Numerical Investigation of Three-dimensional Flow around a Free ...

Proceedings of The Twelfth (2002) International Offshore and Polar Engineering Kitakyushu, Japan, May 26-31, 2002 Copyright 0 2002 by The International Society of Offshore and Polar Engineers ISBN l-880653-58-3 (Set); ISSN 1098-6189 (Set)

Numerical Investigation

of Three-dimensional

Conference

Flow around a Free-Spanned

Pipeline

Bing Chen I and Liang Cheng Department of Civil and Resouce Engineering The University of Western Australia, Australia

Recently, considerable research efforts have been devoted to developing numerical models for local scour below pipelines (Li and Cheng 1999; Brors 1999; Li and Cheng 2000a, 2000b and 2001). The results presented in these studies are encouraging that even time-dependent scour process due to vortex shedding can be modeled with moderate success. Although these numerical models are exclusively twodimensional, the methods used in these models can be extended to three-dimensional situations. Recently the authors presented a study on three-dimensional modeling of laminar flow around free span shoulder of pipeline (Chen and Cheng, 2001). It was found that a spiral type of vortex tube does exist around the span shoulder region, although the scour profile in the perpendicular direction of the pipe was simplified by a horizontal bed. In this paper the research reported by Chen and Cheng (2001) is extended to turbulent flow case.

ABSTRACT: Three-dimensional flow around a free-spanned pipeline is simulated using a fractional step finite element method. The Reynolds number, based on the pipe-diameter and the approaching flow velocity, is set at 500 and 5~10~ to simulate both laminar and turbulent flow; In high Reynolds number case, Smagorinsky SGS closure is used to model the turbulent flow. It was found, from the limited amount of numerical data, that a spiral vortex tube is formed around the span shoulder and extends around the span shoulder, which is quite similar to the assumption made by Sumer and Fredsoe. It is also found that there are significant shear stress concentrations in the span shoulder area. The reasons for the shear stress concentration are discussed. KEY WORDS: three-dimensional; numerical model; scour; free span; span shoulder; pipeline; shear stress

The accuracy of the prediction of local scour will heavily depend on the accuracy of the simulation of turbulent fluid flow. Flow field around a circular cylinder near an erodible seabed has not been investigated, to the present authors’ knowledge. Therefore the purpose of the present paper is to develop a numerical model to solve the flow around a freespanned pipeline and investigate the local flow structure and the seabed shear stress around the pipeline shoulders. It is expected that the present research be considered as the first-step attempt towards numerical modelling of three-dimensional scour around pipelines.

INTRODUCTION

When a pipeline is placed on an erodible seabed, free span may develop as the result of local scour under the pipeline. One of the most important issues in pipeline stability design is the evaluation of the free span length and the rate at which the free span develops along the span direction due to local scour. Both free span length and the rate of scour development depend on flow conditions around the span shoulders. Since most of the scour investigations in the open literature were on the scour depth or scour development in the perpendicular direction of pipeline axis, the current knowledge on three dimensional scour developments around pipelines is very limited. Fredsoe et al. (1988) investigated the three-dimensional scour development of a sagged pipeline and presented a methodology to evaluate the longitudinal dimension of the scour hole. It was speculated that in addition to the tunnel and lee-wake scours, a spiral type of vortex which forms in front of the pipe near the span shoulder of the scour hole may be responsible for the three-dimensional scour development (Sumer and Fredsoe, 1999). Sumer and Fredsoe (1994) also investigated the self-burial and back-filling processes at span shoulders by physical model tests. However no detailed study on the three-dimensional flow around the span shoulder is available yet (Sumer and Fredsoe, 1999). ’ Permanent address: Department

of Civil Engineering,

Dalian University

MATHEMATICAL

MODEL

Governing Equations The governing equations are three-dimensional Navier-Stokes equations and continuity equation for incompressible fluid. In high Reynolds number flow, Smagorinsky’s subgrid-scale (SGS) model is used, the filtered governing equations are:

(1)

of technology,

61

Dalian 116024, China

The seabed and the surface of the pipe are assumed hydraulically smooth, on which the velocities are set to zero (non-slip condition). On the lateral planes at both ends of the pipe and the top boundary of the domain, symmetric conditions are imposed, i.e. normal gradients of velocity and pressure are set to zero on those boundaries. Pressure on the top point of the outer boundary of grid system is used as a reference value and remains constant. At initial time the fluid is set to rest with zero velocity and pressure fields, then a sudden impulse of velocity is imposed on the inflow and outflow boundaries.

In which u, is the velocity component along i directionp is pressure, the subscript i and j represent the direction of coordinates axes, ij=1,2,3; The over-bar indicates the large-scale components of filed variable. In the Smagorinsky model an eddy viscosity vr is assumed other than the kinematics viscosity v:

FLOW AND GEOMETICAL

Where C, is the Smagorinsky filter width proportional

constant, set to 0.1 in this paper; A is the

to the grid size; 3 is the magnitude I I

Flow over horizontal pipeline, placed over a trapezoidal trench, is simulated using the present fractional finite element method. Geometrical symmetry is employed to reduce the calculation costs and the layout of the seabed and pipeline geometry is given in Fig. 1. The width of shoulder B is set to 20, D is the diameter of cylinder. The distances from both the inlet boundary and the top boundary to the center of the pipeline are SD, and that from outlet boundary to the center of the pipeline is 300. The slope angle 0 is set to 24”, which is close to the repose angle of sands in real life situations. The Reynolds number of the flow is defined as Re=UJllv, U, is the maximum velocity of free stream, v is the fluid kinematics viscosity. A lower Reynolds number Re=500 represents a laminar flow, the flow pattern and bed shear stress distribution should be different to high Reynolds number situation, Re=5x105 here represents a turbulent flow, closer to the real situation in ocean. The thickness of boundary layer at inflow boundary 6 is set to 0.5D in this paper.

(3c)

For laminar flow, the eddy viscosity vr in Eq. (1) is set to zero and Eqs. (1) and (2) degenerate to normal N-S equation and continuity equation for laminar flow. A fractional step finite element method is used to solve the governing equations in this paper, i.e. one time step is divided into three sub-steps. The fractional step finite element method initially estimates the velocity components explicitly at the first two intermediate sub-steps, based on the velocities and pressure from the preceding time step, then updates pressure by solving the pressure Poisson equation based on the most update velocities, and finally calculates the velocities based on the new pressure and intermediate velocities. The details of the scheme can be found in the work of Jiang and Kawahara (1993). A preconditioned BiCGSTAB (Feng and Huang et al, 1995) iteration solver is used to solve the pressure Poisson equation.

Grid System The S-node brick element is adopted in this three-dimensional FEM model. Fig. 2 shows the three-dimensional grid system used in this investigation. It should be noted that only the projection of grid on the lateral boundary plane is shown to give a clear image. The grid near the solid bed and the surface of the pipe are reasonably fine to resolve the flow field near the solid walls. The distance from upstream inflow boundary to the center of the pipe is SD, and that from outflow boundary to the center of the pipe is 300. The grid system consists of 183635 nodes and 172676 elements.

Initial and Boundary Conditions a parabolic

velocity

profile

is assumed

in

(44 For turbulent flow, a power law profile is assumed:

ANALYSIS

u yt u,=s 0

At the outflow boundary, pressure is set to be constant and the gradients of velocity components in streamwise direction and velocity components themselves in directions perpendicular to streamwise are set to be zero:

Flow Structue around Span Shoulder The flow field is averaged over a period of time for the purpose presentation. The average of a flow parameterf is defined as

p = constant

au av aw

METHOD

The hydrodynamic forces acting on free spanned pipeline had been discussed by Chen and Cheng (2001), the results showed that the threedimensionality of the flow would fully developed after the dimensionless time greater than 30. The calculated forces will not be presented here and we will pay more attention to flow structure and bed shear stress around the free span.

In which U, is the maximum velocity of free stream, y is the distance from the solid wall of bottom, 6 is the thickness of boundary layer at inflow boundary.

z=dx=dx=O

AND GRID SYSTEM

Flow and Geometrical Parameters

of large-

scale strain-rate tensor:

At the inflow boundary, laminar flow situation:

PARAMETERS

(5)

62

of

Since considerable time is needed to establish regular vortex shedding in the simulation, the average is taken over the period from t=40 to 100, in which the three-dimensional flow are fully developed.

Distribution

pipeline. More physical computational results.

The bed shear stress is an important quantity as long as local scour is concerned. In laminar flow situation, assume a parabolic velocity profile in the boundary layer,

z =w puz

z,

=-

through

the following

b

In turbulent flow situation, assumed as:

the velocity

profile

U+

= 2.44lny+ +4.9

y+ >lO

U+

=y+

y+