Hydrodynamic modeling of flow around bridge piers

Transcription

1 Hydrodynamic modeling of flow around bridge piers E. D. Farsirotou*, J. V. Soulis^, V. D. Dermissis* *Aristotle University of Thessaloniki, Civil Engineering Department, Division of Hydraulics and Environmental Engineering, Thessaloniki, GREECE ^Democrition University of Thrace, Civil Engineering Department, Fluid Mechanics/Hydraulics Division, Xanthi, GREECE Abstract The free-surface flow in the vicinity of a bridge pier positioned normal to a fixed channel bed represents a classical problem of fluvial hydraulics. The present study aims to a realistic two-dimensional numerical modeling of the flow around obstacles such as a cylinder. The depth averaged Navier-Stokes equations describing viscous, unsteady, free-surface flow are solved using an explicit finitevolume numerical scheme in integral form. A grid clustering technique is used in order to form a more dense computational grid in high flow gradients regions. A multi-grid algorithm has been developed to accelerate the convergence solution. The numerical simulation was done for two different inflow discharges. The numerical results of water depth profile show the expected alteration of the flow field around the cylinder. Comparisons with other numerical solution proposed by Yulistiyanto, Zech and Graf [6] and with their experimental results show that the proposed approach is a comparatively accurate and reliable technique. The flow depths at the center line both upstream and downstream from the cylinder are in good agreement. However, very close to the cylinder differences are presented between predictions and measurements. 1 Introduction A satisfactory understanding of the flow field around a cylinder can be obtained either by numerical simulation the flow and/or by performing extensive

2 16 Hydraulic Engineering Software VIII measurements around the cylinder. Richardson and Panchang [4] used a computational fluid dynamics model called FLOW-3D to simulate the flow occurring in the vicinity of a cylindrical bridge pier within a scour hole. Yulistiyanto, Zech and Graf [6] used Saint-Venant equations for 2D depthaveraged flow to simulate flow around a cylinder positioned in an open channel. Numerical solutions were obtained using an explicit,finite-differencescheme according to MacCormack and were compared to appropriate measurements. In the current research work a two-dimensional numerical modeling of the flow around a cylinder is presented. The depth averaged Navier-Stokes equations describing viscous, unsteady, free-surface flow are solved using an explicit finitevolume numerical scheme in integral form. The numerical results of water depth profile are compared a numerical solution proposed by Yulistiyanto, Zech and Graf [6] and along their experimental results. 2 Hydrodynamic flow equations The channel flow will be assumed to be homogeneous, incompressible, twodimensional and viscous with wind and Coriolis forces neglected. Hydrostatic pressure distribution is assumed throughout the flow field. The two-dimensional, unsteady, free-surface flow in channels with fixed bed is described by a system of non-linear, parabolic, partial differential equations. The continuity and momentum equations ( Molls and Chaudhry [3]) are the following: a a(hu) a(hv) (1) a(hu) at a(gh2 a \i, ax 2^, MM, L ax a(huv ay ax J + Sfx) a(hu) a(hv) (2) a(hv) at _a_ ax ay a(hu) ~W' a(huv) ax ax -gh _a_ > >oy vt a(hv) ay ay (3) where x and y represent the Cartesian co-ordinate positions in the longitudinal and transverse directions respectively; t is the time; u and v are the average velocity components in the x and y directions; h is the water depth; g is the

3 Hydraulic Engineering Software VIII 1 7 gravity acceleration; Vt is the eddy viscosity, So% and Soy are the channel slopes and Sfx and Sfy are the friction slopes which are defined as: nt vvur + v^ 4/3 where n is the Manning's flow friction coefficient. By writing the equation for frictional resistance in this way it was assume that all of the resistance is due to bottomfriction,thus neglecting the boundary layers on the side walls. The eddy viscosity, v^ is defined (Yulistiyanto, Zech and Graf [6]) as: (5) where C = n is the Chezy's, friction coefficient for flow. 3 Finite-volume approximation The system of differential eqns (l)-(3) is too complex to be solved by analytical methods. Therefore, an explicit finite-volume scheme developed by Farsirotou [1] is used in this study in order to obtain an approximation solution. The equations of continuity, x-momentum and y-momentum are applied to a series of finite volumes with adjacent volumes sharing a common face. At the end of each time step At the net flux into each elemental volume is zero, so that overall water mass flow is conserved, and the changes in momentum are equal to the forces imposed by the boundaries of the system. The two-dimensional flow equations may be written down as conservation equations for a control volume AV of unit height and for a time step At as: - Ah - f A(hu) Ay + A(hv) Axl L v ' v / J (6) A[2.0vt A(hu)]Ay Ax AxAx {A(hv)Ayl Ax At (7)

4 18 Hydraulic Engineering Software VIII - A(hv) = ~A( gh^ 72+ hv^l Ax + A(huv)Ay 1 -^ + gh(soy + Sfy W - ^ ' V / ^ AvAv \ J -> i A[2.0vt A(hv)]Ax Ay AyAy Ay At (8) XFLUXi i AY Figure 1: Notation for the mass flux across a finite-volume. Figure 1 shows the notation used for mass flux balancing across a finite-volume of the flow. Similar notation is adopted for the balancing of the x-momentum and y-momentum fluxes. Thus, for the water mass flux, a XFLUX at a point i,j is defined as: (XFLUX), j = Ay (9) while the YFLUX, at the same point i,j is defined as:

5 Hydraulic Engineering Software VIII 19 "(Hj -_i+(hj " AY 2 "Nij-i+Mij" Ayg (10) 2 The second term of the above equation comes from the balancing of the mass fluxes into the ABE flow region, see Figure 1.The terms A(hu) and A(hv) of eqn (6) are defined as:.u) = (XFLUX)^-(XFLUX)^._j (11) j (12) Similar differences are applied to all A terms, of eqns (7) and (8). Thus, the water depth difference in each finite-volume is given by, Ahy=j^-(XFLUX).^._i At (13) For the x-momentum flux balance, the corresponding (XFLUX)y and (YFLUX)y are defined as: (XFLUX)j. = Ay -2.0 v 2 Ax Ay (14)

6 20 Hydraulic Engineering Software VIII (YFLUX).. = Ax + A(hu). j_^ 2 Ay 2 Ax Ax 4-2.0V; 2 Ax (15) Thus, the difference A(hu) in each finite volume is given by, "(XFLUX)..-(XFLUX); j A(hu)ij=- At (16) For the y-momentum flux balance, the corresponding (XFLUX)jj, (YFLUX)y and the difference A(hv) for each finite volume are similarly defined. Backward differencing is used for the A/ estimation, while forward differencing is used for the A/ estimation. For the current numerical scheme these differences are necessary for convergence. The solution is independent on the direction of the grid. The slopes Sox and Soy are precalculated and stored at the beginning of each time step, while the friction slopes S& and Sfy are updated continuously. 4 Application The free-surface flow in the vicinity of a cylindrical bridge pier shows that as the undisturbed steady mean flow approaches the pier a bow wave is formed on the upstream side of the pier. In the current application the subcritical flow near a cylindrical bridge pier is numerically simulated with two different categories of hydraulic parameters given in Table 1. The computational domain and the explanation of the symbols used in Table 1 are shown on Figure 2.

7 Hydraulic Engineering Software VIII 21 Figure 2: Geometry of the channel and computational domain. Table 1. Uniform flow variables and channel parameters Run 1 2 D (m) B/L (m/m) 2.0/ /4.0 Q (m'/s) So Id'* n ^outlet (m/s) ^outlet (m) Boundary conditions In the present study, two types of boundary conditions are encountered, the open boundaries and the solid boundaries. For the current subcritical flow entrance at the upstream open boundary a fixed value of the flow rate and a relative flow direction are specified. At the downstream open boundary a uniform across the width water depth is specified. The inflow discharge, Q, is maintained constant with time. The water depth at the channel outlet, houtieb is also maintained constant. The free-slip condition is used along the solid boundaries. The direction of the flow is determined in such a way that the normal velocity of each point at the boundary along the perimeter of the cylinder is zero. 4.2 Computational grid The numerical computation was carried out with a nonuniform grid with 82 cell numbers in the x- direction and 45 cell numbers in the y-direction. The size of the grid is smaller close to the cylinder in order to get more information on flow near the cylinder. Distance intervals Ax have a minimum and maximum value of 0.10 and 0.01m, respectively. Distance intervals Ay have a maximum value of 0.04 and a minimum value of 0.02 based on clustering technique (see next paragraph). A multi-grid algorithm has been developed by Soulis [5] in order to accelerate

8 22 Hydraulic Engineering Software VIII the convergence solution and a 2x2 multi-block grid is used in the current application Grid clustering Flow problems with large gradients of the physical quantities need dense computational grid formation for efficient flow depiction. Then, rather using a uniform grid distribution in the tangential direction, grid points may be clustered in high flow gradients regions. This reduces the total amount of required grid points. Grid nodes need not to be uniform in any direction. The following algebraic equation given by Hoffman and Chiang [2] may be employed for this purpose:,,, x r/ \ / xi y u = y w + y W - y w, (2a a,, ALFA = MzU-, BETA = (18) 1-a b-1 where im is the maximum number of grid points in the transverse direction, b is a clustering parameter with values ranging from 1.05 for dense computational grid and 1.20 for less dense while a defines where the clustering takes place. Using a=0.0 clustering takes place at the upper wall while for a=0.5 clustering takes place at lower and upper wall. In the current application clustering takes place at lower and upper wall with a=0.5 and b=l Numerical results The current finite-volume method computational results were compared with available in the literature measurements performed in a laboratory channel as well as with an explicit MacCormack finite-difference scheme by Yulistiyanto, Zech and Graf [6]. Figures 3 and 4 show the water flow depths for the two different hydraulic conditions, run 1 and 2, respectively. The flow depths at the center line both upstream and downstream from the cylinder, h, were nondimensionalized with reference to the outlet water depth, houtiet- The simulated flow depths clearly showed the existence of a bow-wave in the vicinity of the cylinder. The water level behind the cylinder was lowered. High viscous effects in the regions immediately upstream and downstream to the cylinder give rise to substantial differences between predictions and measurements.

9 Hydraulic Engineering Software VIII current model 23 measured - Yulistiyanto et al. mode Axial distance x(m) Figure 3: Comparison between current method predictions, measurements and Yulistiyanto et al. model along the center line, at Q=0.248 nf/s and O "Er measured Yulistiyanto et al. model 3 O Axial distance x(m) Figure 4: Comparison between current method predictions, measurements and Yulistiyanto et al. model along the center line, at Q=0.149 mvs and houtiet=0.173m.

10 24 Hydraulic Engineering Software VIII 5 Conclusions The Navier-Stokes equations for 2D depth-averaged flow were used to simulate flow around a cylinder positioned in an open channel. Numerical solutions were obtained using an explicit finite-volume scheme with appropriate boundary conditions. The computational model has been applied to predict the flow field around a cylindrical bridge pier. Calculated results are in satisfactory agreement with measurements and with other numerical results. The proposed 2D finitevolume algorithm is able to simulate the existence of a bow-wave at the upstream face of the cylinder and the expected flow alteration due to the presence of the cylinder. References [1] Farsirotou, E.D. Numerical and experimental study of scouring in alluvial channels. Ph.D. in Aristotle University of Thessaloniki, Department of Civil Engineering, [2] Hoffmann, K.A. & Chiang, S.T. (eds). Computational Fluid Dynamics for Engineers, Engineering Education System, Wichita, Kansas, [3] Molls, T. & Chaudhry, M.H. Depth-averaged open-channel flow model. Jowma/q/'//y^mw/zcEMgme^mg, ASCE, 121(6), pp , [4] Richardson, J.E., & Panchang, V.G. Three-dimensional simulation of scourinducing flow at bridge piers. Journal of Hydraulic Engineering, ASCE, 124(5), pp , [5] Soulis, J.V. Multiple grid solution of the open channel flow equation using a marching finite-volume method. Advances in Water Resources, 14(4), pp ,1991. [6] Yulistiyanto, B., Zech, Y. & Graf, W.H. Flow around a cylinder: shallowwater modeling with diffusion-dispersion. Journal of Hydraulic Engineering,, 124(4), pp , 1998.