**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

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Simulation of the Hydrodynamic Loading and Structural Response of a Marine Riser A. Dercksen and F. van Walree Maritime Research Institute Netherlands Wageningen, The Netherlands Abstract The flow around circular cylinders at moderate to high Reynolds numbers is characterized by vortex shedding. Hydroelastic oscillations are caused by the oscillatory in-line and transverse forces, induced by this periodic shedding of vortices. The hydrodynamic loading and structural response of a marine riser is simulated using the results of several flow simulations with the vortex blob method. A three-dimensional vortex blob method is discussed. Operator splitting is used to obtain a formulation in which the viscous effects are incorporated using a stochastic process, while the inviscid flow is described by the evolution of a flow map. The numerical algorithm consists of an accurate and efficient spectral model combined with a variational formulation. A comparison with exper- imental results and semi-empirical models is used for validation. A description of the computational as- pects related to the coupling of the results obtained from the vortex blob method to an existing struc- tural analysis code is given. Finally some computed results of the dynamic response of a flexible riser, a pipe and a cable subjected to environmental forces due to current will be compared with experimental and full scale results. 1. Introduction Hydroelastic oscillations are caused by the peri- odic shedding of vortices from the boundary layer of a flexible cylindrical blunt body, inducing trans- verse and in-line forces. The vortex shedding fre- quency may lock on to the natural frequency of the system, provided the internal structural damping is sufficiently low. The structure extracts energy from the flow resulting in a motion amplitude increase to values of once to twice the characteristic length. An important issue is the increased drag force and risk of fatigue due to the high frequency oscillations of the structure. Several semi-empirical formulae exist for the pre- diction of drag and lift forces. These models all have in common that details of the flow field are not taken into account, i.e. they are chiefly based on mechani- cal and electrical analogous. More advanced models contain tuning parameters to fit the model to exper- imental observations. Again, these parameters have no physical interpretation other than their analogous in other fields. On the contrary, very advanced nu- merical schemes exist for solving the instationary vis- cous Navier Stokes equations. These methods cer- tainly take into account the flow field, but even with present day computing power such complex simula- tions will impose an economical limit on their use for practical problems. The authors stress that these models are invalu- able for research purposes such as understanding more about the details of such complex flows. Attempts have been made to couple the direct flow simula- tion to a structural code by e.g. Hansen et al. ~l]. The limitations on computer power require many sim- plifications and heuristic arguments in the model. The predictive abilities of this approach are there- fore rather doubtful. For practical situations such as hydroelastic oscillations, simplified models must be developed, drawing on the results and insights ob- tained from large scale, systematic computations and experiments. As a first step in this direction, a wake- oscillator model (e.g. Griffin 2], j3],[44) is tuned to the results obtained by performing a number of sim- ulations using a vortex blob method for solving the Na~rier-Stokes equations. The calculations were con- ducted for a two-dimensional flow around a spring mounted circular cylinder. The tuned model is subse- quently used in a structural analysis code to simulate the hydrodynamic loading and structural response of a marine riser, a pipe and a cable in current. 617

2. Theory Vortex Blob Method rR(r', t, to) The equations governing the evolution of a vortic- ity field A, defined as the curl of the velocity field A, in a viscous incompressible fluid in a Eulerian reference frame are commonly known as the viscous vorticity transport equations. They can be summarized as: D~-=~-Vu+Z'V2w (1) with the incompressibility constraint V · u = 0 for the velocity field. In addition the velocity field must satisfy the no-slip boundary condition at a smooth body surface S and approach a uniform velocity field U far ahead of the body. It is therefore advanta- geous to define the total velocity field u as the sum of two components, viz. u = u + use with u the dis- turbance velocity field which approaches a zero value far away from the body, and use the velocity field of the irrotational flow around the body. The set of equations is completed with an initial vorticity field we which is assumed to be tangential to the body surface. The left-hand side of the vorticity trans- port equation is the material derivative of the vor- ticity field, whereas the right-hand side is responsible for vortex stretching and diffusion of vorticity. Note that in two-dimensional flows, the stretching term is absent. The vortex blob method is based on operator split- ting, where all relevant physical processes are mod- elled subsequently in time. The solution of the vis- cous vorticity transport equations is approximated by successively solving the inviscid vorticity transport equations for a small time step, followed by the dif- fusion of vorticity and the creation of vorticity at the body surface, in order to maintain the no-slip con- dition. Several authors have investigated this frac- tional step algorithm as was proposed by Chorin, e.g. Chorin t5], Beale [6],~7],[8~. Recently, Van der Vegt t94 has given a convergence proof of a product algorithm for the approximation of the solution of the viscous vorticity transport equations. In order to state the algorithm, it is necessary to introduce some definitions. The inviscid flow map R is defined as an invertible continuously differentiable mapping from the initial fluid volume DO to the fluid volume Q at time t: R(r',t,tO`): TO ~ Q At ~ [to,t~] (2) with r' the position of a fluid particle at initial time to. The position r of a fluid particle at time t then is given by the relation: 618 (3) If r' is kept fixed, while t varies, equation (3) reps resents the path of a fluid particle initially at A'. The flow map R must have a Jacobian determinant unity due to the incompressibility constraint, t104. Anal- ogous to the velocity field u it is advantageous to separate the total flow map R in a part R - , related to the velocity field of the potential flow around the body and a disturbance flow map R by means of the relations: R(r',t,tO) = R°° 0 R(r',t,tO) (4) with: Rot (r', t, to) = r' + use (tto) (5) where up is the velocity field for the irrotational flow, i.e. the potential flow. The vector stream function A can be defined by relating the curl of the vector stream function to the total velocity field n. The use of the vector stream function has the advantage that this representation immediately satisfies the incompressibility constraint for the flow field. The relation between the fluid par- ticle velocity R and the Eulerian field velocity u now can be expressed as: R (r', t, to) = Vr x A (r, t) = VR X A (R (r', t, to), t) (6) The relation between the vorticity field in both co- ordinate systems can be constructed likewise, [10], yielding: ~ (r, t) = me (a') Vr`R (r', t, to) (7) The introduction of the flow map has transformed the problem of determining the vorticity field by solv- ing the inviscid vorticity transport equation into the calculation of the flow map R and the vector stream function A. The system of equations (6) and (7) must be completed by deriving equations for the vec- tor stream function. This can most conveniently be done by first separating the vector stream function A into several components: A = A + Am + V?,b (8) where the vector stream function Am has a curl equal to up. The function fib which must be twice differ- entiable can always be chosen such that the distur- bance vector stream function A is divergence free. It is, however, not necessary to calculate the function fib

because in the vortex model only the curl of the total vector stream function is required, hence the gradi- ent in equation (8) will give no contribution to the velocity field. The relation between the vorticity field 5~0 and the disturbance vector stream function A now is obtained by using the definition of the vorticity field and the divergence freedom of A: V2A (`r, t) =~ (or, t) (9) The boundary conditions for the vector stream func- tion A can be directly obtained from the no-flux con- dition for the velocity field at the surface S in an inviscid fluid and equation (8), yielding: n. V x A =n. V x Am = 0 at S (10) since An is the stream function for the potential flow around the cylinder. At great distance from the body the disturbance vector stream function A must sat- . , ashy: A ID (11) The whole system of equations for the disturbance flow map, vector stream function and vorticity field (6), (7) and (9) was put by Van der Vegt t9] in a Lagrangian variational formulation for bodies with an arbitrary geometry by means of the following Action Principle: J [Lr' A, to, try = J dtL [R. A, t] (12) with Lagrangian functional: L fR, A, t] = ~ d3r(~0 (or'`) Vr`R (r', t, to) (~-R (T., t, to) x R (r', t, to)A (R. t)) + - ~ d3T TV X A (r t) ~2 (13) 2 n together with the initial condition n · 5~0 at S for the vorticity field and boundary condition n · V x A = Q at S for the vector stream function yields the set of equations for the flow map and vector stream function after variations bR and bA, with t · [A = Q at the body surface S with tangential vector t. Here the integration in the Lagrangian functional is conducted over both the initial fluid volume SO as well as the fluid volume Q at time t. The Eulerian-Lagrangian formulation requires at first sight the determination of both a disturbance flow map and a vector stream function after which the vorticity field can be determined. The distur- bance flow map can, however, be obtained from the Taylor series expansion of the flow map. This is ac- complished by deriving from the Lagrangian formula- tion a Hamiltonian formulation using a Dirac bracket. An extensive discussion of this reduction process can be found in t94. The result, which can be verified by using (7), reads: R (`r', t +^t, t) = R (T', t, t) + At (VR X A (R. t) ) + 1t,t2 (VR X A (R,t)) VR (VR x A (R't)) +(9 (^t3) (14) The determination of the flow map R is the first step in the product formula. The next step is the modelling of the effects of viscosity, i.e. diffusion and creation of vorticity. The model is based on the ran- dom walls interpretation of the incompressible Navier- Stokes equations given by Peskin t114. The effect of viscosity is to generate a random disturbance on the particle paths generated by the flow map R. The stochastic flow map is defined as the sum of the in- viscid flow map and a disturbance flow map which is a random vector from a sphere of radius (~12~T)2, ~ being a small time step. The viscous vorticity evolu- tion operator is represented by the following sequence of maps: ~ ~ ~ ° ET ° DOT (15) where ~ is an operator which reflects particles, which diffuse into the body, across the boundary. The invis- cid evolution operator En is defined by equation (7~. The diffusion operator DO causes a random transla- tion of each point of the vorticity field. The main result of Van der Vegt t9; states that, provided the inviscid flow map exists and is unique, the expectation of the viscous evolution operator con- verges to the evolution operator of the viscous vor- ticity transport equations. Analogously, the expecta- tion of the particle paths, converge to the real particle paths. More details and a proof of convergence can be found in Van der Vegt t94. In the next section, the numerical implementation of the vortex blob model is discussed. 3. Numerical Implementation _ . .. . .. . _ . .. . . . .. .. The numerical implementation of the vortex model described in the previous section requires the dis- cretization of the vorticity field and an algorithm to 619

evaluate the flow map. For two-dimensional flows several schemes have been suggested. In the point vortex method the vorticity field is represented by a set of point vortices as follows: N w(r,t) = ~ri6(r_i(t)) (16) i=1 where N is the number of vortices, r, represents the trajectory of the vortex with index i. The flow map is then calculated by means of the Biot-Savart law of interaction, resulting in the fol- lowing set of ordinary differential equations for the particle trajectories: ~ ~ it ~) z · +B (but) (17) Here _B is an additional velocity due to an onset flow and the disturbance velocity caused by the body. The advantage of this procedure is that it is gridless, so that numerical problems related to grid generation and stability criteria for high Reynolds numbers are avoided. However, the algorithm has an operation count proportional to N2, and suffers from singular behaviour in the induced velocities when two vortices approach each other. The singularities lead to chaotic motions after some time steps. A variation to the point vortex method is the vortex in cell method in which the flow map is calculated on a fixed grid, re- ducing the singular behaviour and gaining numerical efficiency at the cost of accuracy. The discretization used in this paper is the vortex blob approximation: N ~ (r, t) = ~ rid (lori (t)l) (18) i=1 where by is a function of compact support e.g. a Gaus- sian distribution. This method is more suitable for flows with a smooth vorticity field, and eliminates the singular behaviour in the induced velocities. The vorticity field obtained by this discretization, however, does not satisfy the inviscid transport equa- tions. Convergence criteria for the two-dimensional vortex blob methods are given by Hald et al. [12i, t134. In order to calculate the trajectories of the blobs, the stream function is needed. As was men- tioned earlier, the use of Biot-Savart's law is numer- ically inefficient. Improvements are made by solving the Poisson equation for the stream function on a grid fixed in space. This results in an operation count pro- portional to the number of vortex blobs N. plus some overhead for a fast Poisson solver. The procedure discussed in this paper is based on the results from Section 2. For two-dimensional flows, the stream function has only one non-zero com- ponent. The stream function is separated into three parts: A=Ah+AP+A~ (19) where AP is the solution of the Poisson equation in an unbounded domain, Ah is the solution of the Laplace equation with boundary conditions at the body sur- face S. and Am has a curl equal to the uniform onset velocity U°°: V2AP = _~d v2Ah = 0 n.(VxA e*) = (20) (21) n.~(V x APez)a,,, + U - ) (22) The velocity of the surface 5 is given by u'`,. Both stream functions AP and Ah must have a curl equal to zero at infinity. Substituting AP and ~ in the action principle, results in: J [rj,A ] = 2/ dt/dr'~VAP(~')~2 [i / dt ~j / dr ~ (I fir'~ (t)~) AP (r') 2 ( `' j ~ )} in which rj = (Xj,Yj) is the trajectory of a vortex blob with index j in an unbounded fluid. The stream function AP is expanded in Fourier harmonics with period Lz and Ly in x and y-direction: AP (`r'`) = As, Ak exp (ill r') (24) k Substituting this relation in (23) and taking vari- ations with respect to A`, Xj and Yj leads to the following set of equations: LzLy Skid = p (I) ~ rj exp ~ik · r) (25) dXj ~ P (akin ~ At exp (ill · r) (26) clip = '9X P (skip ~ Ak exp (`ik r) (27) where the filter function P is defined as: P(lkl)= / 620 I dr'~ (fir' - ~ (t)~)exp (id 7.') (28)

For each vortex, the number of trigonometric func- tions to be evaluated equals the number of vortices N. This can be avoided by employing cubic spline ap- proximations to the exponentials, which has the extra advantage of being able to use Fast Fourier Trans- forms. Several of the spline approximations were tested, and are discussed in Van der Vegt et al. [14~. After calculating the stream function AP, the bound- ary conditions for Ah are immediately known, and Ah can be determined. The paths of the vortices depend, however, only on the curl of stream function so that it is more efficient to formulate the problem in terms of Oh = V x Ahez. Bearing in mind the product for- mula for the solution of the Navier-Stokes equations, and the fact that the only way to introduce vorticity in a flow is through the no-slip condition at a solid boundary, the following integral formulation was de- rived: P /s of (tip · K2 (P. q)) dSq = /s (I (u,,,q UV X Aqez)) (tp Kit (`p,q`)) dSq (~29) in which lye is the strength of a vortex layer over the body surface S. t is the tangential vector on the sur- face, and ]~i,~2 are given by: Kit (p, q) = Vpln |T<P~ | (~30~) K2 (`P, q`) = ez x Kit (`p, q) (~31) More details on the derivation of the boundary in- tegral representation can be found in e.g. Hunt t15~. The integral equation for fly' does not have a unique solution (Hunt, t154~. When the body surface is dis- cretized by panels with a constant source strength and vortex strength, this leads to a matrix which is nearly singular. Since vorticity can only be intro- duced through the no-slip condition, the integral of w. over the surface must be equal to zero: /~y~dS = 0 (32) Adding this condition to the boundary integral equation poses a problem with a unique solution. The inviscid vorticity transport can now be summarized ate follows: R = uP + ?,,h + U (33) which is solved using a second order Runge-Kutta method. The resulting tangential velocity jump (vor- ticity layer) at the surface, is represented by a set of blobs, which are diffused into the fluid through appli- cation of the random walk. The two-dimensional model as presented here is discussed in more detail in Van der Vegt et al. [9i, t16i, t144. More information about two-dimensional methods for solving the viscous transport equations can be found in e.g. Sarpkaya t17i, Tiemroth t18] and Leonard [194. For simulation of high Reynolds num- ber flows, many blobs and small time steps are re- quired, resulting in a large computational effort. The extension of the vortex blob model to three dimen- sions is not straightforward. Due to the large number of vortices required to obtain a realistic description the flow field, the tracking of individual vortices is not feasible. Furthermore the stretching term in the viscous vorticity evolution equation introduces extra difficulties. For the three-dimensional flows, a more advanced computational scheme is required. Van der Vegt t9] developed a special purpose spectral model to calculate the flow map. The discretization of the vorticity field also requires special attention. A new representation of the vorticity, employing blobs which are modified to account for the presence of a body, is presently under development. Results and theoretical details will be published in the near future. 4. Vortex Induced Cylinder Oscillations The wake of a bluff body is comprised of an al- ternating vortex street. The shedding frequency fv of the vortices is a function of the ambient flow ve- locity U. cylinder diameter D and Reynolds number. This relation was first discovered by Strouhal. The non-dimensional Strouhal number S is defined as: find S= U (34) and is more or less constant over a wide range of Reynolds numbers (100 < Re < 10000): S = 0.2, for smooth cylinders. If the cylinder is flexibly mounted, there are non-linear interactions between the shed- ding frequency and the cylinder motion. Provided the damping is suffiently low, the cylinder can extract energy from the flow, exciting sustained oscillations at a frequency close to, or coincident with, its natural frequency. In water both cross-flow and and in-line excitation occurs. The in-line oscillations are excited at much lower velocities than required for cross-flow excitation. Two dimensionless numbers which are often used in analyzing experimental data are the reduced ve- locity: I/; = U/fnD and the stability parameter: (35) Ks = 2mi/pD21 (36) 621

where In is the natural frequency of the spring mounted cylinder, m is the virtual mass, ~ is the logarithmic decrement of damping, p the fluid density and I the cylinder length. Experiments have shown that the cross-flow excitation range extends over 4.5 < Vr < 10 with a maximum amplitude of 1.5 diameters. For in-line oscillations there are two instability regions within 1.25 ~ Vr ~ 3.8 with a maximum amplitude of 0.20 diameters. More details on experimental re- sults can be found in the review of Sarpkaya t204. During a test program conducted at MARIN, mea- surements were performed on the spring mounted cylin- der. A general arrangement including the co-ordinate system is presented in Fig. 1. ~ Spring - Uni f orm on set flow 1 1 - 1 ~ x Fig. 1. Spring mounted cylinder arrangement Cross-flow and in-line oscillation experiments were performed. A comparison with the semi-empirical wake-oscillator as proposed by Griffin is discussed in [224. The results appeared very promising, and are rendered valuable for validating the vortex blob method. The two-dimensional vortex blob code has been tested on a number of cases of practical inter- est. Calculations on a fixed cylinder and on a cylinder forced to oscillate were presented in t164 and [21~. For the spring mounted cylinder, the method was supple- mented with a module for solving the equation of motion of a spring mounted cylinder. At each time step, the exciting force originating from the flow field is used as the forcing term for the equation of mo- tion of the cylinder, which is then displaced, causing a modification of the flow field. The region which is of practical importance is the lock-in region, where the vortex shedding frequency is near the natural fre- quency of the mass-spring system. The added mass and damping are incorporated in the calculation of the forces from the vorticity field. The equation of motion is calculated using a second order Runge- Kutta method in time, as is the case for the update of the blob positions. 5. Structural Analysis Code The structural analysis code available at MARIN is a general purpose time domain simulation program to compute the three-dimensional behaviour of sub- merged cylindrical bodies excited by end motions, waves and current. Although the program was de- veloped for flexible risers, it may also be applied to other submerged flexible slender structures, such as Bowline bundles, hoses, umbilicals and mooring lines. The mathematical model is based on a discrete el- ement technique known as the lumped mass method. This technique involves the lumping of mass, excita- tion forces and reaction forces at a finite number of nodes along the structure. All forces are formulated in terms of element properties, i.e. position and ori- entation. By formulating the laws of dynamic equi- librium and the stress-strain relations for each node, a set of equations of motion results. Additional equa- tions are derived for the element twist motions due to torsion. These equations are solved in the time do- main using finite difference and iterative procedures. It is assumed that the structural elements have ax- isymmetrical properties with respect to dimension, fluid force coefficients and stiffness. A detailed description of the computer program can be found in Van den Boom et al. t234. In this paper the discussion is limited to the incorporation of vortex induced fluid forces and motions in the model. At each time step the velocity and acceler- ation vectors of the structural elements are known. From these vectors, the cross-flow velocity and ac- celeration components are determined. Using these quantities, equation (37) is solved using a second or- der Runge-Kutta method. The cross-flow hydrody- namic lift force is then known. The drag force is derived from equation (40) and the time history of the structural response. The lift and drag forces are then evenly distributed along each element. From the excitation forces and reaction forces, the program computes the kinematics for the next time step. This process is repeated for each time step. A circular cross-section is assumed in this approach, i.e. ele- ment rotation is assumed to be perpendicular to the flow. The flow along each element is further assumed 622

to be fully correlated, i.e. two-dimensional. This assumption is valid as the amplitude of motion in- creases t204. Although the fluid loading is thus es- sentially two-dimensional on each element, the struc- tural response computational scheme allows for three- dimensional responses. 6. Wake-Oscillator Models As an attempt to collect all observed phenom- ena in a compact model, several semi-empirical ap- proaches exist [20~. One of the models is the wake- oscillator model, in which a non-linear oscillator for the lift force is coupled to the equation of motion of the cylinder. The response parameter SG = 27rS2K5 is the main parameter of importance. The governing equations of the so-called Griffin Model are: Cr + DOCK - ECHO - Cr - (Cr/ws) ~ [wsGCrh~sHCr] = ~sFY/D (37) for the lift coefficient, and: y + 2t,~ny + w2y = (pU2D/2m) CE (38) for the equation of motion in the direction normal to the onset flow. The shedding frequency us follows from the Strouhal relation and the natural frequency from an = ^, where k is the spring stiffness. The dimensionless coefficients F. G and H are to be determined from experimental results. The major drawbacks of the model are the facts that it is based on fluid damping in still water and that there is a continuous phase angle variation be- tween the exciting force and the response of the cylin- der. Additional damping has therefore been added to the Griffin model. This damping is a function of the cross-flow displacement and is given for By > 0.25 by: where: 4m [~/4 + 0 25 (by0 25)] (39) This is a slightly modified form of the empirical equa- tion as proposed by Skop et al., see e.g. Sarpkaya [204. For the drag coefficient of the oscillating cylinder, the following empirical formulation is used: CD = CD + J sin(2wt) (40) CD = CDO (1 + I (~y/D) ) (41) where h' is the frequency of cross-flow motion, By is the standard deviation of the cross-flow motion and I and J are constants to be determined from numerical results obtained with the vortex blob method. CDO is the mean drag coefficient for a stationary cylinder, Red = u~nD2/z' is the Reynolds number based on the oscillation. 7. Discussion of Vortex Blob Calculations Calculations using the vortex blob method, were performed for values of the reduced velocity ranging from 5.0 to 7.0. The Reynolds number ranged from 0.9 x 105 to 1.2 x 105. Table 1 shows a comparison be- tween experimental and computational results for the cross-flow motion of a spring mounted rigid cylinder, with Vr = 5.25, and D=0.1 m: Table 1 Cross-flow results: measured and calculated Quantity _ CD acD ~CD CLma:e ~C,. ACT, Oman my . my Measured 2.26 1.10 18.33 6.77 3.83 9.25 1.27 0.79 9.25 Computed At* - 0.05 3.49 0.59 19.60 4.40 1.70 9.75 0.20 0.13 9.80 l At* = 0.005 4.50 0.52 19.60 8.33 2.50 9.75 0.22 0.10 9.80 The most striking result is the difference between the maximum amplitudes of the motion. The cal- culated value seems to be closer to normal values re- ported in literature, while the measured value belongs to the largest values reported. As was mentioned be- fore, the calculations were not performed to get con- clusive numerical results, but to capture the relevant phenomena involved. The calculated quantities show a reasonable resemblance with the measured data. Of course the comparison above does not take into ac- count the details of the flow field. The vortex blob algorithm contains parameters, some of which must be determined through numerical tests. Interesting items in the calculations were: · start-up time, i.e. dimensionless time passed before the cylinder was allowed to oscillate, · time step, · panel size, ., . · grin spacing, · vorticity reduction scheme 623

In the present simulations, the flow field is largely determined by the motion of the cylinder. For this reason, some precautions must be made with respect to the coupling of the flow field calculation and the motion of the cylinder. In order to prevent the sys- tem from decoupling, the time integration and the start-up time must be chosen correctly. In simula- tions without a start-up time, the system became un- stable in the lock-in region for a dimensionless time step at* = U^t/D larger than 0.05, i.e. the cylinder disappeared out of the computation domain which was 8 cylinder diameters in both directions. A grid of 64x64 was used in all calculations, with 64 panels on the cylinder surface. A start-up time of I* = 20 was chosen, see e.g. Sarpkaya t20~. In order to keep the number of blobs within reasonable limits, i.e. less than 40000, no experiments could be performed with a smaller time step, unless very crude vorticity reduc- tion was applied. Earlier calculations pointed out that the vorticity reduction should not be applied in the close vicinity of the cylinder, to prevent loss of accuracy. The re- duction or clustering process must be applied with great care, since it distorts the flow field, and there- ~f - ''' - Cu rrent vel oc i ty L ~ Y Fig. 2. Vertical riser arrangement x Table 3 fore the shedding process. A bi-cubic interpolation to a fixed grid sufficiently far away from the cylinder did not cause dramatic disturbances in the motion of the cylinder, although it was noticable in the force registration. Sensitivity to changes in grid and panel size were not investigated in this study. Numerical results con- cerning these parameters can be found in e.g. Van der Vegt A. The calculations discussed above were performed on an ETA-1OP232, and a typical simulation required 20 to 30 hours of computing time. 8. Practical Results In this section some practical results obtained from calculations with the structural code will be presented. A first case concerns a vertical marine riser subjected to a uniform incoming flow. For this application, re- sults obtained from another computer program are also available, see Hansen et al. A. The main pa- rameters are shown in Table 2, while Fig. 2 shows a general arrangement. Some statistical results of the vibration properties of the riser are shown in Table 3. Table 2 Vertical riser properties Quantity Unit Value Length m 100 Diameter m 2 Mass kg/m 3220 Natural frequency rad/sec 0.71 Flow velocity m/& 1.0 Tension N 1.6 x 106 Response parameter _ _ _ 0.15 Comparison between calculated results for vertical riser Quantity Unit Calculated Hansen . Xmin m 1.05 1.35 Xma:c m 2.60 2.80 Ymin m 3.05 3.40 Oman m 3.05 3.40 CD _ 1.75 1.60 /\Tma~ N 8.5 x 104 8.0 x 104 624

Typi ca 1 second mode di spl acement Extreme di sD1 acements -2 0 2 In-1 ine displacement (m) Fig. 3. In-line displacements of vertical riser Relatively large displacements are shown which are not unusual for lightly damped structures. The resemblance with results obtained from Hansen et al. [1] is good, although their maximum cross-flow and in-line displacements are larger. This may be caused by the fact that they use no structural damping at all. Using the present structural code this is not possible due to numerical instabilities for very lightly damped structures. Figs. 3 and 4 show the extreme in-line and cross- flow displacements. The in-line displacement is dom- inated by a static part due to the mean drag force. The dynamic in-line displacement is dominated by a second mode vibration. This is due to the frequency of the drag force which is twice the cross-flow force frequency. The cross-flow frequency is pronounced and vibrates in the first excitation mode at a fre- quency close to the natural frequency. The second case concerns a horizontal pipe and cable in a uniform tidal current. Full scale mea- surement data are provided by Vandiver t244. Fig. 5 shows the experimental arrangement and Table 4 shows the pipe and cable particulars. The agreement between calculated and measured vibration data is good, as shown in Table 5. tic 2n r Typical non-symmetrica di spl acement Extreme di spl acement ~ or, l l _ I t 1 , \ _4 0 4 Cros s-fl ow d i spl acement ( m) Fig. 4. Cross-flow displacements of vertical riser Current vel oci ty in x-direction x y Fig. 5. Horizontal pipe and cable arrangement Table 4 Horizontal pipe/cable properties Quantity Unit Length m Diameter m Mass kg/m Natural frequency rad/sec Flow velocity m/& Tension N Response parameter | Reynolds number 62s _ . Value pipe 23 0.0414 3.327 4.53 0.5 3500 0.24 18800 cable 23 0.0318 1.952 5.78 0.5 3500 0.27 14500

Table 5 Comparison between computed results and experimental (full scale) results for horizontal pipe and cable PIPE Calculated 0.121 0.362 1.625 CABLE Calculated 0.015 0.440 2.30 1 Quantity ~Z/D X/D ZD 11 IJnit. 11 ~ Experiment .- 0.125 0.411 1.67 11 | Quantity | ~Z/D aX/D CD - IJnit Experiment 0.063 0.383 2.07 The standard deviation of the in-line cable mo- tion is, however, overpredicted by the structural code. The calculated data show a lock-in behaviour of the pipe and cable which involved the second and third mode of vibration. This behaviour corresponds to the observed vibrations of the experimental results. Fig. 6 shows a plot of a typical pipe response un- der non-lock-in conditions. It is in reasonably good agreement with the observed response. ~ 0. 035 a) a) Q .~ lo 1 ~ 0.035 lo Cal lo l 0.015 0 0.015 I n -1 i ne d i spl acement ~ m) Fig. 6. Pipe response under non-lock-in conditions The results shown here are in a general good cor- relation with experience and experimental data. 9. Conclusions The results presented in this paper support the authors' view that complex flow simulations based on the Navier Stokes equations are invaluable for under- standing flow phenomena. The Navier Stokes solvers should not be used as black box modules. Especially in flow problems which are not fully understood at present, the predictive abilities of computational pro- cedures must not be taken for granted. The insight obtained from the numerical methods should be used, together with experimental data and experience, to obtain simpler models for engineering applications. The wake-oscillator model which was used to de- scribe hydrodynamical loading contains a minimum of information of the flow field. The results, however, are rendered to be sufficient for use in practice. References [1] Hansen, H.T., Skomedal, N.G. and Vada, T., "Computation of vortex induced fluid loading and response interaction of marine risers", Pro- ceeding~ of Eight International Conference on Offshore Mechanics and Arctic Engineering, The Hague (1989~. t2] Griffin, O.M., Skop, R.A. and Ramberg, E.R., "The resonant, vortex-excited vibration of struc- tures and cable systems", Proceedings of o§- shore Technology Conference, Houston, Paper 2319 (1975~. t3] Skop, R.A. and Griffin, O.M., "An heuristic model for determining flow-induced vibrations of offshore constructions", Proceedings of o§- ~hore Technology Conference, Houston, Paper 1843 (1975~. t4] Griffin, O.M., Skop, R.A. and Koopmann, G.H., "Measurements of the response of bluff cylinders to flow induced vortex shedding", Preprints of Proceedings of Offshore Technology Conference, Houston (1973~. t5] Chorin, A.J., "Numerical study of slightly vis- cous flow", Journal of Computational Physics, Vol. 57, pp. 785-796 (1973~. [64 Beale, J.T. and Majda, A., "Vortex methods 1, convergence in three dimensions", Mathematics of Computation, Vol. 39, pp. 1-27 (1982~. j74 Beale, J.T. and Majda, A., "Vortex methods 2, higher order accuracy in two and three dimen- sions", Mathematics of Computation, Vol. 39, pp. 29-52 (1982~. 626

[8] Beale, J.T. and Majda, A., "A convergent 3-D vortex method with grid free stretching", Math- ematic~ of Computation, Vol. 46, pp. 401-424 (1986~. t9] Vegt, J.J.W. van der, A variationally optimized vortex tracing algorithm for three dimensional pow around solid bodies, PhD thesis, Techni- cal University Delft, The Netherlands (1988~. t10] Serrin, J., Mathematical principles of classical fluid mechanic`, Volume 8/1 of Encyclopedia of Physics, Springer Verlag (1959~. t11] Peskin, C.E., "A random walk interpretation of the incompressible Navier-Stokes equations", Communications on Pure and Applied Mathe- matics, Vol. 38, pp. 845-852 (1985~. [12] Hald O.H. and Del Prete, M., "Convergence of vortex methods for Euler's equations", Math- ematic~ of Computation, Vol. 32, pp. 791-809 (1978~. [13] Bald, O.H., "Convergence of vortex methods for Euler's equations 2", SIAM Journal of N?`meri- f22] cat Analysis, Vol. 16, pp. 726-755 (1979~. [144 Vegt, J.J.W. van der and Huijsmans, R.H.M., '`Numerical simulation of flow around bluR bod- ies at high Reynolds numbers", Proceedings of 15th Symposium on Naval Hydrodynamics (~1984~. t154 Hunt, B. Numerical Methods in Applied Fluid Dynamics, Academic Press (1980~. t16] Vegt, J.J.W. van der and Boom, W.C. de, "Nu- merical simulation of flow around circular cylin- ders at high Reynolds numbers", Proceedings of the 4th International Conference on Behaviour of Offlhore Structures (BOSS '85J, Delft (1985~. t174 Sarpkaya, T., "Computational methods with vortices - the 1988 Freeman Scholar Lecture", Journal of Fluids Engineering, Vol. 111, pp. 5- 52 (1989~. t18] Tiemroth, E.C., The simulation of the viscous pow around a cylinder by the random vortex method, PhD thesis, Department of Naval Ar- chitecture and Offshore Engineering, University of California, Berkeley (1986~. t19] Leonard, A., "Vortex methods for flow simula- tion", Journal of Computational Physics, Vol. 37, pp. 289-335 (1980~. [204 Sarpkaya, T., "Vortex-induced oscillations, a selective review", Journal of A pplied Mechanic`, Vol. 46, pp. 241-258 (1979~. t21] Oortmerssen, G. van, Vegt, J.J.W. van der and Walree, F. van, "Forces on cylinders in oscilla- tory flow: a comparison of the results of numeri- cal and physical models", PRADS 1987, Trond- heim (1987~. Vegt, J.J.W. van der and Walree. F. van, "Progress in the prediction of forces on cylin- ders", Proceedings of a Workshop on Floating Structures and Offshore Operations, Wageningen (1987~. t23] Boom, H.J.J. van den, Dekker, J.N. and Elsacker, A.W. van, "Dynamic aspects of off- shore riser and mooring concepts", Proceed- ing~ of the 19th Offlhore Technology Conference, Houston, Paper 5531 (1987~. t24] Vandiver, J. Kim, "Drag coefficients of long flexible cylinders", Proceedings of the 15th Off- ~hore Technology Conference, Houston, Paper 4490 (1983~. 627