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Two-dimensional, multi-grid, viscous, free- surface flow calculation E.D. Farsirotou*, J.V. Soulis** & V.D. Dermissis* *Aristotle University of Thessaloniki, Civil Engineering Department, Division of Hydraulics and Environmental Engineering, Thessaloniki, 54006, GREECE **Democrition University of Thrace, Civil Engineering Department, Fluid Mechanics/Hydraulics Division, Xanthi, 67700, GREECE Abstract A two-dimensional subcritical and/or supercritical, viscous, free-surface flow calculation method is employed. A very important feature of the method is its simplicity and the physical understanding obtained from the solution procedure. The grid used may be irregular and conforms to the physical boundaries of the problem. A multi-grid algorithm has been developed to accelerate the convergence solution. The viscous flow stresses are described using either fixed value eddy viscosity coefficient or values related to the flow properties. Applications regarding subcritical viscous flow near spur-dike and flow in a supercritical channel are reported. Calculated results are in good agreement with measurements and/or other numerical solution results. Convergence histories are presented and the results indicate that substantial computational savings for the solution accuracy can be achieved. 1 Introduction There exist a class of free-surface flow problems which can adequately be described in the context of depth-averaged two-dimensional mathematical models. These simplified representations of a three- dimensional flow are justified where turbulent mixing, due to bottom roughness, effectively generates a uniform velocity distribution over the Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

* Aristotle University of Thessaloniki, Civil Engineering ... · * Aristotle University of Thessaloniki, Civil Engineering Department, ... Fluid Mechanics/Hydraulics Division,

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Two-dimensional, multi-grid, viscous, free-

surface flow calculation

E.D. Farsirotou*, J.V. Soulis** & V.D. Dermissis*

* Aristotle University of Thessaloniki, Civil Engineering

Department, Division of Hydraulics and Environmental

Engineering, Thessaloniki, 54006, GREECE

**Democrition University of Thrace, Civil Engineering

Department, Fluid Mechanics/Hydraulics Division, Xanthi,

67700, GREECE

Abstract

A two-dimensional subcritical and/or supercritical, viscous, free-surface flowcalculation method is employed. A very important feature of the method is itssimplicity and the physical understanding obtained from the solution procedure.The grid used may be irregular and conforms to the physical boundaries of theproblem. A multi-grid algorithm has been developed to accelerate theconvergence solution. The viscous flow stresses are described using either fixedvalue eddy viscosity coefficient or values related to the flow properties.Applications regarding subcritical viscous flow near spur-dike and flow in asupercritical channel are reported. Calculated results are in good agreement withmeasurements and/or other numerical solution results. Convergence histories arepresented and the results indicate that substantial computational savings for thesolution accuracy can be achieved.

1 Introduction

There exist a class of free-surface flow problems which can adequatelybe described in the context of depth-averaged two-dimensionalmathematical models. These simplified representations of a three-dimensional flow are justified where turbulent mixing, due to bottomroughness, effectively generates a uniform velocity distribution over the

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

14 Hydraulic Engineering Software

depth of the flow field. For 2-D, free-surface flows in complex geometryit is convenient to make predictions using non-orthogonal boundary fitted

computational meshes. Numerical solutions have been presented as earlyas in 1965, Liggett et al [1], but the convergence rate that have achieved

was quite low. Molls et al [2] derived a depth - averaged open channelflow model while Molls et al. [3] applied an ADI implicit scheme and theMacCormack explicit scheme, both second-order accurate to numericallysimulate 2-D flow near a spur-dike. Soulis [4] developed an explicitfinite - volume (transformed grid) numerical technique to simulatesubcritical and supercritical depth-averaged free-surface flows.

A multiple-grid algorithm has been developed to accelerate theconvergence solution of the free-surface flow equations to obtain thesteady flow. In the multi-grid application the corrections to the fine-gridpoints are transferred to a coarse grid to maintain the low truncationerrors associated with fine level of discritizations. Dense grid isincorporated wherever high gradient regions are encountered. At thesame time viscous flow stresses are described using either fixed valueeddy viscosity coefficient or values related to the flow properties.Applications regarding subcritical viscous flow near spur-dike and flowin a supercritical channel are reported.

2 Governing Flow Equations

The two-dimensional, unsteady, free-surface flow in channels with fixedbed is described by a system of non-linear, hyperbolic partial differentialequations as follows

3h a(hu) 3(hv)= — — - + — — -

3t dx dy(water continuity) (1)

9y 3x

(x - momentum) (2)

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

Hydraulic Engineering Software 15

a(hv)"~aT ax

^

-+• -gMSn -S, )

,u) , a(hv)-+

ax+3y

'a(hv) a(hv)

' ^T--f-

(y - momentum) (3)

and SQ are the channel slopes while S^ and S^ are the friction

slopes which are defined as

S, =•fx

h/3

S, =fy 4/

h/3

(4)

where n is the Manning's flow friction coefficient. The viscous flow

stresses are described using either fixed value eddy viscosity coefficient

or values related to the flow properties

3 Numerical Formulation and Flow Boundary

Conditions

A major improvement was obtained by solving the equations of flow (1)-(3) in integral form, i.e. by applying the equations of continuity, x-momentum and y-momentum to a series of finite-volumes with adjacentvolumes sharing a common face, Soulis [5]. The two-dimensional flowequations may be written as conservation equations for a control volume

AV of unit height and for a time step At as

-Ah = [A(hu)Ay + A(hv)Ax]At

AxAy(5)

-A(hu) = Ay-f A(huv)AxAt

AxAy

Ax

A[ vA(hu)Ax + vA(hv) Ay]Ax

AxAy

(6)

At

AxAy

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

16 Hydraulic Engineering Software

-A(hv) = Ax+A(huv)AyAt

AxAy

A[vA(hu)Ax+vA(hv)Ay]-AxAy

A[2.0xvA(hv)]AyAx

(7)

At

AxAy

Figure 1 shows the notation used for mass flux balancing across a finitevolume (cell) of the flow. Similar notation is adopted for the balancing ofthe x-momentum, y-momentum fluxes. More details of the appliednumerical technique can be found elsewhere, Soulis[5]

(XFLUX)._. = (hu). . Ay , (YFLUX), . = (hv). . Ax - (huB). . DYB

(hu). .+(hu) . ,(huB). . = ^ ''•' ^ *'•>-* ,

'•> 2

(hu). , +(hu). . (hv). . +(hv). ,(hu). V ''*'•' ;./'•'. (hv). . = ''•' '"-*\ /i,j ^ ij ' v /i,j

Figure 1: Notation for the mass flux across a finite-volume (cell).

Appropriate boundary conditions need to be applied. Once a steadystate solution is obtained, the sum of the fluxes of each conservedvariable over the faces of each finite volume will be zero and hence theconservation equations are satisfied irrespective of how these changesare distributed. It was decided to send half of the information regardingthe changes in water depth h as well in hu and hv to the upstream face ofthe finite volume involved, while the other half is sent to the downstreamface. The iterations continue until the average relative error based on the

averaged axial velocity drops below 10^ .

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

Hydraulic Engineering Software 17

For subcritical flow entrance the following boundary conditions

have been proved valid: along the upstream boundary a relative flow

direction is specified; across the flow field of the hydraulic structure a

fixed value for the flow rate Q is also specified. At the downstream

boundary a uniform across the width water depth hi is assumed. In caseswhere a mixed subcritical-supercritical type of flow is to be encountered(always subcritical entrance) a value for the upstream total head isspecified instead of the flow rate. For supercritical flow entrance thefollowing conditions have been proved to be valid: along the upstreamboundary the transverse flow direction velocity component is specified: auniform across the width water depth is also specified along with thetotal available head. To close the problem, the condition of no mass flowacross the solid boundaries needs to be applied. The normal velocitycomponent across the solid boundary is set equal to zero.

4 The Multi -Grid Algorithm

The multi-grid algorithm described here consists of a fine-grid solutionprocedure and a coarse-grid acceleration scheme. In the current approach

the solution is advanced simultaneously on the coarse and on the fine-grid. Figure 2 shows a 2x2 multi-grid block. Thus, a coarse grid isformed by combining a group of finite-volumes into a block. At the endof every time-step, during the solution procedure, the changes

Ah, A(hu), A(hv) defined by Eqs (5)-(7), are known for each finite-

volume of the fine-grid mesh. The changes for the block, of the abovementioned flow properties, can be found by summing the fluxes aroundits faces. An other way to find the changes for the block is to sum thealready calculated changes for the finite - volume within the block. For atypical 2x2 block size application these changes are determined asfollows,

= Ah + Ah + Ah + Ah (8)

subscripts 1,2,3,4 denote the mesh control volumes, see Figure 2.Similarly,

+ A(hu) (9)

= A(hv), 4- A(hv)2 + A(hv) 4- A(hv) (10)

Thereafter, the changes of the flow properties at fine-grid points arecalculated from formulae,

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

18 Hydraulic Engineering Software

new T fine .(11)

(12)

(13)

where 1^ S is a linear interpolation operator. The calculations using

Eqs (!!)-( 13) are repeated for the remaining 2,3 and 4 finite-volumes of

the under consideration block. With the multiple grid application, the

use of the blocks minimizes the computational work needed topropagate the unsteady waves out of the computational domain so that a

steady state solution is rapidly reached. The convergence of the multi-grid method is typically 3 times faster than a single grid. Figure 3 showsthe convergence histories for supercritical type of flow calculation usingvarious multi-block schemes. The 2x2 multi-block grid gives the fastestconvergence.

Figure 2: 2x2 block-size grid.

O.H

0.01-

0.001-

0.0001-

le-05

no multi-grid

2x2 blocks •

3x3 blocks

5x5 blocks

0 500 le+03 1.5e+03 2c+03, 2.5e+03 3e+03 3.5e+03 4c+03 4.5c+03 5c+03iterations

Figure 3: Typical convergence histories using variable block-size grids.

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

Hydraulic Engineering Software 19

5 Applications

5.1 Subcritical Flow near Spur - Dike

The subcritical flow near a spur-dike, see Figure 4, is numericallysimulated. Tests were conducted in a straight rectangular flume. The

spur-dike consisted of an aluminum plate 3.0 mm thick and 15.2 cmlong, projecting well above the water surface. For the spur-dike problemthe area around the spur and the recirculation zone are high gradientregions. Therefore, the grid is dense near the spur and near the side wallto which the spur is attached. The grid was adequately refined whenfurther refinement did not perceptibly change the computed results. The(79x49) grid used in this study is shown in Figure 5. For simplicity, aconstant eddy viscosity model is used to close the effective stresses. The

eddy viscosity near the spur is determined (v, = 0.0012m/s ) and is

assumed to exist throughout the entire flow field. For a smooth flume,Manning n is typically assigned a value of 0.010. Figure 6 shows thevelocity agreement between current method predictions and predictionsderived by Tingsanchali et al. [6] also using a 2-D depth averaged flowmodel. Main flow features are well predicted. The agreement betweenthe two method predictions seems to be satisfactory. Predicted isodepths

are shown in Figure 7. -Downstream Length-I— lips

Recirculating ZoneSeparation streamline

y=0,9 n

h—Length of recirculation—-jFigure 4: Definition sketch for spur-dike subcritical flow.

Node (79,49)

0,00 x-4,5 n

Figure 5: Computational grid.

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

20 Hydraulic Engineering Software

x/b=0

x/b=2

current model

Tingsanchali

-05 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5U (ni/s)

Figure 6: Comparisons between computed resultant velocities.

X-AxisFigure 7: Isodepths (m) for the spur - dike flow problem.

5.2 Supercritical Flow at Fr = 4.0 after Rouse et al. [7]

The channel expansion shown in Figure 8 was used to test the accuracyof the numerical method by comparing it with the results of Soulis et al.[8] as well as with measurements. The expansion shown wasdesigned for an entrance Fronde number of 4.0. Comparisons between

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

Hydraulic Engineering Software 21

predictions and measurements using n = 0.012 are shown in Figure 9for the curved side, mid-stream line and center line. The comparisons

are considered to be satisfactory.

i 0 10 20 30 40 50 60axial distance X (m)

Figure 8: Rouse et al. [7] channel expansion.1.2

I

? 0.8

j= 0.6

0.2-

0

lower wall

0 10 20 30 40 50 60

center line

current model

00Mac'Cormack

##Experiment

0 10 20 30 40 50 60

I

JS ***'^ 0.6-§ 0.4-

0.2

0

upper wall

0 10 20 30 40 50 60axial distance X (m)

Figure 9: Comparison between predictions and measurements for theRouse et al [7] channel.

6 Conclusions

A two-dimensional, viscous flow computational model has beendeveloped and subsequently applied to predict the flow-field near spur-

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

22 Hydraulic Engineering Software

dikes and expansion channels using the conservative form of the shallowwater equations. The developed algorithm is capable to calculatesubcritical and/or supercritical flows in irregularly shaped flow regions.The viscous flow stresses are described using either a fixed value eddyviscosity coefficient or values related to the mean flow properties. Themulti - grid method substantially accelerated the speed of convergence.

Calculated results are in good agreement with measurements and/or other

numerical solution results.

References

[1] Liggett, J. A. & Vasudev, S. U. Slope and Friction Effects in Two

Dimensional High Speed Channel Flow, International Associationfor Hydr. Research 11*** Int. Congr., Leningrad, 1965.

[2] Molls, T. & Chaudhry, M. H., Depth - averaged open channel

flow model, J. of Hydr. Engineering, ASCE, pp. 453-465, 1995.

[3] Molls, T., Chaudhry, M. H. & K. W. Khan, Numerical simulation oftwo-dimensional flow near a spur-dike, Advances in Water

j, Vol. 18, No. 4 pp. 227-236, 1995.

[4] Soulis, J. V. A numerical method for subcritical and supercriticalopen channel flow, International Journal for Numerical Methods inF/wwk, Vol. 13, pp. 437-464, 1991.

[5] Soulis, J. V. Two dimensional subcritical and supercritical openchannel flow calculation using a time marching method,International Journal for Numerical Methods in Fluids, Vol. 9,

1989.

[6]Tingsanchali,T. & Maheswaran 2-D depth-averaged flowcomputation near groyne J. of Hydr. Engineering, ASCE, 1 16(1),pp.7 1-86, 1990.

[7] Rouse H., Bhoota, B. V. & Hsu En-Yen Design of Channel

Expansions rrmi& ASCE, 116, pp. 347-363, 1951.

[8] Soulis, J. V. & Bellos, K. V. Steady Supercritical, Open ChannelFlow Computations, Proc. 6"' Int. Conf. on Numerical Methodsin Laminar and Turbulent Elow, Swansea, 11-15 July, 1989.

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541