its point of origin. t xs  A t  envelope of V+c intersections F O x (b) (a) Fig. 7.10. The convergence of characteristics and steepening of the wave front in a positive wave When the disturbance is positive, on the other hand, the C1-characteristics converge, as in Fig. 7.10a, and must eventually meet. Such an intersection implies that the depth has two different values in the same place at the same time – an obvious anomaly. What in fact happens is equally obvious: the wave becomes steeper and steeper, as in Fig. 7.10b, until it forms an abrupt steep-fronted wave – the surge, or “bore”. While the intersections of neighbouring characteristics will form an envelope as in Fig. 7.10a, the surge will actually form at the “first” point A of the envelope – i.e. the point having the least value of t. The front of the surge will not necessarily be broken and turbulent; it may, like the hydraulic jump, consist of a train of smooth unbroken waves if the depth ratio is small enough. It follows that a surge would certainly not break at the first point of formation, as at A in Fig. 7.10a, for the depth ratio there approaches unity. Breaking would only occur after subsequent development of the surge beyond the point A; tracing this development would be a matter of some difficulty. However, in all the subsequent argument the term surge will be applied to any abrupt change in depth, as indicated by the point A, whether it is undular or broken. The intersection of any neighbouring pair of characteristics can be located by an elementary geometrical argument. With the terms defined as in Fig. 7.10a, the following results are obtained:  t sin   t sin 2     x s sin  (7-40) xs ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 145
  17. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- and    (tan  ) cos 2  (7-41)  t tan 2  xs   (tan  ) whence (7-42)  V  c 2 xs  d  V  c  dt (7-43) where (V+c) equals the inverse slope of the characteristic, as given by Eq. (7-39). Substituting Eq. (7-35b) into Eq. (7-43), we obtain:  3c(t)  Vo  2c o = (V + c) dx dt  V  c   3c(t)  Vo  2c o  2 2  xs  d  V  c  dt d  3c(t)  Vo  2c o  dt 3c(t)  Vo  2co  2  xs  (7-44) 3dc(t) dt From this equation an envelope of intersections can be traced, given a specified disturbance c = c(t) along the t-axis (Henderson, 1966). If, as in the case of the release of water from a lock, the disturbance is specified as a variation in q = q(t), corresponding values of c and dc/dt can readily be obtained (Henderson, 1966) by the use of Eq. (7-38) and inserted in Eq. (7-44). Once a surge does develop, there is of course an energy loss across the surge, and characteristics cannot be projected from one side of the surge to the other. But it can be shown (Henderson, 1966) that the flow on each side of the surge can be described by a separate system of characteristics. We may also note here that in the case of the negative simple wave, Eq. (7-38) can be written in the more general form: V – 2c = Vo – 2co, i.e. a constant (7-45a) or V  2 gh  Vo  2c o (7-45b) applicable to the whole of the x,t-plane, because a C2-characteristic can be drawn from any point on the plane to the line OF (Fig. 7.6) bounding the zone of quiet. In many cases it is more convenient to solve problems by the direct use of Eq. (7-45) and the concept of wave motion discussed in Section 7.3, rather than by use of the x,t-plane. It need hardly be emphasized that, if a surge forms, Eq. (7-45) could be applied only by using different constants on opposite sides of the surge. The form of the negative-wave profile can readily be deduced by recalling the significance of Eq. (7-39b). This equation gives the speed at which a section of constant depth h moves; since for a given h, this speed is constant, we may replace dx/dt by x/(t - t1) where t1 is the value of t at x = 0 (e.g. as at the point G in Fig. 7.6). Substituting for c, we can then rewrite Eq. (7-39b) as:  3 gh(t1 )  Vo  2 gh o x (7-46) t  t1 where h = h(t1) prescribes the initiating disturbance along the t-axis. Eq. (7-46) defines the wave profile completely in space and time. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 7: UNSTEADY FLOW 146
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