Fluidlrigid body interaction in complex industrial flows D. ~bouri', A. parry1 & A. ~arndouni~ 1 Schlumberger - Riboud Product Center, Clamart, France 2 University of La Rochelle, LEPTAB, La Rochelle, France Abstract Fluid-mechanism interactions occur in a wide range of flow meter categories including turbine and positive displacement systems as well as many flow control devices. This paper outlines computational methods for calculating the dynamic interaction between moving parts and the flow in a flow meter system. The method allows coupling of phenomena without need for access to the source codes and is thus suitable for use with commercially available codes. Two methods are presented; one with an explicit integration of the equations of motion of the mechanism and the other with implicit integration. Both methods rely on a Navier-Stokes equation solver for the fluid flow. The more computationally expensive implicit method is recommended for mathematically stiff mechanisms such as piston movement. The methods are proved against analytical solutions for classical interaction situations and the methods are applied to model real flow meter behaviour. The advances in mesh technology including deforming meshes with non-matching internal sliding interfaces opens up this new field of application for Computational Fluid Dynamics and mechanical analysis in flow meter design. 1 Introduction There are many different groups of flow metering devices [l], some of which include moving parts such as turbines and positive displacement meters and some which are static for example ultrasonic, fluidic and pressure drop based systems.
296 Fluid Structure Interaction 11 In the present note we describe the application of CFD and mechanical analysis in the study of transient fluid-structure interaction in measuring elements or flow control devices comprising moving parts and in particular where the moving parts play a primordial role in the measurement or control. We restrict the explanation to the case where the components that move relative to each other do not undergo internal deformation, that is the components are considered as rigidbodies. However there is no limitation on the displacement of each rigid body. The principles developed are general and could be extended to include small or large internal deformations of the moving components. Applications of the methods described are made for the case of a turbine meter and an oscillating piston meter, a meter belonging to the positive displacement group. A non-linear analysis to predict turbine-forced response is presented in [2] using a coupling method of the fluid and structural models, the structural response is described by a linear model. Blom [3] investigated time-lagged schemes where coupling is included by sequential solutions of fluid and structure models. In an implicit variant the sequential solutions are repeated with interface boundary conditions updated until convergence is achieved. An algorithm is introduced to calculate fluid-structure interaction in a time marching fashlon where both fluid and structure have to be integrated in time simultaneously. In this paper two methods of coupling algorithm are explained. The theory necessary for fluidrigid-body interaction calculations is developed in section 2. The explicit method and more computationally intensive implicit method, requiring the repetition of each time step, are explained. In section 3, the application of the explicit fluidrigid-body interaction method is validated against the analytical solution for the dynamics of rectilinear acceleration of a sphere in highly viscous flows. The case of a tightly fitting piston in a cylinder with incompressible fluid validates the implicit method. A method to treat leakage flow phenomena is indicated. Section 4 describe practical applications of the implicit algorithm for the dynamics of a turbine accelerating from rest and for an oscillating piston meter. The fluid flow analysis software utilised for the calculations included in this paper was Star-CD [4l. 2 Description of calculation methods The flow field variables are calculated from a set of equations which express the conservation of fluid momentum and volume, the Navier-Stokes equations. To cope with large domain deformations we require two capabilities. Firstly, an ability to cope with domain displacement/defonnation. The results of other studies [5][6] showed a successfully application of Arbitrary Lagrangian-Eulerian method to such moving boundary problems The equations of motion are written in a form which accounts for the relative motion of the grid with respect to the fluid. The Geometric Conservation Law is invoked in the formulation in order to avoid errors induced by deformation of control volumes [7]. Secondly, a means of treating sliding interfaces within the calculation domain of the fluid flow.
Fluid Structure Interaction I1 297 Commercially available fluid flow solvers are available with both these essential features for calculating fluid dynamics phenomena in domains undergoing large displacement/defonnation. These methods are usually based on fdte element or finite volume formulations. The rigid-body movement is described by a set of ordinary differential equations (ode's) of the form: where U is the velocity and X is the position of the body. In the context of transient fluid-rigid body interaction, we have the choice of either explicit or implicit time integration of these ordinary differential equations. An example of velocity equation explicit discretization is: U,+, = Un + AtF(Un, xn, tn ) (2) Certain problems are better solved using an implicit discretization, particularly for problems with sensitive force velocity behaviour, known as stiff problems in a mathematical sense. An example of velocity equation implicit discretization is: With the implicit method, we have to repeat the application of the above equations until convergence for each time step. In order to accelerate convergence, it is possible to apply the Newton's method for solving the above eqn (3) as indicated below: U;:; = K+, - p(u;+:,, )] G(K+:,, (4) To realise the repetitions of the same time step we used a restart technique for moving mesh problems automated with an operating system script. The calculation stages comprising initialisation, rigid body dynamic analysis, mesh movement and flow calculation are shown in the flow diagram in figure 1. inttial~satron of fluid held velocrty, presure, boundary cond~tmnd and ilur&ngid-body inteiface forces at 1, and mesh poat~on X, Solution of rigid-body ode's to delermtne 4 U:,, and X:, Bt F Erplc11 method otsp~acement b mesh to &, and update boundary condhons.c Solutmn of the Vow equatms for flurd veluorty, prewre and flu&rtgnf.body rnteifacs forces at time 4,~ Im~liclf Figure 1 : Flow diagram showing stages of fluid structure interaction calculation.
298 Fluid Structure Interaction 11 3 Validation of method in some classical flows 3.1 Validation of explicit method Considering a sphere in a stagnant mfite Newtonian fluid domain undergoing rectilinear acceleration under the Influence of gravity and flow forces. The Reynolds number considered based on sphere velocity, sphere diameter and fluid properties are restricted to the Stokes flow regime with a Reynolds number of 0.1 at terminal velocity conditions. In th~s regime the drag coefficient CD = is given by the expression derived by Stokes Co=24Re for Re pp2 A 2 less then 0.5 where the non-linear convection terms in the Navier-Stokes E( equations are removed or CD = - l +-Re by partially including the effects 6 : ) of the non-linear terms as derived by Oseen [g]. In the case where the fluid density is much smaller than the density characterising the sphere and where the pressure gradient far from the sphere is zero we obtain the following classical momentum conservation equation for the sphere: Assuming the Stokes drag coefficient to prevail, we obtain the following exact solution for the sphere accelerating from rest under the influence of gravity and flow forces: Table 1 shows analytic and calculated steady flow CD values. One observes that the calculated values are closer to the Oseen approximation than that of the Stokes approximation. It was necessary to ensure sufficient distance between the rigid body and the flow domain outer boundary to ensure minimal errors due to domain truncation. Table 1. Steady flow CD at low Re. The curves shown in figure 2 allow comparison of the exact solution, assuming Stokes drag law, and that given by the explicit fluidfrigid-body interaction algorithm. The error at terminal conditions is about IS%, which when considering the difference noted above between Oseen and Stokes steady drag
Fluid Structure Interaction I1 299 values is satisfactory and leads us to believe that the algorithm works for this case. Figure 2: Comparison of Stokes solution and fluid structure interaction. 3.2 Validation of implicit method We describe the problem of a piston in a straight two-dimensional channel, in which the piston moves in response to channel Inlet velocity variation. In the limit of low leak flow rates between the piston and the channel walls there exists an exact solution for the piston movement and the system becomes mathematically stiff. These two facts permit the demonstration and verification of the implicit coupling algorithm. Let us consider the dynamics of the piston in the channel shown below. Further, let us impose a channel inlet velocity varying with time. If we consider that the piston is not fixed it will move in response to the forces acting on it due to the fluid and any contact resistances according to Newton's laws of motion. Pressure outlet cond~tion & f-- f-- Piston Figure 3: Piston in a channel. f-- < Inlet weloclty For the limiting case when the piston is a good fit in the channel and the fluid is incompressible the piston velocity would closely follow the inlet velocity. Figure 4 shows the evolution of piston velocity inside the channel with time as well as the variation of inlet velocity. For ths case the piston width is 50mq piston stream wise length is lmm and 0.lrnrn gap width between the piston and each channel wall. Ramp velocity condition of 120.5 m/s2 was imposed at the entry.
300 Fluid Structure Interaction 11 The fluid was water and the piston density was ten times less than that of water. The fact the gap between the piston and channel walls is small forces the piston velocity to follow closely that of the inlet velocity. This phenomenon is well captured with the implicit algorithm. This stiff problem is very difficult to solve using an explicit integration technique. Figure 4: Evolution of piston and Inlet velocity in a plane channel. In the above calculation the leak paths between the moving piston and the walls were included in the fluid domain and consequently fine meshes were used in these regions as compared to the rest of the domain. In general th~s means of treatment of leak paths is restrictive, in particular as the leak width tends to zero. Local flow rates in leak paths can be modelled conveniently with expressions relating flow rate to pressure difference either side of the leak, leak path geometry, wall velocities and fluid properties. The expressions can emanate from experiments, sub-computations or exact solutions to the Navier-Stokes equations. Also it is straightfonvard to show that the piston dynamics is primarily governed by the pressure field. The fact that the leaks can be modelled by algebraic expressions and that the piston dynamics is governed by the pressure field, enable us to carry out the computations with leak paths much larger in width than the real case, thus alleviating the large computational resources otherwise necessary. The values of inlet velocities imposed for the computations can be corrected by post-treatment for the difference between the calculated leak rate and that obtained by the algebraic expressions with the real leak path properties imposed. 4 Application in industrial flows For problems in which the sensitivity of the rigid-body motion to flow forces is high we need to use implicit time integration techniques for the fluidhigid-body interaction. Such sensitive interaction occurs in positive displacement type flow meters and is the subject of the remainder of the publication.
4.1 Dynamics of a spinner in a conduit with water flow Fluid Structure Interaction I1 301 The algorithm is used to calculate dynamic response of a spinner placed in a conduit used to indicate flow velocity. A lumped parameter analysis of such a spinner gives the classical equation: where Ieff is the effective moment of inertia including the mass of the fluid in the cylindrical volume swept by the blades,,l? is the blade outlet flow angle, is the spinner angular velocity, u is the flow velocity, F is the effective blade radius, A is the area of a plane disc normal to the flow and bounded by the blade root and tip radii and p the density of the fluid. The solution of eqn (7) compares well with the results from the algorithm. As shown in figure 5, the error at terminal conditions is about 5%. 0 001 002 003 OM 005 006 007 008 009 0 I time [S] Figure 5: Dynamic response of the spinner (fluid is water, flow velocity =l ds, initial spinner velocity=o rpm,p=60 degrees, r=8 mm). 4.2 Oscillating piston flow meter [9] The moving element consists of a hollow cylindrical piston with a horizontal web, contained within a cylindrical working-chamber provided with a cover as we see in figure 6.
302 Fluid Structure Interaction 11 Figure 6: 3D view of Working Chamber and Oscillating Piston. A top view of an oscillating piston flow meter composed of a slotted piston which oscillates in a working chamber comprising a partition / guide plate is shown figure 7. In one cycle the angle 6 undergoes one revolution. In fact it will be seen that the piston is always moving in the same direction and each revolution permits a definite volume of fluid to pass through the meter. Figure 7: Schematic of an oscillating piston meter. 4.2.1 Equations of motion of the mechanism, treatment of friction The problem can be schematised by a slider-crank mechanism represented in figure 8. The connecting rod PQ is part of the oscillating piston.
Fluid Structure Interaction I1 303 Figure 8: Free-body diagram of oscillating piston. Where G = resultant body force due to weight according to the principle of Archedes, F = resultant hydrodynamic force, MF = resultant hydrodynamic moment about point 0, R = reaction forces acting on piston at contact lines, MC = reaction force moment about point 0, r~p = distance between 0 and P, m, I = masse, moment of inertia about P of piston. The equations of motion in normal (n), tangential (t) and axial (z) coordinates about 0 are: F" +Gn +R" = rnrop62 F\ G' +R' =mope (9) M: +M: = -jzp + rnrzpe (10) The friction in the n, t plane on the piston bottom or top surface can be expressed using the hypothesis that the repartition of the normal reaction force in the z direction is uniform. The choice of this treatment of friction for the plane contact on the bottom or top of the piston has proved to be useful when compared with experimental data. 4.2.2 Results Inlet and outlet ports are positioned on the ends (top andlor bottom) of the working chamber to allow the 'positive displacement' of fluid. The guide plate serves also to isolate incoming and outgoing fluid. Figure 9 shows the mesh interface between two domains of fluid, one static (the Inlet and outlet parts) and one deforming (the annular chamber with piston). (8)
304 Fluid Structure Interaction 11 Figure 9: Evolution of mesh interface. Below are images of calculated results in an oscillating piston flow meter. Figure 10 shows at left the velocity vectors in a plane through the meter and at right the contour pressure in a plane through the meter. We note the hgher pressure (dark colour) in the inlet volumes to overcome piston friction, inlet/outlet losses and inertial effects. Figure 10: Evolution of piston, velocity and pressure in a plane. The information available for the positive displacement meter using the implicit approach are simulated calibration curves, pressure drops, forces acting on the components and behaviour of meter in a time varying consumption profile. 5 Overall conclusions The explicit treatment of fluid rigid-body interaction in unsteady flows has been validated for a highly viscous flow around a sphere undergoing rectilinear acceleration. To treat stiff systems we can expect to encounter numerical difficulties with the explicit method. Thus the implementation of an implicit approach is imposed for certain problems. A successful implicit implementation was realised for the case of piston movement in a tube. This necessitated several
Fluid Structure Interaction I1 305 repetitions of the same time step with new values of mesh displacement and boundary conditions. Application of the implicit approach has been shown to the acceleration of a turbine from rest and a positive displacement type of flow meter, namely an oscillating piston flow meter. References [l] BS-7405 Selection and application of flowmeters for the measurement of fluid flow in closed conduits. British Standards Institution, 1991. [2] Sayma, A. I., Vahdati, M. and Imregun, M. Turbine forced response prediction using an integrated non-linear analysis. Institution of Mechanical Engineers, Multi-body Dynamics Part K Vol214 No KI, 2000. [3] Blom, F. A monolihcal fluid-structure interaction algorithm applied to the piston problem. J. Comput. Methods Appl. Mech. Engrg. 167, 1998. [4] STAR-CD. Methodology & User Guide. Computational Dynamics Limited. [S] Sarrate, J., Huerta, A. and Donea, J. Arbitrary Lagrangian-Eulerian formulation for fluid-multi rigid bodies interaction problems. Computational Mechanics, 1998. [6] Nomura, T. and Hughes, T. J. R. An Arbitrary Lagrangian Eulerian finite element method for interaction of fluid and rigid body. Comput. Methods Appl. Mech. Engrg. 95, 115-138, 1992. [7] Dernirzic, I. and Peric, M. Space Conservation Law In Finite Volume Calculations Of Fluid Flow. Int. J. Numer. Methods in Fluids, 8, pp.1037-1050,1988. [S] Lamb, B. H. Hydrodynamics. 6th ed., Dover, New York, 1945. [9] Linford, A. Flow Measurement & Meters. N. Tetlow, London, 1949.