SELECTION OF OPTIMA MATHEMATICAL MODELS FOR FLUVIAL PROBLEMS Angel N. Menéndez INA (Instituto Nacional del Agua) Autop. Ezeiza-Cañuelas, Tramo J. Newbery km 1.62 C.C. 21, B1802WAA Aeropuerto Ezeiza, Argentina Fax (54 11) 4480 4500, Ext. 2359 e-mail:
[email protected], web page: http://www.ina.gov.ar
ABSTRACT A methodology of analysis is presented for selection of the optima hydrodynamic and sedimentologic models for any flow situation in fluvial problems. It only requires the appropriate selection of scales by the user. Then, an inter-comparison among them is performed to identify all possible simplifications. Applications to specific problems are presented. 1 INTRODUCTION Modeling sediment transport problems and the associated morphodynamics requires selection of the appropriate theoretical model. The literature dealing with this subject offers a huge variety of mathematical models, with specific abilities aimed to solve particular situations. This turns the matter of choosing a model an issue far from being trivial for the applied engineer. The temptation of using the most general model, which hopefully includes all possible phenomena, faces quite fast with practical obstacles related to high software cost, large computational times, and high requirement of input data. A more basic objection to this approach lies in the fact that, in this way, no identification of the main mechanisms driving the system dynamics is performed (unless a sensitivity study is undertaken through extra -and costly -- numerical experiments). As a consequence, there is a lack of physical insight about the problem, a quite common cause of mistakes through wrong interpretations. The recommended alternative is to undertake a model selection based on a sound set of criteria so as to choose the most efficient model, i.e. the one that includes only (but all) the relevant mechanisms. An attempt in this direction for fluvial dynamic problems was already presented by the author some years ago (Menéndez 1997), based on the well-known technique of identifying and comparing different scales of movement. In this paper, using the same idea, a much more general approach is introduced and applied to fluvial dynamic problems. It starts with the hydrodynamics, which is the driving force of sediment transport, to continue then with sediment dynamics. Applications to erosion problems are also presented.
2 CHOOSING THE HYDRODYNAMIC MODEL Classes of scales Five classes of scales will be distinguished, according to their association with one particular aspect of the problem: o Flow scales: These are the spatial scales related to the particular flow phenomenon that is going to be modeled. In the present case, the phenomenon could be just open-channel flow. o Morphology scales: These are the spatial scales related to the morphology of the physical surfaces that limit the flow. In the present case, these surfaces are the bottom and margins of the channel. o Forcings scales: These are the temporal scales related to the forcings which modulate the flow. In the present case, the main forcing is the incoming flow discharge, though tidal, wind and wave action could also be considered. o Observer scale: This is the temporal scale of analysis of the observer. o Model scales: These are the spatial scales of analysis of the observer. Three main spatial directions will be distinguished: longitudinal (in the flow direction, basically horizontal), lateral (normal to the flow direction and in the horizontal plane) and vertical (normal to the latter two directions). They will be denoted by x, y and z, respectively. For open-channel flow, the lateral (Lfy) and vertical (Lfz) spatial scales of the flow are easily identified as the scales for the channel width B and the water depth h, respectively. The corresponding longitudinal (Lfx) spatial scale of the flow arises from the scale for the flow slope S, and is defined as h/S, as a detailed analysis of waves in open-channels shows (Menéndez and Norscini 1982, 1986). Associated temporal scales of the flow can be defined if the main momentum transport properties along each spatial direction are used: Longitudinal time scale of the flow: T fx ≡ L fx / U , where U is the flow velocity scale. Lateral time scale of the flow: T fy ≡ L fy / ε y , where εy is the lateral diffusivity scale. 2
Vertical time scale of the flow: T fz ≡ L fz / ε z , where εz is the vertical diffusivity scale. 2
The longitudinal spatial scale of the morphology (Lcx) is determined by the longitudinal extension of bed forms (dunes, bars, meanders, etc.). The lateral spatial scale (Lcy) is linked to the lateral extension of the margins or bed forms (pools, bars, etc.). The vertical spatial scale (Lcz) is determined by the height of bed forms. Associated temporal scales (Tcx, Tcy, Tcz) can be defined in the same way as for the flow scales. For open-channel flow, the longitudinal time scale of the forcing (Trx) is determined by the temporal evolution of the flow discharge. The lateral time scale (Try) is linked to the temporal evolution of the margins. The vertical time scale (Trz) is determined by the temporal evolution of the channel bottom. Associated spatial scales can be defined using the transport properties: Longitudinal spatial scale of the forcing: Lrx ≡ UTrx . Lateral spatial scale of the forcing: Lry ≡ ε yTry . Vertical spatial scale of the forcing: Lrz ≡ ε zTrz . The time scale of the observer (To) is the temporal step for which flow observations are of interest. Associated spatial scales (Lox, Loy, Loz) can be defined in the same way as for the forcings scales. The spatial scales of the model (Lmx, Lmy, Lmz) are the spatial windows for which flow observations are of interest. Associated temporal (Tmx, Tmy, Tmz) scales can be defined in the
same way as for the flow scales. Model definition The relationships between scales of different classes or of the same class determine the relative significance of the dynamic mechanisms and, hence, define the appropriate, mostefficient hydrodynamic mathematical model. The relation between the spatial scales of the flow and of the observer determines the dimensionality of the problem, according to the following criteria: 1D Problem: Lox L fy , L fz
2D Problem: Lox ∼ L fy L fz or Lox ∼ L fz L fy 3D Problem: Lox ∼ L fy ∼ L fz For open-channel flow, the basic hydrodynamic model for the 1D Problem is the 1D Longitudinal Model, i.e. the de Saint Venant equations (Chow 1959, Henderson 1966, Pujol & Menéndez 1987):
∂Ω ∂Q + =0 ∂t ∂x ∂U ∂U ∂z gn 2U 2 +U + g + 4/3 = 0 ∂t ∂x ∂x R
(1)
where t is the time coordinate, Q the flow discharge, Ω the area of the cross-section, R the hydraulic radius, U the section-averaged velocity, g gravity, z the water surface level and n the Manning roughness coefficient. Two sub-cases can be distinguished for the 1D Problem: Fine Section: Loy L fy and Loz L fz Gross Section: L
