Comparison Of 2D Numerical Schemes For Modelling Supercritical And Transcritical Flows Along Urban Floodplains

Transcription

1 City University of New York (CUNY) CUNY Academic Works International Conference on Hydroinformatics Comarison Of D Numerical Schemes For Modelling Suercritical And Transcritical Flows Along Urban Floodlains Yared Abayneh Abebe Solomon D. Seyoum oran Vojinović oland K. Price Follow this and additional works at: htt://academicworks.cuny.edu/cc_conf_hic Part of the Water esource Management Commons ecommended Citation Abebe, Yared Abayneh; Seyoum, Solomon D.; Vojinović, oran; and Price, oland K., "Comarison Of D Numerical Schemes For Modelling Suercritical And Transcritical Flows Along Urban Floodlains" (014). CUNY Academic Works. htt://academicworks.cuny.edu/cc_conf_hic/149 This Presentation is brought to you for free and oen access by CUNY Academic Works. It has been acceted for inclusion in International Conference on Hydroinformatics by an authorized administrator of CUNY Academic Works. For more information, lease contact AcademicWorks@cuny.edu.

2 11 th International Conference on Hydroinformatics HIC 014, New York Cit USA COMPAISON OF D NUMEICAL SCHEMES FO MODELLING SUPECITICAL AND TANSCITICAL FLOWS ALONG UBAN FLOODPLAINS YAED A. ABEBE (1), SOLOMON D. SEYOUM (1), OAN VOJINOVIĆ (1), OLAND K. PICE (1) (1): UNESCO-IHE Institute for Water Education, Westvest 7 - P.O.Box 3015, 601 DA, Delft, The Netherlands Urban floodlains usually have irregular geometry due to different obstacles, urban infrastructures and sloe conditions. This may change the flow regime from subcritical to suercritical flow conditions, and vice versa. Imlementation of the full momentum equation in D shallow water equations (SWEs) is not trivial in mixed flow conditions as subcritical and suercritical flows require different boundary conditions and hence different solution algorithms. Some models ignore the convective acceleration term (CAT) to simlify imlementation of the momentum equation for mixed flow conditions. This work tried to investigate the effect of neglecting CATs by testing two D models which imlement - full SWEs and comletely reduced CAT. The models' erformances were then tested by setting u hyothetical case studies with changing flow regimes. Simulations results were comared to each other by setting the solutions of the method that solve the full equations as a reference. Findings of the numerical tests showed that, in the cases, results of the model which ignore CATs fully were very similar comared to solutions of the model which imlement full SWEs. Hence, simlified models which ignore CATs may be used to model urban flood lains without significant loss of accuracy. INTODUCTION The governing equations in D models are the so-called shallow water equations (SWEs). The system of D SWEs is obtained by integrating the Navier Stokes equations over deth and relacing the bed stress by a velocity squared resistance term in the two orthogonal directions. The assumtions used in this case are: uniform velocity distribution in the vertical direction, incomressible fluid, hydrostatic ressure distribution, and small bottom sloe (Yoon and Kang [10]). This system of equations consists of three equations: one equation for continuity and two equations for the conservation of momentum in the two orthogonal directions (Mignot et al [6]). The advantages of D models include: more accurate solution of the governing equations; two or three orders of magnitude higher resolution outut; flowaths do not have to be redefined; vastly more accurate maing of flood inundation, flood levels and flood hazard. (Verwey [9]; Syme [8]). However, the rimary disadvantages of D models are the longer simulation times (Evans []; Syme [8]) and comlicated comutation of D unsteady flows due

3 to need for efficient solver routine and the inclusions of roer boundary conditions (Fennema and Chaudhry [3]). Various methods can be introduced in discretizing SWEs to better simulate urban flooding and to reduce comutational time so as to use the modeling tools for real time alication. For instance, exlicit finite difference schemes which cature shocks can be used to discretize SWEs (Liang et al [5]) though they suffer a conditional stability roblem in the use of larger time stes. As a result, for examle, a number of commercial software roducts solve the full flow equations using imlicit finite difference schemes which are unconditionally stable. However, discretizing the full flow equations using imlicit finite difference schemes is not easy to imlement theoretically as it incurs comlications in the alication of boundary conditions in case of transcritical flows. One way of tackling the boundary condition roblem is ignoring the convective acceleration terms (CATs) in the momentum equations of the flow equations. The basic argument for this assumtion is that these terms are small comared to the other terms in urban floodlain flows, which means a subcritical flow condition is assumed. Hence, one boundary condition at each end (i.e., ustream and downstream) is rovided and the same solution algorithm is used in both subcritical and suercritical flows. Ignoring the CATs also favors the model by reducing comlexity of the equations and hence the simulation time. With this background, this work mainly tried to investigate the merits and drawbacks associated with ignoring the CATs in D suercritical and transcritical flow conditions. Two methods were used for this urose: MIKE1 flow model and a Non-Inertia D model. Finall numerical exeriments using hyothetical case studies which somehow reresent urban floodlains were carried out to test the methods. MODELS DESCIPTION MIKE1 The following descrition of MIKE1 commercial software ackage is based on MIKE1 Flow Model Scientific Documentation (DHI [1]). The hydrodynamic module in the MIKE1 Flow Model (MIKE1 HD) is a general numerical modeling system for the simulation of water levels and flows in estuaries, bays and coastal areas. It simulates unsteady D flows in one layer (vertically homogeneous) fluids and has been alied in a large number of studies. The conservation of mass and momentum equations that describe flow and water level variations are given as: q d t x y t (1) t x h fvv q y h x h w g q gh x C h ( x a ) 0 1 ( h xx ) ( h xy ) w x y q ()

4 q t q y h fvv q x h y h w gq q gh y C h ( y a ) 0 1 ( h w y yy ) ( h x xy ) (3) where h( x, is water deth; d( x, is time varying water deth; ( x, is surface elevation;, q( x, are flux densities in x and y directions; C ( x, y) is Chézy resistance; f (V ) is wind friction factor; V, Vx, Vy ( x, is wind seed and comonents in x and y directions; ( x, y) is Coriolis arameter, latitude deendent; a ( x, is atmosheric ressure; w is density of water; and xx, xy, yy are comonents of effective shear stress. In area of high velocity gradients, that is, for flow at high Froude numbers, selective introduction of numerical dissiation has been used to imrove the robustness of the numerical solution, and to rovide MIKE1 with the caability to simulate locally suer-critical flows. The numerical dissiation is introduced through selective "u-winding" of the CATs, as Froude number increases. To ensure that the dissiative effects of u-winding are only included when necessar a Froude number deendent weighing factor has been introduced where: 0, Fr ( Fr 0.5), 0.5 Fr 1.0 (4) 3 1, Fr 1.0 The weighing factor is alied to the convective momentum terms, such that: x h j (1 ) (5) x h x h j j1 This brings the effects of u-winding in gradually as the Froude number increases from 0.5 to 1.0. For Froude numbers of one or more, the CAT is fully u-winded. MIKE1 HD makes use of the so-called ADI technique to integrate the equations for mass and momentum conservation in the sace-time domain. The equation matrices that result for each direction and each individual grid line are resolved by a double swee algorithm. Non-Inertia D Model This modeling software was develoed in UNESCO-IHE as art of a PhD research. The descrition of the model is based on Seyoum et al [7]. The system of D SWEs is obtained by integrating the Navier Stokes equations over deth and relacing the bed stress by a velocity squared resistance term in the two orthogonal directions. The continuity equation for the D flood lain flows is formulated as h ( hu) ( hv) 0 t x y (6)

5 Neglecting eddy losses, Coriolis force, atmosheric ressure, wind shear effect and lateral inflow, the momentum equations in x and y directions can be written as ( hu) ( hu ) ( huv) H gh gc f u u v 0 (7) t x y x ( hv) ( hv ) ( huv) H gh gc f v u v 0 (8) t y x y where H is the water level; u and v are the velocities in the directions of the two orthogonal axes (the x and y directions); and the coefficient C f aearing in the friction terms is normally exressed in terms of the Manning n or Chézy roughness factor C. Two-dimensional flow over inundated urban flood lain is assumed to be a slow, shallow henomenon and the CATs can be assumed to be small comared with the other terms; and therefore, can be ignored. Exressing the velocities in terms of the discharges and using Chézy roughness factor, the simlified momentum equations in x and y directions can be written as: t h Yg g x C 1 Y 1 X (9) t h Xg g y C 1 Y 1 X (10) where and are the discharges in the directions of the two orthogonal axes (the x and y directions); x and y are the grid sacing in the x and y directions; and and are the water deths at the cell boundaries. The ADI finite difference method is imlemented for the numerical solution of the governing equations. The PDEs of the governing equations are transformed to difference equations on a regular Cartesian grid. NUMEICAL EXPEIMENTS The D numerical exeriments were conducted based on steady flow tests for suercritical and transcritical flow conditions. The tests were conducted on rismatic channels with constant and variable sloes. Each test was conducted for three different channel bed resistances defined by Chézy coefficients of C 1 10, C 8 and C For the MIKE1 model set u, an initial condition of 8 cm water deth was used, whereas, the non-inertia model started the comutation from a dry bed. The ustream boundary condition used in all the D exeriments was a steady flow of 5 m 3 /s which flows for a eriod of one hour. As a downstream boundary condition, a normal deth boundary for subcritical flows and a critical deth boundary for critical and suercritical flows were used. In addition, a 1 m by 1 m DTM and a time ste of t 0. 1 second were used in all exeriments.

6 Test 1 Prismatic channel with constant sloe This exeriment was conducted on a 100 m long and 10 m wide hyothetical rectangular channel as shown in Figure 1. The bed sloe of this channel was S 0. 0 which was constant along the channel length. The channel bed was made horizontal across the channel width. Figure 1. a) Side view and b) Plan view of a rectangular hyothetical channel used for numerical exeriment The first simulation was for a channel bed resistance of C In this test, the Froude number was 0.45 such that the flow is subcritical. The second and third simulations were for C 8 and a Froude number of 1.64 and C 3 45 and a Froude number of.03. Since the Froude numbers were greater than one, the latter two flows were suercritical. Figure. Longitudinal water deths 30 minutes after simulation started Figure shows simulation results of the two model setus half an hour after simulation started. The flow deths shown in the figures are taken along the longitudinal rofile. Since the channel bed is horizontal across the channel, the flow in that direction is insignificant. In the

7 figures, it is shown that the non-inertia model results are quite similar to the MIKE1 model results even in those flows which are characterized by very high Froude numbers. Though the effects did not roagate further, there were discreancies in the model results at the ustream and downstream ends. Those mismatches were due to the imlementation of the modeling tools while treating the given boundary conditions. Consequentl in this case, solving the D SWEs which comletely ignore the CATs give almost the same result as the full D SWEs in simulating ure suercritical flows. Test Prismatic channel with variable bed gradient This case study was designed to test the caability of the models to handle changes in flow regimes. Similar to the revious exeriment, this exeriment was conducted on a 100 m long and 10m wide hyothetical rectangular channel. Nevertheless, as illustrated in Figure 3, the channel was divided into three reaches of length L1 300 m, L 600 m and L m with a bed sloe of S , S 0. 0 and S resectively. Once again, the channel bed was made horizontal across the width. Figure 3. a) Side view and b) Plan view of a rectangular non-rismatic channel used for numerical exeriment Three tests were conducted in this case also. The first one was with C 1 10 and Froude number range of to 0.45; the second one was with C 8 and Froude number range of to 1.64; and the last one was with C 3 45 and Froude number range of to.03. The higher Froude numbers were registered in the relatively steeer second reach. In the first test, the flow was subcritical in the three reaches. In the second test, the flow was subcritical in the first and third reaches while it was suercritical in the second reach. Whereas in the third test, the flow was suercritical in all reaches. The results of the models are resented grahically in Figure 4 a, b and c. The figures show flow deths along the channel half an hour after simulation started and are taken at the centre of the channel width. Since the channel bed is horizontal across the channel, the flow in that direction is insignificant. The figures demonstrate that, in deth basis, the non-inertia model results are quite similar to the MIKE1 model results in all flow conditions ure subcritical, transcritical and ure suercritical flows. In this case again, the discreancies at the two ends were due to the imlementation of boundary conditions by the modeling tools. However, the deth results from the two models showed slight variation at/around the critical and shock oints. As shown in Figure 3, x 300 m and x 900 m mark the breaks in the bed sloe. It is observed in Figure 4 that the deth oututs from the two models were almost the same excet at the critical and shock oints. In addition, the discreancy between the two model oututs increased when the flow became more suercritical. For instance, the difference

8 between the two model oututs at the shock oint was 0.05%, 1.40% and 19.00% for the resective roughness of C 1 10, C 8 and C This finding is in line with findings of Hunter et al [4]. The authors concluded that even though flows in urban environments are characterized by transitions to suercritical flow and numerical shocks, the effects are localized and they did not aear to affect overall wave roagation. Figure 4. Longitudinal water deths 30 minutes after simulation started Figure 5 also shows zoomed rofiles of the deths around the two oints for C 8. The figures demonstrate that the results of the two models were different around the critical and shock oints. Figure 5. Deth rofiles at critical and shock oints (left and right ictures resectively) As a result, in this case also, solving the D SWEs which comletely ignore the CATs give almost the same result as the full D SWEs in simulating ure suercritical and transcritical flows.

9 CONCLUSIONS The comarison results showed that there are cases in which the cost of comletely neglecting the convective acceleration terms (CATs) from the D SWEs is minor. This is esecially shown on the results of simulations of flows in channels with uniform bed gradient. If one is interested in simulating urban floods which are characterized by relatively flat surface, it may be enough to use those modeling tools which does not imlement CATs with a ossible advantage of reducing simulation time. On the other hand, at critical and shock oints, the results of the noninertia model record differences comared to the MIKE1 results. This shows that, if one is interested in designing hydraulic structures with breaks in bed gradient, it may be necessary to trace secial features like hydraulic jums; and in this case, the use of modeling ackages which better treat the CATs and cature shocks are more suitable. Besides, the CATs may still have higher imortance in dam break analysis, modeling tsunami wave or modeling flows characterized by reflected waves. In conclusion, it seems that the use of the full SWEs is not a strongly binding rule in modeling urban floodlains. It rather deends on the circumstances even if suercritical states dominate the flow. EFEENCES [1] DHI, MIKE1 Flow Model - Hydrodynamic Module Scientific Documentation, DHI, Hørsholm, Denmark, (01). [] Evans B., A multilayered aroach to two-dimensional urban flood modeling, Doctor of Philosoh University of Exeter, (010). [3] Fennema.J. and Chaudhry M.H., Exlicit methods for -D transient free-surface flows, Journal of Hydraulic Engineering, Vol. 116, No. 8, (1990), [4] Hunter N.M., Bates P.D., Neelz S., Pender G., Villanueva I., Wright N.G., Liang D., Falconer.A., Lin B., Waller S., Crossley A.J. and Mason D.C., Benchmarking D hydraulic models for urban flooding, Proceedings of the ICE - Water Management, Vol. 161, No. 1, (008), [5] Liang D., Falconer.A. and Lin B., Comarison between TVD-MacCormack and ADItye solvers of the shallow water equations, Advances in Water esources, Vol. 9, No. 1, (006), [6] Mignot E., Paquier A. and Haider S., Modeling floods in a dense urban area using D shallow water equations, Journal of Hydrolog Vol. 37, No. 1-, (006), [7] Seyoum S., Vojinovic., Price. and Weesakul S., Couled 1D and Noninertia D flood inundation model for simulaiton of urban flooding. Journal of Hydraulic Engineering, Vol. 138, No. 1, (01), [8] Syme W., Flooding in Urban Areas - D Modelling Aroaches for Buildings and Fences, 9th National Conference on Hydraulics in Water Engineering, Darwin Convention Centre, Australia, 008. [9] Verwey A, Hydroinformatics suort to flood forecasting and flood management, 4th Inter-Celtic Colloquium on Hydrology and Management of Water esources, Guimaraes, Portugal, (007). [10] Yoon T.H. and Kang S.K., Finite volume model for two-dimensional shallow water flows on unstructured grids, Journal of Hydraulic Engineering, Vol. 130, No, (004),