A simplified method for the fatigue assessment for part ially strake -covered pipeline spans is provided and calibrated so that for t he bare pipe case it is consistent with the current DNVGL Recommended Practice (DNVGL -RP-F105). It is based on a Rayleigh -Ritz approximation using the first undamped mode shape. In that sense it is equivalent to a single -mode analysis using Shear7, excep t that it accounts for frequency - dependent added mass, and the formulation is presented in a compact and convenient non -dimensional form. The method presented may also be consider ed a simplification of that in Vivana in that it does not account for propag ating wave effects, and is therefore named “Sivana”. Like any frequency -domain method Sivana is based on the assumption of harmonic response. This requires excitation and added ma ss functions that define a harmonic hydrodynamic force as a function of the harmonic motion. On the other ha nd, current practice by DNVGL -RP-F105 is based on a response function, which gives VIV amplitude as function of reduced velocity and the Scruton ma ss-damping parameter K s. To bridge this gap , an inversion technique is deve loped whereby the excitation function for bare pipe is derived from the DNVGL - RP-F105 response function, such that for bare pipe the Sivana calculation will lead to the same respon se amplitudes as DNVGL -RP-F105. The Sivana method is illustrated by an exam ple involving a single span with different degrees of strake coverage centered at midspan. Although the Sivana method might be considered a natural and straight -forward extension o f current span assessment practice to include partial strake coverage, cauti on is in order for certain approximations in the DNVGL -RP-F105 response function may be less appropriate for the use in this paper, than for the use envisioned by the developers of the DNVGL guidance. 1 Introduction Pipeline spans can arise from scour, as well as from laying the pipe on an uneven seabed. The former can involve high overall costs to monitor span formation and remediate when necessary, to prevent a fatigue failure due to VIV. For scour -induced spans th at become too long, providing a support near midspan may temporarily alleviate the problem by dividing the span in two shorter ones, but further scour can quickly lead to two spans of excessive length. The supports also disrupt a natural cycle whereby spa ns finally become so long that the pipe sags into the trench it creates. A better alternative can be to cover the span with strakes, partially, to the extent needed to limit VIV fatigue damage. The method presented in this paper can not only account for partial strake coverage, but also for other changes along the length of the span, such as changes in current velocity and/or pipe diameter. The response function approach is described in detail in DNVGL -RP-F105 [ 1]. By fitting VIV test data on flexible pipes or spring -supported rigid cylinders, the response function provides the VIV amplitude as a function of the component of current velocity normal to the pipeline , the still-water undamped na tural frequency, the mass damping ratio, and the hydraulic diameter of the pipe. In addition to the above primary parameters, secondary parameters that affect the VIV response amplitude for each mode calculated according to F105 [1] include: For cross -flow: certain effect s of seabed proximity and scour trenches on VIV and the frequency ratio for modes that may “compete” . For in -line: the turbulence of the incoming flow, and the angle between the flow direction and the pipeline . In addition, for both in-line and 1 Peek Solutions, Sant Andreu de Llavaneres, Spain. 2 Sintef Ocean, Trondheim, Norway 3 Crondall Energy Subsea, Newcastle, UK. 4 Shell Global Solutions International, B.V., Rijswijk, The Netherlands. 5 Shell Philippines Exploration, B.V., Manila, Philippines. 2 cross -flow, there are the unsteady flow parameters “α” and “KC” in F105 arising from including wave as well as steady current effects. Finally there are parameters that are “not explicit”6 in F105, in the sense that they are considered in developing the response function but the sensitivity to variations in that parameter in the range of application has apparentl y has been neglected, presumably deemed to be adequately covered by safety margins included explicitly (in the form of safety factors) or implicitly (e.g. in the form of a response functi on that provides an envelope to the available data) . These include t he Reynolds number , pipe roughness, the pipe mass (except to the extent that the pipe mass influences the still -water natural frequency ), the mode shape and “ in-line induced cross -flow”, which refers to how the presence of an in -line line mode can influenc e cross -flow response. A drawback of the response function approach is that changes in current velocity, pipe diameter, and strake coverage along the length of the pipeline cannot be acco unted for, except if the response function is constructed from tests involving the same changes along the length of the pipeline . The alternative is to rely more on theory and less on tests, as is done for risers, using codes such as Shear7 [ 2], VIVA [ 3], or Vivana [ 4,5]. For this, the empirical basis comes from forced vibration tests, which represent a single cross section . The motion is assume d to be harmonic and either horizontal (in -line) or vertical (cross - flow). This is captured in unidirectional, harmonic force d-vibration tests on a rigid cylinder such as those of [ 6] for cross -flow, and [ 7] for in -line. The excitation and added mass functions resulting from such tests are then used in the aforementioned codes to calculate VIV re sponse. This is referred to as the excitation function approach. Sivana [8] is a simplified version of this excitation function approach , based on a Rayleigh -Ritz approximation using the undamped, still -water modes . For pipeline spans responding in the lowest -frequency modes with negligible propagating wave effects this provides a good approximation. In Section 3, the energy balance condition for Sivana is given in a convenient non -dimensional form. For pure in -line VIV, the excitation function approach has been quite successful: using the excitation and added mass functions from tests by Aronsen [ 7], a good approximation to the in-line response function from F105 [1] is obtained with Sivana for the conditions considered in [ 8] (including a mass ratio of 1.23, low tu rbulence, normal flow and no seabed proximity effects) , as shown in Figure 1. However, it gets more complicated at current velocities beyond the onset of cross -flow VIV, since cross -flow (“CF”) and in -line (“IL”) VIV interact: Due to the hydrodynamic interaction7, the CF motion typically includes a component of IL motion at double the CF frequency [22, 23]. Further, the IL component can enhance the CF excitation leading to higher CF response than woul d occur otherwise. To account for this and other effects , cross -flow excitation functions in Vivana are not obtained directly from forced vibration t ests such as [ 6]. They have been adjusted to account for these effects [24,5]. Despite thi s adjustment, carried out mainly with riser response in mind , the excitation and added mass functions from Vivana [4] do not provide a good match to the DNVGL -RP-F105 cross -flow response functio n is seen in Figure 2. To investigate CF-IL interactions, tests involving forced simultan eous forced motions in both directions have been performed [ 7]. However, even if one assumes that the in -line frequency is double the cross -flow frequency, and neglects Reynolds number effects, it still takes 4 dimensionless pa rameters to define such a bidirectional test : an IL amplitude and a phase difference are added to the CF amplitude and frequency . With 4 instead of 2 parameters , it requires orders of magnitude more tests to empir ically define the excitation and added mass function s in 4 variable s, since enough experimental points in the 4 -dimensional space are needed so that one can reasonably interpolate between those points. For these reasons , bi-directional forced vibration te sts may yield interesting and useful insig hts, but they are not the most direct path to a practical yet reasonably accurate assessment procedure , nor are they intended as such. Another approach to obtain excitation and added mass functions is based on the response of a densely 6 Only the Reynolds number is stated to be “not explict” in F105 [ 1], but the other parameters mentio ned also do not influence the F105 calculated response, and it is assumed that they been considered in a similar way. 7 In addition to hydrodynamic interaction, there can be structural i nteraction between IL and CF directions, to the extent that the drag force makes the pipe hang in in inclined plane, rather than a vertical plane as it would under gravity force alone. The modes are then in -plane and normal to the plane in which the pipe hangs. Further interactions can occur due to geometrically nonlinear dynamic structural behaviour. However, here reference is made only to the hydrodynamic interaction, which is also present when the dynamic behavior remains purely in the linear domain w ith non IL -CF interactions. 3 instrumented flexib le pipe, by deducing the hydrodynamic forces from the stiffness and motion of the pipe. Thus, Wu [ 9] estimated hydrodynamic coefficients along the pipe from a limited number of acceleration and strain measuremen ts using a finite element beam model for a pinned -pinned beam under constant tens ion. The forces on the pipe are adjusted to optimize the fit to the experimental results, with the dynamic equilibrium equation imposed as a constraint . From the resulting hydrodynamic forces , local excitation and added mass coefficients are estimated . However, the application of the method to the free spanning pipe needs to be further investigated , especially for low reduced velocit ies, and the associated increased uncertain ty due to small measured response amplitudes . For th e above reason s, the approach used here to estimate the excitation function is a different one : It build s on established engineering practice that has already been e xtensively calibrated against experim ental results. Instead of obtaining the excitation function from for ced vibration tests, it is obtained by inversion of the response function in DNVGL -RP-F105 [1], which is based on VIV tests on flexible and/or spring supported pipes, and for cross -flow already accounts for the bi -directional, non -harmonic components of VIV motion. The direct inversion is enabled by the Sivana simplifications and presented for cross -flow in Section 5. It leads to an excitation function that exactly reproduce s the response predicted by DNVGL -RP-F105 [1] for uniform pipe and current velocity . Combining this effective excitation function for bare pipe with existing results for straked parts , then yields the cross -flow VIV response for the partially straked pipe. Seabed proximity effects can be important [10-15], but are not considered here. All test results used, including forced vibrat ion tests, and tests on flexible pipes, are for pipes or cylinders “towed” in stationary water at a sufficient distance to boundaries to make the boundary effects negligible. Similar principles can be applied with seabed proximity effects, but that would require a repeat of test programs such as [ 6,7] in flowing water at different distances from the model seabed, possibly also considering different roughness of the seabed. Here only the effect of par tial strake coverage on VIV is addressed without considering seabed proximity effects at the same time. 2 Nomenclature 2.1 Terms and Abbreviations 2D test refers to a VIV test on a spring -supported rigid cyl inder that is free to move with constant displacement along its length. bare “bare pipe ” refers to the pipe that may be coated but is not covered with strakes CF cross -flow, refers to component of VIV motion normal to the flow direction and the pipe F105 refers to the DNVGL Recommended Practice for span assessment [ 1] F105 added mass function : refers to added mass function given in Section 4.4.15 of [1]; with the add ed mass coefficient under flowing conditions, C a, given as a function of reduced vel ocity V R = 1/fs*. F105 response function: response function A*=A*(VR, K s) recommended in Section 4.6 of [1] for in -line VIV, and in Section 4.4 of [1] for cross -flow VIV. Specifically it refers to th is function for steady currents, with all safety factors set to 1, no seabed proximity effects, low turbulence, flow normal to the pipeline an d no mode competition . There are no other secondary parameters that influence the calculated response in [ 1], as described in the introduction . IL in-line, refers to component of VIV motion in the flow direction (where the flow is not normal to the pipe i t refers to the component of VIV motion normal to the pipe and the CF direction of mo tion) normalized velocity: current velocity expressed in pipe diameters per per iod of VIV oscillation under flowing conditions, 1/f* reduced velocity: current velocity expressed in pipe diameters per natural period of oscillation in still water, VR = 1/f s* response function: VIV response function is a function A*=A*(VR, K s) such as those defined in [ 1] providing the normalized VIV response amplitude A*=A/D as a function of reduced velocity V R = 1/f s*, and the Scruton damping par ameter K s = π2 mes,avg* ζ. Secondary parameters (see below) may also influence response amplitude . If so, fixed given values of these secondary parameters are considered. 4 secondary parameters: parameters other than reduced velocity V R and the Scruton mass d amping parameter K s that may influence maximum VIV amplitude, including but not limited to mass ratio (m ratio), mode shape (ϕ), Reynolds number, pipe roughness, and turbulence and/or unsteadiness of incoming flow . Sivana simplified version of Vivana [4] introduced in [ 8] and formulated in Section 3 straked : strake -covered , althou gh the general principles may be applied to any type of strake, where reference to specific quantitative results is made, it applie s to strakes with 3 strands around the circumference at a pitch of 17.5 D and a strand height of 0.25 D. transverse: “Transverse displacement” is vertical for cross -flow VIV, and horizontal (la teral) for in -line VIV. These are considered as separate case s. Interactions between vertical and lateral motions are not considered in the formulation but included in estimating effective value s of the excitation coefficient. Vivana Frequency domain VI V response code [ 4,5] based on the assumption of harmonic motion and that such motion gives rise to a harmonic hydrodynamic force on the pipe, which is defined by excitation and added mass functions, and the influ ence of VIV amplitude on added mass is neglected. Vivana default excitation and added mass functions: this refers to the functions defined in [ 4]. These functions have been updated in [ 5], but [ 5] is only available to holders of a license f or the Vivana code. 2.2 Latin Symbols A = amplitude of harmonic vibration at the location x along the span where it is a maximum A* = normalized amplitude A, given by A* = A/D, c orresponds to maximum of the local amplitude Y*, which typically (but not necessarily) occurs at midspan A*(VR, K s) = VIV amplitude response function, in particular, in Section 5, this refers to F105 response function for cross -flow VIV as defined under terms above Au*(VR) = undamped response function, A u*(VR) = A*(VR, 0) Ca = Ca(f*) = added mass coefficient under flowing conditions . As in [ 4], this is assumed to be a function of dimensionless frequ ency f* only, i.e. C a=C a(f*). Cas = added mass coefficient in still water , the values used here are Cas = 1 for bare pipe, C as =1.51 for straked pipe. For Implementation A , the straked or bare values are used, as applicable, but for Implementation B , the bare value is used on strake -covered portions of the pipe as well as bare ones. This Implementation B provision only applies for the still water added mass and not for the added mass under flowing conditions, defined above. Ce = C e(f*,Y*) = excitation coe fficient. Also referred to as “lift coefficient” for in -line VIV. The function C e(.,.) is the excitat ion function. Cen = C en(r) = normalized excitation function, see Eqs. 12 and 13 Cen2D = C en2D(r) = normalized excitation function when calibrated to the response function from a 2D test , see Eqs. 9 and 10. D = pipe outer diameter including any coatings Ein = net energy input into the span per cycle of harmonic motion Ein* = Ein/(2m w L U2) = dimensionless energy input EI = effective flexural stiffness of t he pipe, including contribution from coatings, if applicable f = frequency of VIV oscillati on f* = dimensionless frequency of VIV oscillation , f* = fD/U fs = undamped natural frequency in still water , for Implementation A it is the natural frequency of th e partially strake -covered pipe, for Implementation B it is the natural frequency of the ba re pipe. fs* = fsD/U = 1/V R = dimensionless , still -water , undamped natural frequency ; 1/f s* is the reduced velocity V R k = soil spring stiffness (force per unit leng th along pipe per unit deflection), refers to stiffness of vertical or horizontal springs for cross -flow or in -line VIV, respectively 5 Ks = π2 mes,avg* ζ = Scruton damping/stability parameter. For implementation A the value for the partially strake -covered pipe; for Implementation B the value for the bare pipe. Ks(.) = inverse function to R K(.) defined in Eq. 8 L = span length or any other reference length (e.g. overall length of a zone of multiple interacting spans) mes,avg* = ∫span (mratio + C as) ϕ2 dx/{∫ mode ϕ2 dx}=M es*/ϕrms2 = modal effective mass ratio in still water; for Implementation B, or for 100% bare or 100% straked pipe this is simply m ratio + C as. mp = mass of pipe, coatings, and content per unit length of pipe , for Implementation A this also in cludes the mass of the strakes, for the straked portions, but for Implementation B, the mass of the strakes is excluded mratio = mp/mw = mass ratio mw = mass of water displaced by the bare pipe per unit length of pipe ( ¼ π D2 ρ) N = effective axial force in the pipeline where it spans, positive for tension Nc = buckling load; absolute value of N at which buckling in compression occurs Nr = number of points used in the discretization to solve the integral equation arising from the inversion problem r = ratio of damped to undamped VIV amplitude, r = A*/Au* RK(.) = function such that R K(Ks) provides the factor by which the VIV amplitude is reduced by the damping Ks according to the F105 cross -flow response function, defined in Eq. 7 U = current velocity, mor e precisely , the magnitude of the component of current velocity normal to the pipeline at the pipeline elevation that would occur there if the pipeline were not present , assumed to be constant along the length of the span VR = 1/f s* = reduced velocity , for partially straked pipe it is based on the natural frequency with the strakes for Implementation A, and without the strakes for Implementation B. x = coordinate defining location along the pipeline Y = Y(x) = A ϕ(x) = amplitude of transverse displacement at location x Y* = Y*(x) = Y/D = A* ϕ(x) = normalized amplitude of transverse displacement at location x y(x,t) = Y(x) cos(ωt) = A ϕ(x) cos(ωt) = transverse displacement at location x and time t 2.3 Greek & Mathem atical Symbols ϕ = ϕ(x) = still water mode shap e; normalized so that ϕ(x)=1 at the location x where the modal displacement is a maximum in absolute value; for Implementation A , this is the undamped mode shape for the partially straked pipe; for Implementa tion B it is the mode for the bare pipe ϕrms = root mean square modal displacement, ϕ rms2 = L-1 ∫mode ϕ2 dx ρ = density of seawater ω = 2πf = angular frequency of VIV oscillation (under flowing conditions) ωs = 2πf s = undamped natural frequency of vibrati on in still water ; for the partially straked pipe for Implementation A, but for the bare pipe for Implementation B. ζ = modal damping ratio in still water due to soil and structural damping, not including any hydrodynamic damping ; for the partially strake d pipe for Implementation A, but for the bare pipe for Implementation B. ∫mode (.) dx = integral of (.) covering the span(s) considered plus soil -supported portions of the pipe where the modal displacement is significant ∫span (.) dx = integral of (.) cove ring the span(s), excluding soil -supp orted portions of the pipe 6 3 Sivana Assumptions and Formulation Whereas the basic principles of Sivana and results from applying these to in -line VIV are given in [ 8], details of the assumptions and formulation are given in this section but not in [ 8]. As in [ 4], linear behavio ur of the pipe - soil system with viscous damping an d harmonic response are assumed. Further, in “Sivana” , a Rayleigh -Ritz approximation is used, based on the undamped, still -water vibration mode ϕ=ϕ(x). T he frequency is obtained from the Rayleigh quotient, and the damping from an energy balance condition . For partially straked spans, t here are two possible implementations of the Sivana approach , depending on whether the undamped, still -water modes with or wit hout the strakes are used in the Rayleigh -Ritz approximation . These are referred to as Implementa tion A and B , respectively. The same equations apply for both implementations, but the definition of certain symbols differs as specified in the Nomenclature section. For instance, ϕ=ϕ(x) denotes the undamped, still -water mode shape with the strakes for Implementation A and without the strakes for Implementation B . The dynamic properties f s, ζ, and K s apply to the pipe with strakes for implementation A, but without the strakes for Implementation B. All results presented in this paper are based on Implementation B. As in Vivana [ 4], it is assumed that , at any cross section x, for a given harmonic transverse displacement Y(x) cos(ωt), the hydrodynamic force per unit length exerted by the water on the pipe is also harmonic and given by F m cos(ωt) – Fe sin(ωt), where Fm = ω2 Y Ca mw is the component of hy drodynamic force in -phase with the motion, and Fe = C e ½ ρ U2 D is the component 90° out -of-phase with the motion . The latter provides excitation when positive or damping when negative . Here C e=C e(f*,Y*) and C a=C a(f*) are the dimensionless excitation and added mass coefficients, respectively. Neglecting Reynolds number effects, it f ollows from dimensional analysis that these coefficients depend only on the dimensionless frequency f*=fD/U and amplitude Y*=Y/D. A further approximation is to neglect the effect of the amplitude Y* on the added mass coefficient C a. Both these approximat ions made in Vivana [ 4] as well as here. To extend the validity of the expression F e = C e ½ ρ U2 D to negative values of Y, the excitation coefficient Ce=C e(f*,Y*) must be an antisymmetric function of the amplitude Y*, since a 1 80° phase change in the motion must bring with is a corresponding 180 ° phase change in the hydrodynamic force. Such negative values of Y arise where the mode ϕ=ϕ(x) is negative at certain locations x. Henceforth C e(f*,Y*) is only defined for positive values of Y* with the understanding that for negative values C e = -Ce(f*, |Y*|). Following the Rayleigh -Ritz approach, the pipe motion is assumed to be of the form: y(x,t) = ϕ(x) A cos(ωt) (1) The frequency of such motion is obtained from the Rayleigh quotien t, considering only the forces in -phase with the motion. This yields (ω/ω s)2 = (f/f s)2 = (f*/fs*)2 = ∫mode (mratio + C as) ϕ2 dx / {∫mode (mratio + C a) ϕ2 dx} (2) Here the subscript “s” indicates still -water, as opposed to flowing conditions. Thus, ω=2πf is the angular frequency under flowing conditions, whereas ωs=2πf s is the corresponding value under still -water conditions. (See also the Nomenclature section, especially for differences in symbol definitions for Implementations A and B.) Since C a=C a(f*), Eq. (2) can be solved for the dimensionless VIV oscillation frequency f* before the amplitude A* is known. For the 90° out -of-phase components of force , considering the structural and soil damping (captured by the Scruton number K s)8 as well as the hydrod ynamic excitation or damping, the energy balance condition c an be written in dimensionless form as Ein* = A* (1/L) ∫ span Ce(f*, A* ϕ ) ϕ dx - 4 π f* fs* Ks A*2 ϕrms2 (3) 8 To derive the energy dissipated by str uctural and soil damping, it has been assumed that the system in still water has classical modes. I.e. that the undamped mode shapes are orthogonal with respect to the damping matrix, as well as the stiffness and mass matrices. If that is not the case, t he result still applies as a Rayleigh - Ritz approximation. This approximation is tacitly made for such cases in the guidance provided in Section 4.6 of DNVGL -RP-F114 [ 16] to estimate damping. 7 where Ein* = Ein/(2m w L U2) is the dimensionless energy input required to achieve energy balance , and ϕ rms is the root mean square modal displacement, as defined in the Nomenclature . Thus , the amplitude of VIV is calculated as the value of A*=A/D for which E in* from Eq. (3) is zero. Alternatively, with E in=0, solving Eq. (3) for K s, one obtains Ks = (1/L) ∫ span Ce(f*, A* ϕ ) ϕ dx / {4 π f* fs* A* ϕrms2} (4) for the damping K s for which the nor malized vibration amplitude is A*. If the assumption C a=C a(f*) were relaxed, allowing C a=C a(f*,Y*), the same equations still apply, but one would need to solve Eq. 2 and Eq. 4 simultaneously for f* and A*, instead of solving first Eq. 2 for f* and then Eq. 4 for A*. All results presented in this paper have been obtained for the case when the amplitude Y* does not affect the flowing added mass coefficient C a. 4 Applicati on to In -Line VIV Application of the Sivana approach to pure in -line VIV is given in [ 8]. Therefore only the key points are summarized here. 1) The excitation and added mass functions for bare pipe are from forced vibration tests in [ 7], as smoothed and inter polated in [ 4]. For straked pipe , they are available from recent forced vibration tests reported in [8]. 2) For the pipeline spans considered, undergoing first mode respo nse, good agreement between the Vivana and Sivana predictions of in -line VIV is obtained. This confirms that propagating wave effects that are included in Vivana but not in Sivana are not important for the cases considered , which involv e first mode respon se. 3) Good agreement is also obtained for in-line response of bare pipe between the Sivana and the F105 response function of [1], as shown in Figure 1. 4) The Sivana results are not sensitive to the mass rati o m ratio or the mode shape ϕ, provided that the Scruton Damping parameter K s does not change. (Changing m ratio at constant damping ratio ζ means that Ks also changes , and that does change VIV response amplitude. ) 5) Even without soil and/or structural dampin g, 14% strake coverage centered at midspan is sufficient to suppress pure in -line VIV for the example considered in [ 8] with a mass ratio of m ratio=1.23 . For many pipeline spans, in -line fatigue d amage dominates in assessments performed according to [ 1]. This provided the motivation in [8] to first look at in -line VIV. However, in view of the high eff ectivenes s of even a small amount of strakes to suppress in -line VIV, longer spans become permissible, and it is necessary to extend the methodology also to cross -flow VIV, despite the challenge that arises because cross -flow VIV mostly also involves a com ponent of cross -flow-induced in -line motion. 8 Figure 1 Comparison of F105 in -line VIV response function [1] with that predicted by Sivana using excitation and added mass functions from Aronsen [7,8], for bare pipe with mass ratio of m ratio = 1.23, and the mode shape from the analytical solution in [17,18] for a single spa n with an axial force of N/N c = -0.20 and a normalized span length of L(k/EI)1/4 = 19.13 . 5 Cross -Flow Excitation Function by Inversion of F105 Response Function Nothing new is needed to apply Sivana to calculate cross -flow VIV amplit udes of partially strake d pipes: For straked pipe , the excitation (or “lift”) and added mass functions are available from [ 19], and for bare pipe, they are available from forced vibration tests by Gopalkrishnan [ 6], with th e Vivana implementation thereof described in [ 4]. The problem is tha t, for bare pipe, the above -referenced excitation and added mass functions do not give a good predictions of VIV response amplitude compared to the F105 respons e function from [1] that comes from VIV tests on flexible pipes , as shown in Figure 2. Possible explanation s include: 1) IL-CF interaction is not considered, though CF mostly also involves a component of IL motion that is not present in the forced vibration tests from which the e xcitation function is derived. Such interactions are clearly evident from the efforts in [ 9] and [ 25] to determine hydrodyna mic local forces and thus the excitation coefficients from the response of a well -instrumented flexible pipe. Although force d vibration tests involving in -line as well as cross -flow motion are possible, a nd have been performed by Aronsen [ 7], the results of such tests depend on a larger number of input parameters, not just on f* and Y*. This makes it much more demanding to cover all relevant combinations of the input parameters in an experimental program of bi -directional force d vibration tests. 2) Assuming that the added mass coefficient depends only on frequency f* and not on amplitude Y*, may be a worse approximation for cross -flow VIV than for in -line. The Vivana default added mass function corresponds to Gopa lkrishnan ’s resul ts at an amplitude of Y*=0.5. Picking a different amplitude or including the effect of amplitude in the added mass f unction would lead to different results. Amplitudes below Y* = 0.15 are not captured in Gopalkrishnan’s data, and these could involve very high added mass, especially near the Strouhal frequency. 3) In addition to interaction between vertical and horizontal motions, interactions with higher harmonics and higher modes could be significant [26]. 4) VIV motion is not harm onic as assumed. Even a narrow -band motion can be sign ificantly different, exhibiting beating for instance. Forced vibration tests for beating motions in [ 6] show that this has a significant effect on hydrodynamic forces. Thi s is more so for cross -flow, where both the Strouhal frequency and the pipe natural frequency play a role. 9 For these reason s, the excitation function from forced vibration tests in [ 6] have been adjusted in [ 4] to get the Vivana default functions used for Figure 2 [27], [24]. The recent development in the excitation functions and the response frequency model focus on m ulti-frequency/mode riser response s [28]. Modifications to the existing hydrodynamic load models, including both the excitation and the added mass functions, are need ed to improve the VIV response prediction for free span pipelines. Here a different calibration of the excitation function is developed, to match pipeline response as predicted by the F105 response function in [1]. This is done by mathematically inverting the Sivana analysis to go from respo nse function to excitation function, rather than vice -versa. I.e., the excitation f unction is constructed to make the differences between grey and black lines in Figure 2 vanish. The inversion process involves two steps: First t he added mass functio n C a=C a(f*) is constructed considering the response frequencies from VIV tests on flexible pipes as well as the forced vibration test results from [ 4,6] (Section 5.1). Then, the excitation function C e = C e(f*,Y*) is construct ed to match the F105 cross -flow response function from [ 1] (Section 5.2). There is a complication: A ccording to Sivana , the VIV amplitude depends not only on the reduced velocity VR = 1/f s* and the Scruton Damping parameter K s, but also on the mass ratio m ratio and the mode shape ϕ=ϕ(x), yet these effects are not reflected in the F105 response function . To address this, the inversion is performed for the calibration -basis parameters shown in Table 1, which are chosen to be representativ e of the conditions for which the F105 response function is calibrated. The same excitation function is then also used for other values of the mass ratio and mode shape. This inversion appro ach is applied only for the bare pipe. For any straked segment s, the excitation and added mass functions available in [19] are used. Figure 2 Cross -flow response function A*=A*(VR, K s) from Sivana using Vivana default excitation and added mass functions compared with F105 cross -flow response function . (This Sivana response function is calculated for the mass ratio and mode shape of Table 1. For F105, the same response function applies for a range of mass ratios m ratio and mode shapes ϕ.) 10 Description Symbol Value Mass Ratio mratio 1.32 Mode Shape ϕ = ϕ(x) Determined from analytical solution for a single span from [ 17,18] with N/(k EI)1/2 = -0.01508 and L (k/EI)1/4 = 27.47. Plotted in Figure 8. Table 1 Cross -Flow calibration -basis parameters for which the excitation function is calibrated to return the F105 response function 5.1 Bare Pipe Cross -Flow Added Mass Function For bare pipe, bot h C a and C as are constant with respect to location x along the span, so that Eq. 2 reduces to (f*/fs*)2 = (m ratio + C as) / (m ratio + C a) (5) Measuring the frequencies from tests on flexible pipes in [ 20] and calculating the cor responding C a using Eq. 5, leads to Figure 3. Also shown i s the added mass function recommended in [ 1] to count cycles of fatigue damage. This is expressed as a function of reduced velocity V R = 1/f s*. This will be referred to as the “F105 added mass function”. Using Eq. 5 to transform the F105 added mass function of Figure 3 to the form Ca=C a(f*) leads to Figure 4. For comparison, the Vivana default added mass function from [ 4] is also included . Observations and r emarks about this plot are: 1) The F105 added mass function depends on the mass ratio. It should not. At least, the results of a forced vibration test shou ld not d epend on the mass ratio. For a given imposed motion of the cylinder, the mass of the pipe will influence the total force that needs to be applied to the cylinder to impose the motion, but it does not influence the hydrodynamic force . 2) The F105 relation bet ween f* and C a exhibits looping behavior. I.e. , between points D and E Figure 4, Ca=C a(f*) is multiple -valued. I.e. there is a relationship between C a and f*, but C a is not a function of f*. A forced vibration test at a given (f*,Y*), should yield just one C a value. However, a Sivana type model with Ca=C a(f*,Y*) could explain why the apparent added mass functions obtained from tests on a flexible pipe , or a flexibly supported rigid cylinder exhibits looping behavior. 3) In the mapp ing from Figure 3 to Figure 4, labelled points A to F serves to indicate which points map to where. At point D the dimens ionless frequency f* is a minimum. This occurs at a reduced velocity of about 6. At higher current velocities (from D towards E) , the increase in oscillation frequency is less than proportional to the increase in current velocity , according to the F105 a dded mass function. The above observations do call for further investigation and sug gest that enhanced versions of Vivana and/or Sivana that allow C a = C a(f*,Y*) instead of being limited to C a = C a(f*) may be appropriate. However, the objective here is no t to revisit the effort that went into the development of the F105 response and adde d mass functions from VIV test data, but rather to build on those results, in order to extend the F105 guidan ce, at least in the interim, for partially straked pipes , sticking with the Vivana/Sivana assumption that the added mass is a function of dimension less frequency only. For that purpose, the added mass function shown by the heavy black line in Figure 4 is used in what follows. It agrees with the F105 function for the calibration -basis mass ratio up to normalized velocities ( 1/f* ) of around 6, and then transitions steeply to the Vivana default function. 11 Figure 3 F105 added mass function from [ 1] com pared with added mass calculated from response frequencies in VIV tests on flexible pipes in [ 20], with a mass ratio of m ratio = 1.3 for test series “75xx” and m ratio = 1.3 for test series “42xx”. Figure 4 Cross -flow added mass function from F105 [ 1] (grey lines) compared with Vivana default [ 4] and chosen added mass function used here 12 5.2 Bare Pipe Cross -Flow Excitation Function The section presents the inversion process to get the cross -flow excitation function C e = C e(f*,Y*) for which the F105 response function is matched when using Sivana wi th the calibration basis mass ratio m ratio and mode shape ϕ=ϕ(x) of Table 1. All calculations described in this section are done for the calibration -basis parameters of Table 1, even if the resulting excitation function is to be applied for a different mass ratio and/or mode shape. The F105 cross -flow response function can be written as A*(VR, K s) = A u*(VR) R k(Ks) (6) where A u*(VR)=A*(VR,0) is a piecewise linear function defined in F105 and shown in “Figure 4 -2” of [1], that gives the undamped (K s=0) response amplitude, and RK(Ks) = 1 – 0.15 K s for K s ∈ [0,4] = 3.2/K s1.5 for K s ∈ [4,∞) (7a,b) RK(.) is a monotonic function . The corresponding inv erse function K s(.) given by Ks(r) = (3.2/r)2/3 for r ∈ (0,0.4] = (1 – r)/0.15 for r ∈ [0.4,1] (8a,b) The Sivana -based inversion to get the excita tion function from the response function is particularly simple for a 2D test. This refers to a test involving a spring -supported rigid cylinder, where the mode shape is ϕ(x)=1 ∀ x. For this case, Eq. 4 yield s directly Ce,2D(f*, A*) = 4 π f* fs* Au* Cen2D(A*/Au*) (9) where Cen2D(r) = r K s(r) is referred to as the normalized excitation function , plotted in Figure 5, and given by Cen2D(r) = (10.24 r)1/3 for r ∈ (0,0.4] Cen2D(r) = r(1 – r)/0.15 for r ∈ [0.4,1] (10) To evaluate C e,2D(f*,A*), write Eq. 5 a s, fs* = f* [(m ratio + C a(f*))/(m ratio + C as)]1/2 (11) and use it to calculate f s* and V R=1/f s*, then obtain the corresponding undamped response amplitude Au*=A u*(VR) from the F105 undamped cross -flow response function , then C e,2D(f*,A*) from Eq. 9. More generally, considering the mode shape ϕ=ϕ(x) for a flexible pipe, inversion requires solving an integral equation that arises from the energy balance condition, Eq. 3 . In what follows only the results are given. They can readily be verified, by subs tituting back into Eq. 3 to check that it is satisfied. (That is easier than finding the solution when one does not know it. ) The result is Ce(f*, Y*) = 4 π f* fs* Au* Cen(Y*/Au*) (12) where the function C en(.) is the solution to the following integral e quation (1/(L ϕ rms2)) ∫span Cen(rϕ) ϕ dx = C en2D(r) ∀ r ∈ [0,1] (13) To evaluate C e(f*, Y*), proceed in the same way as for C e2D(f*,A*): use Eq. 11 to obtain f s* and VR=1/f s*, then Au*=A u*(VR) from F105 [1], and finally C e(f*, Y*) from Eq. 12. The integral equation , Eq. 13, needs to be solved only once for the calibration -basis mode shape. For r ≤ 0.4, the solution is Cen(r) = C en2D(r)/Φ 1 for r ∈ [0,0.4] (14) where Φ1 = [∫ span |ϕ|4/3 dx]/[ L ϕ rms2] = 1.1835 for the calibration -basis mode shape (15a,b) For r > 0.4, the integral equation is solved numerically, by a collocation method as follows: 1) A parametrized trail function Cen(r) is defined by Eq. 14 for r ≤ 0.4, and for r > 0.4, C en(r) it is assume d to be piecewise linear and continuous function of r between evenly -spaced points at r = ri = 0.4 + 0.6 i/N r for i = 0,1,2,...,N r. Thus the problem of determining a continuous function C en(r) is reduced to determining a finite number of unknowns C en,i = C en(ri) for i=1,2,...,N r. 13 2) The values o f C en,i for i=1,2,...,N r are obtained by enforcing Eq. 13 at r = ri for the same range of i values. This can be done sequentially for one unknown C en,i at the time, as follows: a) Suppose the solution is available up to C en,i-1. (Initially, for i=1, this is the case from Eq. 14.) b) Substitute r=r i into Eq. 13. Since the integral does not extend beyond r i, this equation involves a single unknown, C en,i. c) Take the unknown C en,i outside the integral . Calculate t he rema ining 2 int egrals numerically . Solve the remaining equation for C en,i. d) Repeat this for i=1,2,...,N r to complete the solution for 0.4 < r ≤ 1. To check for convergence, this has been done for N r = 12, 24, and 48 . The results could not be distinguished on a plot, so finally only the N r=48 plot is included in Figure 5. This sugge sts the solution is accurate, even if it does not prove convergence of the numerical solution scheme . (Indeed for very large N r the scheme was found to diverge, probably because of the approximate numerical evaluation of the integrals . This remained limited even as the mesh for the trial functions for C en(r) was refined. Thus, for the numeri cal integration, the solution can become non -unique above a limiting N r value. ) The shape of C en(r) in Figure 5 is similar to that for the 2D excitation function C en2D(r), but there are quantitative differences: the peak value is down from C en2D @max = 5/3 = 1.667 to C en@max = 1.366. The location of the peak decreases from r=0.5 for the 2D case to approximately9 r=0.434. The value of r where C en=0 is down from 1 for the 2D case to 0.8354. This means that , for the calibration -basis mode shape, the undamped amplitude is 1.197 times that of a 2D test . For the calibration basis mode , there is never a need to evaluate C en(.) for r > 1. Adding strak es only reduces the r value. However, for a different mode shape, Sivana could encounter r > 1 values. Even for the calibration basis mode r > 1 could occur during the Sivana iterative solution process while solving for the unknown vibration amplitude. For such cases , linear extrapolation from the last two points ( at r=1, and r=1 -Δr) is used . Contour plots of the resulting excitation function are shown in Figure 6. It is clear that the steep transition zone in the added mass function introduced in Figure 4 has a st rong impact on the excitation function that goes with it to match the F105 response function. (The transition zone is indic ated in Figure 6 by dashed lines , and covers the range 0.16 ≤ f* ≤ 0.165 in Figure 6 which corresponds to 6.06 ≤ 1/f* ≤ 6.25 in Figure 4.) 9 This approximate location of the peak is obtained by fitting a parabola to the numerically calculated points at r = 0.425, 0.4375, and 0.45. 14 Figure 5 Normalized cross -flow excitation function s for which Sivana matches the F105 response function for the calibration -basis mass ratio and mo de shape of Table 1 (continuous line). The dashed line gives the same for the mode shape of a 2D test from Eq. 10 instead of the calibration basis mode shape. 15 Figure 6 Contour plots of excitation funct ion C e=C e(f*,Y*) for Cross -flow VIV constructed to match the F105 response function for the calibration basis mass ratio and mode shape, and the added mass function of Figure 4. (Lower shows the tra nsition zone of the added mass f unction at a magnified scale.) 6 Applications to Cross -Flow VIV 6.1 Bare Pipe By construction, applying Sivana with the calibration -basis mass ratio and mode shape of Table 1, the F105 response function is reproduced exactly. However, f or Sivana the mass ratio does have some effect on the response as shown in Figure 7. The effect is only s ignificant at high reduced velocities, beyond the response function plateau. Although this mass ratio effect is not included in [1], experiments covering a wide rang e of mass ratios, as in [ 21], do show a significant effect of the mass ratio on the bandwidth of the response function. 16 Figure 7 Sensitivity to the cross -flow VIV amplitude from Sivana to the mass ratio for values of the Scruton mass damping parameter of K S=0 (in black) and K S=2 (in grey). Mass ratios are 1.6, 1.32, and 1, decreasing in the direction of the arrows. For the calibrat ion-basis mass ratio of 1.32 the result agrees with the F105 response function , as it should . To also examine the sensitivity of the Sivana results to the mode shape, the response functions are calculated for the mode shapes shown in Figure 8. It was found that the effect of the mode shape does not depend on reduced velocity. I.e., the mode -shape effect is accounted for by a factor on amplitude that does not depend on t he reduced velocity. Values of this factor are shown in Table 2 for two values of the Scruton mass damping parameter K S. This shows that the mode shape effect is small in the range between the simply -supported and fixed -fixed mode , which should encompass most single -span cases in practice. Th e maximum amplitude is between 1.18 and 1.20 times that for a 2D test .10 10 This refers t o a 2D test simulated with Sivana using the excitation function derived from the calibration basis mode shape of Table 1 and Figure 8. 17 Figure 8 Mode shapes co nsidered for mode shape sensitivity analysis, top to bottom: (1) a 2D test in which the displacement is constant along the pipe, (2) simply -supported conditions for which the mode shape is sinusoidal, (3) calibration -basis mode shape of Table 1, and (4) the mode shape for a pipe that is fully fixed at both ends, with zero effective axial force KS = 0 KS = 2 Simply -Supported 1.178 1.179 Calibration -Basis 1.197 1.200 Fixed -Fixed 1.195 1.199 Table 2 Factors to be applied to the amplitude from a 2D test10 to get the maximum amplitude for each o f the mode shapes of Figure 8 for values of the Scruton damping pa rameter K S shown. These factors do not depend on the reduced velocity. 6.2 Partially Straked Pipe Applying Sivana for a partially straked span with the strake coverage centred at midspan gives the results in Figure 9 for various levels of strake coverage. Structural and soil damping is assumed to be zero (K S=0), and the mass ratio and mode shape are the calibration -basis ones of Table 1. The reduced velocity shown is that for the bare pipe. With the strakes, the natural f requency in still water is a bit lower so the reduced velocity is a bit higher. This contributes to the apparent shift in the reduced velocity at whic h the largest VIV amplitudes occur in Figure 9. Figure 10 shows similar results for K S=2 as Figure 9 shows for K S = 0. There is a general lowering of the response due to the incre ased non -hydrodynamic damping. It is clear that the benefit of strakes for cross -flow is less than for the in-line case , especially on the left side of the response function . E.g. 14% strake coverage fully eliminates in -line VIV for the example in [8], but for cross -flow, the response is only reduced by about 40% at a reduced velocity of V R=4 increasing to about 50% at VR=6. That strakes produce more reduction in cross -flow response in the left s ide of the response function than on the right side is in part because the strakes affect the natural frequency, and the reduced velocity shown in the plot is based on the natural frequency of the bare pipe. 18 Figure 9 Effect of va rious levels of strake coverage on cross -flow VIV for the undamped case with the calibration - basis mass ratio and mode shape of Table 1. Figure 10 As Figure 9, but for K s = 2. 7 Closing Remarks This paper does not present a new theory, experimental results, nor a systematic evaluation of existing theories and experimental results. Instead it offers a practical, quantitative way to assess the effectiveness of strakes to suppress VIV in pipeline spans, build ing on the well-established F105 response function from [ 1], which is based on a broad review of experimental results that is periodically updated as new results bec ome available . It is intended for curr ent velocities up to the peak response of the first mode. 19 The two most common ways to estimate VIV amplitudes are the response function approach typically used for pipelines and recommended in [ 1], and the excitation function a pproach such as Vivana [4], Sivana , and SHEAR7 [ 2], more typically used for risers. Both have advantages and disadvantages. The advantage of the response function approach, is that it is more direct, not relying o n the assumption of harmonic and uni -directional motion, and there is a larger set of experimental data from which the response function can be developed , which has been exploited in [ 1]. The advantage of the excit ation function approach is that it can account for varying conditions along the length of the span based on theory, without requiring a set of VIV tests for every such condition. Also the mass ratio and mode shape effects are accounted for. The inversion technique presented allows the advantages of both approaches to be exploited, by enabling the calculation of an effective excitation function from VIV tests of flexible pipes or flexibly -supported rigid cylinders at different damping ratios to be converte d into an excitation function, by solving an integral equation for which an explicit numerical solution technique is given. Sivana is merely a simplified version of Vivana [4] without propagating wave effects . It is also simila r to a single -mode analysis in SHEAR7 [ 2], except that in Sivana the difference between flowing and still -water added mass is accounted for, whereas that difference is not addressed in [ 2]. Neverthel ess, the Sivana energy balance condition is given in a simple, dimensionless form not found in the referenced works . The inversion technique has been applied to the F105 cross -flow response function from [1] to get an effective bare pipe excitation function . This together with the excitation function for strakes from [ 19] yields a simple metho d to estimate the VIV response for partially strake -covered pipes that is consistent with the current practice for bare pipe , with illustrative results shown in Figure 9. Similar results for the pure in-line case have been presented in [8] without inversion . Instead, forced vib ration tests from [ 7] and [ 8], are used directly to define the excitation and added mass functions for both bare and straked pipe, respectively. In contrast to the cro ss-flow case, for in -line, the excitation function from the forced vibration tests already provides a good fit to flexible pipe response, as shown in [8] and in Figure 1. The results indicate that a small amount of strake coverage is already sufficient to suppress in -line VIV, but more strakes are needed if the current velocity goes above the threshold for cro ss-flow. Although inversion builds upon well -established F105 gui dance in [ 1], care must be taken, as approximation s therein that are good for one purpose may not be good for another . There remains a need for further work, including the following: 1) The F105 response function could be improved , or at least its relation to experimental data better documented in the public domain, especially at high damping values. In that process, the inversion technique presented here can help in making the best of available experimental results, e.g. to compe nsate for mass ratio or mode shape effects. 2) There are two issues associated with the added mass function in the form C a=C a(f*): a) Firstly, neglecting the effect of the vibration amplitude Y* becomes an important drawback, e.g. at low reduced velocities when the pipe response has low amplitude and may occur at a frequency closer to the Strouhal frequency than the natural frequency of the span, implying a very high added mass that is not captured assuming added mass is a function of normalized frequency f* only. b) Even for flexibly supported single degree of freedom rigid cylinder tests such as [ 21], the response is not always harmonic, as assumed. There may be components of the response at the Strouhal frequency (implying a considerab le change in added mass) as well as at the resonant frequency (for which the added mass remains close to the still -water value). 3) There remains a need for validation by tests on flexible, partially straked pipes exhibiting typical low -mode pipeline VIV resp onse. Although t here are several VIV model test campaigns whe re the pipe model is partially or fully straked [ 29-31], these exhibit high -mode mode response more typical of risers. The only such test s with low -mode response the authors are aware of are reported in [32] but they involve a different strake profile for which excitation/damping and added mass functions from forced vibration tests are not available . As a result, a direct comparison between the prediction and the model test for low mode response of a partially strake -cover ed span is not feasible with currently available data . 4) Finally, perfect strakes are assumed there. In practice these may be affected by marine growth, damaged , or deformed [34-36]. It remains to incorporate the effect of this on the excitation and added mass functions, so that it is accounted for in the assessment. 20 A number of the above issues could be addressed by going back to the experimental data involved in the calibration of the F105 response function, and calibrating the bare pipe added mass and e xcitation functions directly from the data instead of indirectly from the F105 response function. The inversion technique presented in this paper facilitates that process. 8 Acknowledgements This effort was partially sponsored by Shell Philippines Explorati on, B.V. (“SPEX”). Permission to publish it is gratefully acknowledged. Together with [ 8,18,37-39] this paper arose from the original initiati ve of Jeff rey Rimmer at SPEX, to address the integri ty threats to a major deepwater pipeline on an active seabed. Knut Vedelt of DNVGL reviewed the manuscript and provided valuable comments. DNVGL also incorporated Sivana in a version of their “FatFree” software produced for SPEX. However, only the author s, and not the organizations for which they work, or individuals mentioned in these acknowledgements are responsible for the content of this paper. 9