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Über dieses Buch

What is Computational Hydraulics? Computational hydraulics is one of the many fields of science in which the application of computers gives rise to a new way of working, which is intermediate between purely theoretical and experimental. It is concerned with simulation ofthe flow of water, together with its consequences, using numerical methods on com­ puters. There is not a great deal of difference with computational hydrodynamics or computational fluid dynamics, but these terms are too much restricted to the fluid as such. It seems to be typical of practical problems in hydraulics that they are rarely directed to the flow by itself, but rather to some consequence of it, such as forces on obstacles, transport of heat, sedimentation of a channel or decay of a pollutant. All these subjects require very similar numerical methods and this is why they are treated together in this book. Therefore, I have preferred to use the term computational hydraulics. Accordingly, I have attempted to show the wide field of application by giving examples of a great variety of such practical problems. Purpose of the Book It is getting a normal situation that an engineer is required to solve some engineering problem involving fluid flow, using standard and general-purpose computer programs available in many organizations. In many instances, the software has been designed with the claim that no numerical or computer-science expertise is needed in using them.

Inhaltsverzeichnis

Chapter 1. Introduction

Abstract
Computational hydraulics is one of the many fields of science in which the application of computers gives rise to a new way of working, which is intermediate between purely theoretical and experimental. It is concerned with simulation of the flow of water, together with its consequences, using numerical methods on computers. There is not a great deal of difference with computational hydrodynamics or computational fluid dynamics, but these terms are too much restricted to the fluid as such. It seems to be typical of practical problems in hydraulics that they are rarely directed to the flow by itself, but rather to some consequence of it, such as forces on obstacles, transport of heat, sedimentation of a channel or decay of a pollutant. All these subjects require very similar numerical methods and this is why they are treated together in this book. Therefore, I have preferred to use the term computational hydraulics. Accordingly, I have attempted to show the wide field of application by giving examples of a great variety of such practical problems.
Cornelis B. Vreugdenhil

Chapter 2. Water Quality in a Lake

Abstract
In the next few chapters, the main ideas of this book are introduced in a very simple example, where most of the equations can be solved analytically. The type of model is called a “box” model and it is governed by ordinary differential equations. In the simplest case, there is just one first-order equation and the system is accordingly called a first-order system.
Cornelis B. Vreugdenhil

Chapter 3. Numerical Solution for Box Model

Abstract
For simple cases the differential equation for the box model describing the water quality problem of Chapter 2 can be solved analytically. If the inflow or the waste discharge varies in an arbitrary way, this is no longer so. Moreover, in many applications of box models the equations will not be so nicely linear. In general, you will need numerical techniques and these can be illustrated very well for the water quality example.
Cornelis B. Vreugdenhil

Chapter 4. Transport of a Dissolved Substance

Abstract
Suppose that a certain quantity of some substance is discharged into a river, how is it going to be transported? For the time being, diffusion is neglected. This is not very realistic, but you will find more about that in later chapters. The consequence is that the substance is carried with the packet of water in which it was discharged, see Fig. 4.1. If the discharge takes a time δt, the length of the packet is uδt, where u is the flow velocity. After a time t, the entire packet has been transported over a distance ut. In the meantime, the substance may have decayed with a relaxation time T, just as in Chapter 2 (now only due to degradation, as there is of course no in- or outflow for the packet of water). In fact, you could use the same formulation as in that chapter in a frame of reference moving with the flow velocity. However, that approach cannot be easily generalized to more complicated cases. Therefore, it is better to consider the mass balance for a (stationary) elementary control volume shaped as a slice which covers the entire river cross-section A over a length Δx.
Cornelis B. Vreugdenhil

Chapter 5. Explicit Finite-Difference Methods

Abstract
If you restrict yourself to using the same grid points as those in section 4.2, a general explicit method can be written as
$$\frac{{c_j^{n + 1} - \frac{1}{2}\alpha \left( {c_{j + 1}^n + c_{j - 1}^n} \right) - \left( {1 - \alpha } \right)c_j^n}}{{\Delta t}} + u\frac{{c_{j + 1}^n - c_{j - 1}^n}}{{2\Delta x}} = 0$$
(5.1)
where α is a free parameter that can be manipulated for stability and accuracy. An exercise for this chapter shows that this is indeed a general method for approximation of the simple-wave equation without a decay term. If α=1, you get the method of Lax; therefore the general case can be called a modified Lax method. The difference equation used in section 4.2 is a special case α=0.
Cornelis B. Vreugdenhil

Chapter 6. Kinematic Waves

Abstract
In the theory of long waves in rivers, to be discussed in chapter 15, an extreme case can be considered in which the momentum equation reduces to a simple equilibrium between bottom friction and gravity (for a more detailed discussion see that chapter):
$${c_f}\frac{{Q\left| Q \right|}}{{RA_s^2}} = gi{\text{ }}$$
(6.1)
Cornelis B. Vreugdenhil

Chapter 7. Diffusion

Abstract
Consider the flow of groundwater in a more or less horizontal porous layer of soil (Fig. 7.1). The mass balance for a column of soil is similar to that for a river:
$$\frac{{\partial a}}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {au} \right) = w/n$$
(7.1)
where
$$\begin{array}{*{20}{c}} {a = thickness\;of\,the\,water\,layer\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {u = flow\,velocity\,averaged\,over\,the\,layer\,\,\,\,\,\,} \\ {w = ra\inf all\,in\,volume\,per\,unit\,surface\,area} \\ {n = porosity\,of\,the\,soil\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \end{array}$$
It has been assumed that rainfall reaches the groundwater layer very quickly; temporary storage in the unsaturated soil above the groundwater layer is not taken into account.
Cornelis B. Vreugdenhil

Chapter 8. Numerical Accuracy for Diffusion Problems

Abstract
To study the numerical accuracy for diffusion problems, it is no longer sufficient to consider one sinusoidal wave, as in previous chapters. You must take into account that an arbitrary function, acting as initial condition, can be thought to be built up from a series of sinusoidal functions, which is called a Fourier series. As an example take a block function with length 2L (Fig. 8.1).
Cornelis B. Vreugdenhil

Chapter 9. Diffusion Model for Coastline Development

Abstract
For the global behaviour of a coastline there exists a simple theory which, without going deeply into the physics, can give a useful insight in the consequences of breakwater construction, beach nourishment etc. The approach is originally due to Pelnard-Considère.
Cornelis B. Vreugdenhil

Chapter 10. Consolidation of Soil

Abstract
If soil, saturated with water, is loaded, e.g. by constructing a building on top of it, a deformation can occur, even if water and grains are considered incompressible. The reason is that the grains can move relative to one another, such that the pore volume changes. The water in the pores has to flow in or out, which takes some time. This process of consolidation is therefore time dependent. For a more comprehensive discussion see Verruijt (1983). Here, we consider a simple case where the process occurs in the vertical dimension only (Fig. 10.1).
Cornelis B. Vreugdenhil

Chapter 11. Convection—Diffusion

Abstract
In Chapter 4, the transport of a dissolved substance was discussed without taking diffusion into account. This is not very realistic as there are a number of causes by which a substance (such as salt or a waste material but also temperature) is spread out in addition to being transported with the mean flow: molecular diffusion, turbulent mixing, and (very importantly) variations of flow velocity over the river cross-section. For a detailed discussion see, e.g., Fischer et al. (1979). You will find it intuitively plausible that a substance tends to be transported by these processes from locations where the concentration is high to locations where it is small, such as shown in Fig. 11.1. This is called diffusion here, whatever the cause. For the spreading by velocity variations, the term dispersion is often used, which has, however, other meanings as well. The transport with the mean velocity is called convection and the combined phenomenon convection—diffusion.
Cornelis B. Vreugdenhil

Chapter 12. Numerical Accuracy for Convection—Diffusion

Abstract
In order to study the numerical accuracy for the convection-diffusion equation, you could follow the same approach as in Chapter 8. Unfortunately, this gets quite complicated, as an additional parameter (the Courant number) comes in. For practical purposes, it is often sufficient to consider diffusion and convection separately, and in that order. The procedure is then:
(a)
determine relevant length and times scales for diffusion, as described in sections 8.1 and 8.2.

(b)
determine the numerical parameters for diffusion as described in section 8.3.

(c)
for the shortest relevant wave length found in step (a), determine the numerical parameters for pure convection, as described in section 5.5 (taking into account, of course, the correct speed of propagation). The idea is that shorter waves will be damped by diffusion, so there is no point in trying to “convect” them accurately. Moreover, the shortest wave is the critical one for numerical accuracy.

(d)
use the smaller of the time steps and grid intervals, found in steps (b) and (c).

Cornelis B. Vreugdenhil

Chapter 13. Salt Intrusion in Estuaries

Abstract
In a tidal river, salt water from the sea tends to penetrate due to its slightly greater density. The form in which this happens depends on the tidal influence. If there is hardly any tide, the salt water will penetrate underneath the fresh river water as a “salt wedge”. Far more common is the situation where a strong tidal action takes care of a mixing process, such that the water in a particular cross-section has an almost uniform salt concentration; however, this concentration varies in longitudinal direction. This is called the well-mixed case. The salt concentration is now governed by the convection-diffusion equation (11.4) with two complications:
(i)
the hydrodynamic variables A and Q are now functions of space and time, to be computed as discussed in Chapter 16;

(ii)
the diffusion coefficient is influenced by the longitudinal gradient of salt concentration. This is a rather complicated matter (cf. Thatcher and Harleman 1972),that is not discussed here.

Cornelis B. Vreugdenhil

Chapter 14. Boundary Layers

Abstract
The idea of boundary layers can be illustrated very well using the example of sediment transport in suspension. If water in a river flows over a sandy bottom at sufficiently high velocity, sand will be picked up and carried with the flow, even though it tends to fall back. The process by which sediment particles are kept in suspension is turbulent diffusion. Just as in chapter 11, turbulence causes an effective transport from regions with high sand concentration (near the bottom) to those with low concentration (near the water surface). If you consider a steady-state situation, the mass balance for suspended sand is
$$u\frac{{\partial c}}{{\partial x}} + \frac{{\partial s}}{{\partial z}} = 0$$
(14.1)
in which the sediment flux in vertical direction is
$$s = {w_s}c - D\frac{{\partial c}}{{\partial z}}{\text{ }}$$
(14.2)
where x and z are the horizontal and vertical coordinates, D is the turbulent diffusion coefficient and ws the settling velocity or fall velocity of the sand particles.
Cornelis B. Vreugdenhil

Chapter 15. Long Waves

Abstract
Tidal waves in rivers and seas, flood waves in rivers, but also oscillations in harbour basins are long waves relative to the water depth. A mathematical formulation can be obtained by integrating the general hydrodynamic equations over the depth or over a river cross-section. To understand the principles, it is sufficient to use a very much simplified set of equations in this chapter and the next one. For completeness, some of the corresponding results for the complete equations are given in appendix 1. For further reference, you can consult books such as Jansen (1979) or Cunge et al (1980).
Cornelis B. Vreugdenhil

Chapter 16. Numerical Methods for Long Waves

Abstract
From the simplified Saint-Venant equations (15.1) and (15.2), you can easily eliminate one of the unknowns, so that you obtain one differential equation with one unknown. However, for the general equations given in appendix 1, this is much more complicated and moreover not very useful. The straightforward way is to discretize the equations directly. To this end, a grid in the x-t plane is chosen with spatial grid size Δx and time step Δt (Fig. 16.1).
Cornelis B. Vreugdenhil

Chapter 17. Long Waves in Two-Dimensional Areas

Abstract
The theory of long waves applies also to tidal waves or storm surges in the sea, in shallow coastal areas and estuaries, to flood waves in rivers with flood plains and similar situations. In many such cases, the wave length is so much larger than the water depth that a two-dimensional, depth-averaged mathematical model is adequate. The formulation is essentially the same as in Chapters 15 and 16, if the dependence on two horizontal coordinates x, y is taken into account. See Fig. 17.1 for definitions.
Cornelis B. Vreugdenhil

Chapter 18. Finite-Difference Methods for Two-Dimensional Long Waves

Abstract
In order to solve the two-dimensional long-wave equations numerically, the region of interest is covered by a rectangular grid. Recently, curvilinear grids that fit to the boundaries have been used but this is outside the scope of this book. There are various ways of arranging the variables in the grid, three of which are shown in Fig. 18.1.
Cornelis B. Vreugdenhil

Chapter 19. Potential Flow

Abstract
The flow of an incompressible fluid is generally described mathematically by the Navier-Stokes equations (14.9) and (14.11) (in two dimensions; otherwise there is an additional, similar equation), together with the equation of continuity
$$\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0$$
(19.1)
The numerical solution of these equations is not discussed in this book. There are, however, certain situations in which they can be considerably simplified. Consider a quantity called the vorticity $$\omega = s\partial u/\partial y - \partial v/\partial x$$, which is a measure of the rotation of a fluid particle about its axis (normal to the plane of flow; in 3-d the vorticity is a vector).
Cornelis B. Vreugdenhil

Chapter 20. Finite-Difference Method for Potential Flow

Abstract
For a numerical solution of the potential equation, the region in the (x, y) plane is covered by a grid with sizes Δx, Δy (not necessarily equal); an example is given in Fig. 20.1. If you limit yourself to regular and rectangular grids, the boundaries of the region have to be represented by broken lines through the nearest grid points; the same difficulty was met in Chapter 18. Here, there is no question of staggered grids, as there is only one unknown. If you replace each term in the potential equation (19.4) by the straightforward finite-difference expression (there is not too much of a choice), you get for the special case of Δx= Δy:
$${\phi _k}_{ + 1}{,_j}{\text{ }}{\phi _k} + {\phi _{k - 1,j}} + {\phi _{k,j + 1}} + {\phi _{k,j - 1}} - 4{\phi _{k,j}} = 0$$
(20.1)
If you mark the weighting coefficients for each grid point in Fig. 20.1, you obtain a “finite-difference molecule” connecting each grid point to its four direct neighbours. Each equation (20.1) contains five unknowns, so that the method is implicit and you will have to solve the set of such equations for all grid points together.
Cornelis B. Vreugdenhil

Chapter 21. Finite-Element Method

Abstract
If you want to compute flows in regions of a nicely rectangular form, there is no problem in using regular grids as in the previous chapter. By means of “telescoping grids” (Fig. 21.1), it is even possible to concentrate grid points in certain parts of the region where you want to have higher accuracy, for example because you expect the higher derivatives in the truncation error (eq. 20.4) to be large. However, for regions with a complicated shape such modifications are insufficient. What you would like to have is a grid that (i) is adapted to the boundaries, and (ii) has the possibility of local refinement for better accuracy. In recent years, such techniques have been developed in finite-difference form (see, e.g. Thompson et al., 1985). Another very powerful method is the finite-element method, of which a very simple variety is discussed here, based on a triangular “grid” (Fig. 21.2). There are several good books available, both for beginners and specialists (Pinder & Gray 1977, Taylor & Hughes 1981, Baker 1983, Chung 1978).
Cornelis B. Vreugdenhil

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