Issue 
La Houille Blanche
Number 8, Décembre 1967



Page(s)  847  860  
DOI  https://doi.org/10.1051/lhb/1967059  
Published online  24 March 2010 
Quelques aspects des écoulements presque horizontaux à deux dimensions en plan et non permanent, application aux estuaires
Some aspects nearhorizontal twodimensional unstead flow and application to estuaries
E.D.F., Direction des Etudes et Recherches, 6, quai Watier  (78) Chatou.
This paper deals with an investigation of nearhorizontal free surface flows with a view to finding digital computer solutions for hydraulics problems in coastal waters. PART I.  This sets up the equations for the problem, with the following basic assumptions : Uniform velocity distribution along a vertical. Hydrostatic pressure distribution. Little bed and surface slope. The unknowns to be determined in terms of space variables x, y and the time variable t are as follows : h = depth of water u, v, = velocity components [Eq. (1), (2), (3)] or : z : free surface level ; U: uh ; V : vh [Eq. (I), (II), (III)]. PART II.  In this part, the above equations are made linear, which holds good if, taking the mean free surface level as the datum level, Z remains small compared to depth D =  Zf (where Zf is the bed level) and if water column displacements are small compared to the considered area [Eq. (I a), (II a), (III a)]. Where these assumptions no longer apply in full, equations can be considered which are no longer purely linear, in which only the convection terms are ignored. PART III.  In this part, the characteristics theory is applied to the quasilinear system (I), (II), (III) in order to show up the boundery conditions required for the calculation, and to establish a numerical solution scheme. After stating the equations in form (1 b), (II b), (III b) and then in matrix form (4), the condition is sought for which the unknown vector R. = (Z, U, V) is determined near a surface (S) whose eqnation is t = Φ (x, y) and on which R (x, y, t) = R [x, y, Φ (x, y)] = R0 (x, y) are given. This condition requires that the determinant of system (4), (6), (7) whose unknowns are ∂R/∂x, ∂R/∂y, ∂R/∂t be different from zero. ∂R/∂x and ∂R/∂y are eliminated for simplification and equation (8) is obtained, in which the determinant to be consideredcalled the 'charaeteristic determinant' is det A = det (I  A p  Ayq), where p = ∂Φ/∂x and q = ∂Φ/∂y. If this determinant is zero, relations (9) and (10) are obtained between the director cosines of the normal to surface (S). Surfaces (S) satisfying these relationships form two families of surfaces referred to as 'characteristic surfaces'. It can he established by local inspection that tangent planes t = px + qy to these surfaces at an assumed origin have the following envelope : First family : the cone whose equation is (11) or (11') ; Second family : the straight line which is the locus of circular sections of the above cone through planes t = constant. As the determinant of system (8) is zero the latter is impossible or inedeterminate. It is indeterminate if the equations of (8) are not linearly independent : if (8) is written down in the form AX = Ythere is a vector t such that tA = 0 and tY = 0. The result of this is (l2), i.e. a linear combination of equations (4). Hence, this relationship is a compatibility condition which is satisfied by the unknowns on the characteristic surfaces. Vector t and the corresponding charactcristic relationship arc made explicit for each family of surfaces by the subsequent ca1culations; equations (13') and (14) emerge, which can be written as (13') and (14') in wich only internal differentiation operators for the considered surface appear. These relationships connect a space point (x, y, t) to neighbouring points belonging to characteristic surface passing through that point ; the latter, therefore, is related to the envelope of these surfaces. The notions of a 'domain of influence' and 'domain of dependence of a point' emerge, i.e, the inside of the future (or past) characteristic cone having that point at its apex ; the state of the point is determined by that of the points within its 'domain of dependence' and it enters into the determination of points belonging to its 'domain of influence' , The final paragraph of Part III gives the results of the application of the characteristics theory to the linear equations ; for the first family, the envelope of the characteristic surfaces is a cone whose equation is (15), and it is the axis of this cone for the second family. The characteristic relationships are (16) and (17), or (16') and (17'). PART IV.  Discusses boundary conditions. System (I), (II), (III), is to be solved in a cylindrical domain D = Ω x [0, T]. As the unknown vector R = (Z, U, V), is given in Ω for t = 0 (initial conditions), it is necessary to introduce boundary conditions at each instant t0 on the contour Γ of Ω. There are at least as many of these conditions as there are characteristic halfplanes (t < t0) originating from outside D and based on Γ X [0, T], Except for shooting flow, therefore, one or two conditions must be set, depending on whether the flow is from or into Ω. If the equations to be solved are in linear cone only one boundary condition is required, as fol1ows : Along the coast : an impermeability condition: Wn = 0, where W is a velocity vector and n is normal to the shore. Along maritime or permeable boundaries : normal discharge or water level vs. time, or a relationship between the three unknowns. In particular, this relationship can represent an incident wave. PART V.  Two explicit numerical integration schemes are suggested. One is derived from the characterietics theory and relies on the relationships holding good on the characteristic surfaces of the first family (tangent planes to the cone) for θ = 0, + π, ± (π / 2). These relationships are approximated by finite difference equations concerning values of vector R at the mesh points in a predetennined mesh network. This seheme is stable if the CourantFriedrichsLevy condition is satisfied, The secondmore conventionalscheme relies on centred differences to calculate unknowns at staggered points but is governed by stricter stability conditions.
© Société Hydrotechnique de France, 1967