Buletinul Ştiinţific al Universităţii POLITEHNICA Timişoara Seria HIDROTEHNICA TRANSACTIONS on HYDROTECHNICS Tom 59(73), Fascicola 2, 2014 DISCUSSION ABOUT THE SHEAR STRESS DISTRIBUTION TERM IN OPEN CHANNEL FLOW David Ioan 1 Sumălan Ioan 1 Ștefănescu Camelia 1 Abstract: The modelling of open channel flows as 1D scheme is usually performed by advanced numerical models developed in the last years, in which an essential problem is the approach of the shear stress at the channel bed. Starting from fundamental equations of the hydrodynamics, in the paper, the basic equation for 1D open channel flow will be obtained and discussed concerning the approaches of the term which contain the shear stress for both literature approach and numerical models. Keywords: open channel flow, fundamental equations, shear stress, hydraulic radius, resistance radius 1. INTRODUCTION In the modelling of the flow in the open channels the theory is based on the fundamental equations developed for a current tube in the 1 D case, with the mean velocities on cross sections. Based on these equations, the later development has a general character in the sense that the term referring to shear stress at the rigid walls of the current tube is not specified. For this reason there are significant differences to appreciate the term representing shear stress by different authors. These differences are encountered even in the advanced developed numerical models on all over the world, for example HEC-RAS or Mike 11. The paper aims a discussion about the term referring the shear stress at the rigid wall of the current tube in the open channel flow by comparing approaches from the specialty literature used especially in the advanced numerical models existing on all over the world. 2. THE DYNAMICS EQUATION FOR STREAM TUBE Technically, the pipe flow and the open channel flow are stream tubes. The hydrodynamic equation for such a stream tube can be obtained from the basic equation of the hydrodynamics for a control volume. This equation in its turn is obtained on the base of the linear momentum balance (dynamics equation) for a moving fluid body and shows that the time rate of change of the total linear momentum of a fluid body which at the time t occupied the spatial volume Vt is equal to the resultant force acting on the considered body [1], [2]: t t t V V S d vdv f dv t dS dt ρ = ρ + ∫ ∫ ∫    (1) where ρ is the water density, f  is the intensity of the mass forces (e.g. gravity forces f g =   , v( r ,t )  is the fluid velocity in the spatial point r(t)  , t  is the external stress vector (i.e. surface traction force per unit area which acts on the boundary S t ) and V t is the moving fluid body as material system which at the time t occupied the space volume V t limited by his boundary surface S t . Using the Reynolds Transport Theorem i.e. the derivation under the integral sign over V t from (1) the following dynamic equation will be obtained [1], [2]: t t t t V S V S ( v)dV v ndS fdV tdS t ∂ ρ + ρ⋅ = ρ + ∂ ∫ ∫ ∫ ∫      (2) n  is the outward unit-normal vector on the boundary surface S t . In the applied hydrodynamics like open channel flow it is quiet useful to use specific spatial region called control volume denoted V C bounded with its S C called control surface. If the control volume V C which coincides at the time t with the space volume V t (i.e. V C ≡ V t ) the dynamics equation (2) maintain their mathematical form [1], [2]: C C C C fa V S V S vdV v ndS fdV tdS t ∂ ρ + ρ ⋅ = ρ + ∂ ∫ ∫ ∫ ∫      (3) In this equation fa v  represents the absolute velocity of the fluid crossing the boundary surface Sc. The dynamics equation (3) can be extended for a deformable spatial control volume V Ct which at the time t is occupied from the moving fluid (i.e.V Ct ≡ V t ) 1 Politehnica University of Timisoara, Department of Hydrotechnical Engineering, G.Enescu Street, no.1A, Timisoara, Romania, e-mail address:Ioan.David@gmx.net; ioan.sumalan@upt.ro; achim_camelia@yahoo.co.uk 5