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1~6 R.J. H<ln/:l' rl "I. / JOlt"",f tJf H.,'dm/"j" 2/6 (J999IIU-IJ6 cenlml 10 all modelling projects, but is often over- looked. Furtller, in tile current applic:uions of hydrau- lic and hydro]ogical Computafional Fluid Dynamics codes. me~h resolulion is the only unbounded para_ meter value, where specific boundaries and error bands h:lVe not hitherto ~en considered. This is demonstmled in Fig. I. which shows three tinite elemenl discretizalioos for simulating free surface flows in a hypothdical river/floodplain system. There are no a priori objective rules for mesh construction. so even using Ihe best available know]- edge of process length scales in compound channel Iiows (lht typical mesh gener-uion criteria in fluid dynamics applications). one cannal define which of the~ meshes is oplimum. Each discretiZ:lIion could Olus plausibly provide a suliSfaClOry solution 10 the defined problem (as of course could many others nOI i!lustraled here). This is in contr:lSt 10 calibration para- meters, such as surface roughness. which are effec- tively bounded by mown physically realistic ranges and error hands (e.g. Chow. 1959). In the past model- Itrs have lended 10 look for !be minimum mesh reso- lution at which numerical convergence could be achieved (e.g. Dietrich et aI.. 1990; Lardner and Song. 1992; Weslerink el al.. 1994) or used mesh construction criteria based on appredalion of the length scales wilhin the flow (e.g. Gray and Lynch. 1971; Le Provost and VillCent, ]986; Luetlich et al.. 1992; Bales and Anderson. 1993) rather than rigor- ous]y examining mesh resolution impacls. Those studies on the effects of mooel spatial resolution that hal'e been undertaken in hydrology and hydraulics (e.g. Balhurst, 1986; Farajalla and Vieux, 1995: Bruneau et al.. 1995; Bates et aI., 1996) demonstrate the sensitivity of model response to changina resolu- tion but only consider bulle flow outputs from such schemes rather than the fully distributed results. While we may assume that the ruShest resolution provides the best result., neither this, nor the possibi- lity that yet higher nodal densities would give a 'furtherimprovement',isevertypicallytestcd. 2. Spallal resolulion Impacts on model results Owing to Ihe heterogeneity of natural systems, there is a tendency to a~ume that an increase in the number of elements (increased spatial resolution) will improve the realism of the modeJ's predictive ability. as acknowledged by Farajalla and Vieu.'\: (1995). The detinition of spatial resolution being applied in this anicle is the size of the grid cell (element) within the domain, and this will always be referenced to as anaClUaI fie]dscale(m\An increase in spatia] reso- ludon .....ill result in an increase in the number of elements, thus decreasing the average element size. The hypofhesis thai a model's predictive ability increases as the spatial and temporal resolution increases.sfemsfromthreeavenuesofthought: I. expecled impro\'ements in solution stability as the grid spacing tends towards the uue continuum le\'e]; 2. the ability of high resolution models fO facilitate complex, and thereby more realistic parameteriza- tion of the code (cf. Beven, 1989); 3. a closer correspondence between field measure- ment model scales (cf. Bathurst and Wicks, 1991). To date these arguments have not undergone expli- cit testing. This is the central aim of this paper where we present a comprehensive analysis of the effect of spatia] resolution on a typical non-linear numerical scheme. The code selected for investigation, TELE. MAC-2D. isa t.....o-dimensionaltinite e]ement hydrau_ lic mode] which soh'es the depth averaged Shallow Water Equations and invokes the Boussinesq assump- tion to represent turbulent flows. This non-linear equation system is I)'pical of many panial differential equations employed in environmental numerical modelling and has the advantage that the parameter- ization consists of only two variables (boundary fric- tion and turbulent viscosity) and is therefore relatively simple and well bounded. Moreover, the use of computationaUy efficient and stable numerical algo- rithms in the code allows a wide range of mesh discre- tizatiOlls to be constructed for a given problem thus enabling a full investigation of spatial resolution effect$. This model was applied to a typical hydraulic problem, the simulation o{ free surface flow in a compound meandering river channel. and the impact of Changing mesh resolution analysed in terms of the ability of the scheme to simulate bulk flow eharacter. istics, inundation extent and dbtributed hydraulics. The relative dominance of mesh resolution and typical calibration parameters was also examined. Although no single study can perhaps fully R,J. Hllmy ~I Dl./JOIlmtllufH.lJm/og,' 216 (1999) JU-/J6 t~l Meoll T;d>lc I A qu:ullitative summ:uy or !he ~hc$ uppl;ed;1I otdcr '" idtnt;fy a ""itable w()(~ill' rewlu"Oft """, '" "" t982 21158 "'. ,." ... inCh. 36.89 35.45 40.26 38.80 3J.4S "... Ettme/lt$ ''''' ~2S4 J824 5S78 7110 91~8 Mu. ~601.5t !S51.J6 t981.42 25~8,SO ttJ6.02 t59l.97 Min. 31.04 20.83 11.11 1.4t 6.11 4,63 AI'S. lU4 58.13 UlJ J!.~4 30.48 2U8 Sld.lk\" t8.59 23.34 21.92 23.~4 16.71 18.l8 illustrafe a general problem. this initial investigation. using a model fully representative of its class, should be able 10 provide a considerable insight that can be used to define funh.:r, more comprehensive, research programmes. For example. this investigation should be able to detennine whether increasing spatial reso- lution provides model results consistent with the controllina equations and process representation; whether guidelines for the appropriate level of spatial resolution can be provided for specific conditions. and finally whether new model inter-comparison methods are required to facilitate a full eva]uation of the impacI of spatial resolution. 3. Methodology The hydraulic model applied in this sfUdy is the TELE.\.fAC.2D modelling system. TELB.-L\C-2D soJ\'es second-order panial differential equations for depth avemsed free surface flow derived from the full three-dimensional Navier Stokes equations as (ollows: ~ + u.grad(h) + II div(u) :: q ., . ~ + u'grad(lI) + I~ - div(v.grad(Il)) at ax -so _.'71 . il.-r' ~ + u.gr:ad(v) + I~ - div(v.grad(v)) at OJ' '71 = 5-" - 'ay' """ lU3 It 890 616.31 '" 24.11 12.37 aT "it + u.gr:ad(7) - div(Vrgmd(n) = 5T, (4) (I) where II is the depth of the water(m), II, v are the velocity components(m S-I), TOle non-huoyant traCer (-),g is the acceleration owing 10 gravity (ms-l). V, VT are momentum and tracer diffusion coefficients (m! s -I), Zfis the bed elevation (m), t is the time: (s), x, y are the horizontal space co-ordinates (m), q is the introduction or removal of fluid (ms-I) and 5 the source term (ms-l). The model thus calculates water depth and velocity in the x and y directions at each computational node. A complete mathematical description of the modelling system is presented by Hervoeut and Van Haren (1996) while modifications implemented for the appli. cation of the modelling system to a river floodplain, and the effect of different solver techniques are discussed by Bates et aI., (1995). The analysis was pc:rlonned on a purely hypotheti. cal example, although both the domain considered and the input hydrographs were scaled to real events that have been considered in past analyses. Real examples were not considered for several reasons: (i) A simple, computationally efficient domain was needed to enable a large number of simulations to be completed. (ii) Boundary conditions and topography needed to be controlled so only the effect of mesh resolution was considered. (iii) Comparison against tield data was not believed to be beneficial as we do not wish to analyse the model's predictive ability for a panicular reach and the data required for such a study is unlike]y to exist. The dimensions of the domain were 2000 m x 800 m wiOl a 20 m wide, 2 m deep sinuous channel (2) (3)