2 mathematical models as tool for hydraulic experiments

A numerical flume

J. Versteegh

SvasekBV, Coastal and Harbour Engineering Consultants,

Rotterdam, The Netherlands

Abstract

A description is given of a combination of a bathymetric data processingsystem and a 3-D mathematical flow model wich together are used as anumerical flume. With such a system fluid dynamics experiments forengineering and/or educational purposes can be performed on a ordinary PC.

1 Introduction

This article describes a general purpose "numerical flume", consisting of acombination of a 3-D numerical flow model and a grid-based bathymetric datahandling system. This experimental tool has been developed over a number ofyears in an engineering environment and should give a fair impression of whatis needed in practical engineering and educational application of such anexperimental facility.

In a numerical flume experiment the experimenter has full and easy controlover the model parameters. Although it might be a problem to model realityin a true way, for instance in the case of turbulent flow, at least it is exactlyknown what is modelled. This can almost never said of a physical flume.

Another very attractive point in the favor of a numerical flume is the easeand accuracy of the "measurements" in such a flume. In order to assess theresults of a computation one uses post processing programs, which thus can beconsidered as numerical measuring instruments. This point stretches theimportance of good and versatile postprocessing software. As an exampleconsider the "measured" kinetic and potential energy in fig. 10. This wouldhave taken a considerable effort in a physical model, but required only a bitof extra programming in the numerical case.

Transactions on Ecology and the Environment vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

262 Hydraulic Engineering Software

2 Mathematical models as tool for hydraulic experiments

Just as in the case of a physical model, the process of building a mathematicalmodel for a specific project is often a difficult and time-consuming task. Formajor projects it is a neccessary and accepted step, although the time neededfor the set up and tuning of such a model is often underestimated. If such amodel is to be used to do experiments, say for educational purposes, it isimportant that the building process is quick and flexible. Of course, these twoaspects are opposed to each other : a general-purpose, easy-to-use model givesas a rule less accurate results than one that is specially build to solve a specifichydraulic problem.

Here we will describe a specific 3-D mathematical model (FL-3) that canbe used, in combination with a numerical bathymetric system (GriSys), as aversatile experimental tool, and which can be run on ordinary PC's, even forrather large problems. Because of the efficient use of memory the possibleamount of grid points on a typical PC with 8 Mb RAM is in the order of250,000 points. For instance, the numerical flume could have the followingdimensions (based on a horizontal grid size of 2 m. and a vertical grid size of1 m.) : length 300 m., width 80 m. depth 20 m.

The execution time for a computation of this size may run into severaldays, but costs are negligible in most cases. In order to have a clear overviewof the schematisation, all computations are done on a regular block shaped gridsystem. In general, employment of such a grid will not lead to the mostefficient use of memory, but a great advantage is the concurrence of themathematical array position with the actual geometric position in the grid. Thisrectangular shape acts as the basic shape of the numerical flume.

3 Description of the "Numerical Flume"

The numerical flume described here might be a good compromise between thedifferent possibilities mentioned above. The system is based on four distinctparts. One part consists of a bathymetric data processing system called GriSys,which is used to gather or generate depth data and to produce plots and viewsof the bathymetry. A second part combines this bathymetry with structural data(like bridge piers, floating objects) and computes the porosity coefficients (seebelow). The third part consists of the actual flow program (FL-3), while postprocessing or presentation of the computational results makes up the last part.This last part comprises routines from both systems FL-3 and GriSys. The dataflow is indicated in fig.l.

3.1 Bathymetry

The GriSys system is basically a tool to convert a set of general 3-D (x,y,z)data points into a regular two dimensional depth grid Z(i,j). This is done byinterpolating between the given points in a neighborhood of the grid point (i,j).


Hydraulic Engineering Software 263

G r i S y sS

\/\

FL

C 3-D

-3

jrTd \H,U,V.V

contour plotsperspective v i e w

: vector plot: p a r t i c l e tracks

figure 1: functional diagram

The 3-D data points may beproduced by (automated) fieldmeasurements or may be generatedby the user, for instance byspecifying depth contours. The 2-Dgrid produced by the program cansubsequently be used to generatedepth contour maps or perpectiveviews of the bottom geometry.Additionally, volumetriccomputations can be performed.The bottom topology representedon this grid is used directly by theflow program, in addition togeometrical data of structures, if present.

3.2 Flow program

In the field of river- and coastal engineering a wide range of mathematicalmodels to simulate 2- and 3-dimensional flows is available. These models areeither based on finite elements (FE), finite volumes (FV) or finite differences(FD). In the latter case the calculation is carried out on a fixed, rectangulargrid. FE and FV methods are based on variable sized elements, which offerthe user a more flexible way to represent the physical features to be modelled.Building of such a FE or FV model can be a major task, in particular in 3-D.

The flow program FL-3 is based on a simple rectangular FD grid, butavoids the inherent "coarsness" of this kind of grid near the boundaries byusing the porosity formulation (as e. g. used by the PHOENICS program(Spalding[l]). See also Hirt[2]. This method changes the FD formulation nearthe flow boundaries into a rudimentary FV formulation by computing theactual open fraction of the cells that partially cross these boundaries.If, for instance, the model geometry divides a particular grid cell in a "dry"

and a "wet" part, this cell appears in the calculation as having a smallervolume and smaller face areas than the unit cell, in the ratio of the wet part.In this manner the actual mass and flux per cell can be used in thecomputation, which thus can be made mass- and momentum conserving. Theactual location of the boundary is not noted in the formulation. This results ina very efficient method which eliminates to a large extend the disadvantagesof FD along the boundaries. (See for example fig. 9). In order to use thismethod in a flexible way some sort of grid generation system is necessary, butbecause the basic grid is geometrically simple this is not nearly as complicatedas in the case of 3-D FE grid generation.

The structural contours are given by the user together with the bottomtopology generated by the GriSys system. This geometry is projected on thegrid and from the intersections with the grid lines the porosity coefficients are



determined by the grid generation program. The four porosity coefficients pergrid cell (3 face area's and one volume) are stored in coded form in a kind ofcontrol word associated with each grid cell.

Because the three dimensional flow is calculated a special mathematicalformulation is needed to cope with the incompressibility of water. 2-D modelsgenerally do not need this, being based on the so-called "shallow water"equations, which means that the flow to be modelled is not purely twodimensional, but contains the waterdepth as a variable. Instead of being acomplication, this feature actually simplifies the solution of the flow problem.The free surface permits an exchange of potential en kinetic energy that froma mathematical point of view changes the incompressible medium into acompressible one. In stationary 3-D calculations the same approach can beemployed, using the so-called artificial compressibility method (e. g.Chorin[3], Briley & Mcdonald[4]). This method is chosen because itsformulation is basically a 3-D extension of the well-known shallow waterequations. In the latter the three degrees of freedom per computational grid cellare the U- and V velocity and the water level. In the 3-D artificialcompressibility equations we have the U-, V- and W-velocities and a pressurein each cell. One can even express this pressure as a piezometric level, thusincreasing the similarity with the 2-D case. This similarity is onlymathematical, because in the 3-D case the transient pressure waves are not inreal time : only the resulting stationary solution is physically meaningfull.Thus, for steady flow problems, this mathematical trick of artificialcompressibility removes the basic difference between the 2- and 3-D equations,and the same code can be used for 2- and 3-D (steady flow) problems.

Emphasis has been put on efficient use of RAM memory at the cost ofexecution time. The computation proceeds iteratively towards a stationaryequilibrium. The full Navier-Stokes equations are used but only a simple eddyviscosity formulation is incorporated. The solution technique isstraightforward. The flow variables are defined on a staggered grid : the U-,V- and W-velocities are located on the cell faces, the pressure in the cellcenter. Together with the porosity factors, this faciliates a mass- andmomentum conserving formulation.

In each time like iteration and in each direction a semi-explicit convectiondiffusion step is followed by an ADI calculation of the (artificial) pressurewave. Updating proceeds by direct substitution, so for each degree of freedomonly one memory location is used. For details see Versteegh[5,6]. Thecomputation proceeds with non-physical time steps until equilibrium is reached.Anomalies in the input and/or boundary data are generally detected in an earlystage and can be corrected.The free surface is represented by a frictionless upper boundary. The

pressure that the flow exhorts against this "rigid lid" is used as a measure ofthe actual free water level. As long as the variation in waterlevel is small inrespect to the vertical gridsize, this method is acceptable. On the other closed



boundaries a slip factor is prescribed which depends logarithmically on thelocal velocity.

4 Computational environment

To research workers in the larger universities and research institutes thepossibility to run the numerical experiments on an ordinary PC may not seemvery important. But while in the past decade the PC has become an normaloffice tool, present on virtually every desk, the link between this personalmachine and a centrally located large computer is less general. The formertrend towards larger and larger central computing facilities has reversed. Manypeople prefer an personal independent facility, which even in the case of fluiddynamics has become possible by the computing speed and RAM size of themodern PC. Thus, a mathematical model that can be run on a PC in a standalone mode is an attractive proposition in many cases.

The main routines that together comprise the "mathematical flume" arewritten in standard Fortran with a separate plotting interface (which faciliatesadaption to other graphic standards), but many ad hoc routines can be addedin any preferred language.

If a good windows or multi-tasking system is available, the calculation canbe done in background, using the idle moments of the PC.

5 Description of present incorporated features

5.1 Generation of input geometry

If an existing bottom is to be studied the most general way in which measureddata may be available is in the form of x,y,z-points, i.e. depths (z) on arbitrarylocations (x,y). In some cases these locations are grouped together in soundinglines, or form a more or less regular array across the field of interest. Bymeans of the GriSys system all these different data forms can be interpolatedto a regular 2-D grid which has the dimensions and grid spacing of the flowcalculation to be performed.

For many experiments however, the geometry is to be generated by theuser. In order to build a 3-D geometry the following tools are useful:- generation of data points along bottom contour lines. By "sketching"contourlines on the computer screen or digitizer a wide variety of realisticbottom topographies can easily be generated. Of course, individual (x,y,z)-points can be given "by hand" as well. They are converted to a regular 2-Dgrid as described above.- the building of structural elements by means of tri- and quadrangles. In thelatter case, a useful idea is the use of bilinear quadrangles, because then anychoice of the cornerpoints results in a uniquely defined surface. These surfacesare described by 3 or 4 corner points in 3-D space and can be automaticallycombined with bottom data. In this way engineering elements (quay structures,



bridge piers) can be modelled in combination with natural river- or seabottoms.

5.2 Presentation of the input geometry

In order to be useful as a research instrument the system of programs shouldinclude an extensive and flexible way to present the geometrical data to beused as input for the flow model. At the present stage of development, the FL-31/GriSys system offers the following possibilities :

- Presentation of measured data points (fig. 2).- Plotting of contour lines flume bottom (fig. 3).- Plotting of perspective view of same (fig. 4).- The same featues for the structural data and the 3-D grid (figs. 5 and 6).

5.3 Presentation of the results

For this purpose a number of numerical and graphical tools are available.Because of the large amount of data involved in the 3-D solution, it is veryimportant that a clear overview of the results can be obtained. A usefultechnique is to draw arrows indicating the direction and strength of thehorizontal components of the velocity and different colors to show the sign andstrength of the vertical component.

At the other hand the possibility must exist to check the result of acomputation by looking at the actual numbers. Not all possibilities can bedemonstrated here, but some are shown in figs. 7-11.

6 Examples

Two examples are presented intended to give an impression of the practical useof the numerical flume. The first is an engineering application in which theinfluence of the shape of a bridge pier on the flow carrying capacity of a riveris tested. The second example is a more theoretical application.

6.1 Flow around bridge pier

In this example the starting point is a set of artificially generated x,y,z-pointsrepresenting a river bed. In this bed two different designs of a bridge pier aretested. The first step is to interpolate the x,y,z-points to a regular 2-D gridZ(i,j) (fig. 3 and 4). In the case shown here the bottom was generated on thePC screen by specifying depth contours. The bottom generating programproduces x,y points along the specified lines with a given depth z. Then the 3-D grid is generated combining the structural data of the pier design with the2-D bottom grid. In this step the porosity factors are generated.



§ g •••

figure 2: x,y,z-points from field figure 3: depth contours ofmeasurements or generated on PC interpolated Z(ij) bottom

figure 4: perspective view of the figure 5: the two pier designs testedbottom Z(i j) of fig. 3

Two separate flow calculations were made, on a grid with the dimensions138*96*6 grid cells. Figs. 2-4 are produced by GriSys and show the bottomtopography. Fig. 6 gives the (bottom boundary of) the flow domain in the 3-DFL-3 computation. The combination of the structural data (fig. 5) with the gridof fig. 4 is done automatically on the basis of heigt comparison of each gridpoint. In fig. 6 the result for the case of pier PL1 is shown. It can be observedthat most of the base of this pier is submerged in the river bottom. The rateof submersion is controlled by the z-coordinates of the corner points of PL1in respect to the bottom elevation. For each pier design the resulting velocitydistribution is given in figs. 6 and 7. In these figures lines of constant absolutevelocity are given. Values of the velocities of course can be printed in thefigure but are omitted here. However corresponding lines might be recognizedin each figure.

Comparing these two figures it can be observed that in the deep part of thechannel "north" of the bridge pier the flow maintains a higher velocity over a



figure 6: pier PL1 embedded in 3-D grid

considerable length downstream of the bridge in the case of the square shapedpier PL2. The use of color would of course enhance the clearity.

figure 7: velocity distribution PL1 figure 8: velocity distribution PL2

6.2 Flow over a sloping bottom

This is an example of a more theoretical experiment concerning non-uniformchannel flow over a down sloping bottom section. An impression of the set upcan be obtained from fig. 11, where the flow is towards the reader. As theflow is symmetrical only half the channel is modelled. The water surface is notshown in fig. 11, but must be imagined as a more or less horizontal plane at



the top of the grid. Only the boundary of the grid is drawn in fig. 11. In fig.9 an overview of the result is given.

plan view average veloc i ty

v ert, p I an eTve I oc I fy i n m I cTchan

figure 9: velocity vectors used asprofile indicator

1: E k - t o t 2:Ek-mean 3:Ek-rot 4:Epot

figure 10: energy transport throughcross section

It can be seen that the flow contracts towards mid channel. This result can beexplained on the basis of conservation of circulation, but here we only wantto emphasis that this kind of experiment can be done very conveniently in anumerical flume. Notably, the ease of "measuring" in such a flume isdemonstrated in fig. 10 where the different components of the kinetic andpotential energy as transported through cross sections are plotted against thestreamwise direction. To obtain this result in a physical flume would have beena complicated and difficult task.

Finally an exampleof the particle trackfacility is given in fig.11.

The time needed torun these examples isshort in terms ofworking hours. Thefirst example took halfa day to set up, thesecond only a fewhours. The flowcomputation itself took(on a PC) a couple of

nights for the first and a single night for the second example. The timeinvolved in appreciating the results depends of course on the informationwanted.

References

1. Spalding, D. B., A general purpose computer program for multi-dimensional one- and two phase flow, Mathematics and computers insimulation, IAMCS, 1981, XXIII

figure 11: 3-D tracks near bottom



2. Hirt, C. W., Num. methods for bluff body aerodynamics, von Karmaninstitute for fluid dynamics, Lecture series, 1984, 6

3. Chonn, A. J., A numerical method for solving incompressible viscous flowproblems, J. Computational Physics, 1976, 2

4. Briley, W. R. & McDonald, H., On the structure and use of linearizedblock implicit schemes, J. Computational Physics, 1980, 34

5. Versteegh, L,The numerical simulation of 3-D flow through and aroundhydraulis structures, Thesis TU-Delft, 1990

6. Versteegh, J., The numerical calculation of 3-D flow, (ed C. Taylor, W.G.Habashi, M.M. Hafez), pp 2-1955 to 2-1965, Proc. of the 5th Int Conf. onNum. methods in laminar and turbulent flows, Montreal, Canada, 1978,Pineridge Press, Swansea, 1978


2 mathematical models as tool for hydraulic experiments

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