Vortex induced vibration is a major concern in Offshore. Engineering. The flow ... what we would tentatively call LES with a numerically implicit sub- grid model. What is ..... The authors kindly acknowledge the Center for Parallel. Computation of ...
Proceedings of the Tenth (2000) International Offshore and Polar Engineering Conference Seattle, USA, May 28-June 2, 2000 Copyright © 2000 by The International Society of Offshore and Polar Engineers ISBN 1-880653-46-X (Set); ISBN 1-880653-49-4 (Vol. III); ISSN 1098-6189 (Set)
Simulating Vortex Shedding at High Reynolds Numbers P.A.B. de Sampaio Instituto de E n g e n h a r i a N u c l e a r / C N E N , Rio de Janeiro, Brazil A.L. G.A. Coutinho COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
ABSTRACT But the sub-grid model does not need to be explicit: it can be inherent to the numerical scheme. Indeed, Hughes (1995) has shown the relationship between stabilized finite element formulations and the sub-grid modeling of the unresolvable scales.
The turbulent vortex shedding arising from the cross flow past a circular cylinder is analyzed using a Large Eddy Simulation (LES) procedure. In particular, no explicit sub-grid stress model is employed. Rather, the unresolved scales are dealt implicitly by a stabilized Petrov-Galerkin finite element formulation used in conjunction with a time-space adaptive scheme. The numerical results are compared with available experimental data on force coefficients and vortex shedding frequency.
Breuer (1998) has proposed the name LES without sub-grid model to what we would tentatively call LES with a numerically implicit subgrid model. What is important to bear in mind is the distinction between such an approach and a Direct Numerical Simulation (DNS) of turbulence, where the time and space discretizations are fine enough to resolve all turbulence scales. Furthermore, note that the design of a LES with a numerically implicit sub-grid model becomes the design of the numerical method itself. This includes not only the formulation used to obtain the discretized equations, but also the adaptive schemes and other algorithms that affect the way the unresolvable scales are treated (implicitly modeled) by the computation.
KEY WORDS: vortex shedding, large eddy simulation, stabilized finite elements, adaptive methods, circular cylinder. INTRODUCTION Vortex induced vibration is a major concern in Offshore Engineering. The flow around marine structures causes vibrations that may lead to failure by fatigue. The problem is of particular importance in deep water oil exploitation systems, where risers and mooring lines can be viewed as flexible slender cylinders subjected to cross flow. Moreover, for deep water systems the variation of water currents from the ocean surface to the seabed may be large. There is some controversy among researchers on what is the most appropriate way to tackle the problem (Franciss, 1999).
In this work, instead of adopting an explicit sub-grid stress model, the effect of the unresolved scales is accounted for implicitly through the use of a stabilized Petrov-Galerkin formulation. The simulation of the cross flow past a circular cylinder is performed using the adaptive parallel/vector program developed by De Sampaio and Coutinho (1999) for the analysis of incompressible viscous flow. The discretized equations are obtained using a finite element Petrov-Galerkin formulation that automatically introduces streamline upwinding and allows equal order interpolation for velocity and pressure. A remeshing scheme, guided by the estimated error on the viscous stresses, is combined with a local time-stepping procedure, leading to a time-space adaptive computation of the turbulent vortex shedding problem.
We think that the development of Computational Fluid Dynamics (CFD) will have an increasingly important role in the analysis of the vortex-induced vibrations of these slender structures subjected to depthvarying currents. Although the coupled transient three-dimensional simulation of flow and structure needed to address the problem directly is still a colossal task, the time is ripe for paving the way towards such an ambitious goal. In this paper we tackle the more basic, but challenging problem, of simulating the turbulent vortex shedding around a stationary cylinder at the high Reynolds numbers typical of Offshore Engineering applications.
Turbulent analyses with Reynolds number in the range from 104 t o 106 are presented. The quality of these analyses is assessed by comparing the numerical Drag and Lift coefficients, and the numerical Strouhal number, with available experimental data.
A Large Eddy Simulation (LES) procedure is used to study the turbulent flow around a circular cylinder. In a Large Eddy Simulation the large turbulence scales are resolved by the discretization while the small scales are taken into account through the so-called sub-grid models.
THE CONTINUUM AND DISCRETE MODELS We present here the finite element formulation used for the simulation of incompressible viscous flow. The problem is defined on
461
whilst the pressure-continuity equation is
the open bounded domain ~ , with boundary F, contained in the nsddimensional Euclidean space. The governing equations are the incompressible Navier-Stokes equations. These equations are written using the summation convention for a =I . . . . . nsd and b=1 ..... nsd as
(:,o+
PC +t
:uo~ : r (:'a+:U, lt+:P =0
ub~')-'Ox~-LI't~x b ~ G ) J
At 8N~ 8 ~ "+1/2
! :xo :x---T- m : -rat+N'&:xo pa; :~;+x~dnI N,
(1)
a
+ ~ S A t :IVi +
(2)
d u "-=O cgx a
d,-- I
r~
+x.
,o,
:xo :x,
8Xb
(4)
[//(:u:+Dtx:l]d.Q t.:x,
In the above equations ft,
where p and /2 are the fluid density and viscosity, respectively. The velocity and pressure dependent variables are denoted by u~ and p, respectively.
:
axoj]
and ~ are the discretized fields,
interpolated using standard C o finite elements whose shape functions are denoted by N~. The superscripts n, n+1/2 and n+l indicate the time-level. Note that the terms explicitly written as summation of element integrals vanish altogether for the linear shape functions used.
The model is completed introducing boundary conditions and an initial velocity field. Velocity and traction boundary conditions are given data g, and ?a • They are prescribed on the boundary partitions
The problem is solved using a segregated solution procedure. Pressure is computed first, then the velocity field is updated. Eqs. (3)(4) lead to symmetric positive definite matrices, allowing the use of preconditioned conjugate-gradient solvers.
F.~ and F,~, such that F.ouF,~=F and r ~ . n F , . = ~ . Pressure and mass flux boundary conditions are associated to the mass balance and are given as ff and G on the boundary partitions Fp and F c , such
Discretizations similar to that presented here have been obtained by (Zienkiewicz and Wu, 1991) following alternative approaches. The procedure can be extended to the analysis of natural convection problems with the inclusion of the energy balance and accounting for buoyancy forces in the equation of momentum (De Sampaio and Coutinho, 1999).
that F p ~ F o = F and Fpr'~Fo=O . The continuum model is discretized using linear triangular elements to approximate velocity and pressure. Such a choice of interpolating spaces is not acceptable within the mixed-formulation framework, as it violates the Babu~ka-Brezzi condition (Brezzi and Fortin, 1991). However, the Petrov-Galerkin formulation we shall present next avoids such a difficulty through the introduction of extra stabilizing terms (Hughes, Franca and Ballestra, 1986), (De Sampaio, 1991). The formulation also leads to better approximations of convection dominated flows, for it generates streamline upwinding (Brooks and Hughes, 1982), (De Sampaio, 1991). The reader is referred to (De Sampaio and Coutinho, 1999) for a detailed description of the method used in this work.
LES WITH A TIME-SPACE ADAPTIVE PROCEDURE In this work we combine a remeshing scheme with the local timestepping algorithm introduced by De Sampaio (1993). This algorithm sets local time-steps based on the time-scales of the convection diffusion processes resolvable on a given mesh. These time-scales are estimated according to local values of velocity and physical properties, and according to the local mesh resolution. The local time-steps are chosen according to the expression,
The following discretized model is obtained by least-squares minimization of the time-discretized momentum balance with respect to the velocity and pressure degrees of freedom. The pressure-continuity equation Eq. (4) results from the combination of the least-squares momentum minimization and the requirement of flow incompressibility Eq.(2). The discretized momentum balance is given by
he
(5)
a,_- = Ilu"ll where
Air
+-'2-
