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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