VISCOUS FLOW COMPUTATIONS ON SMOOTH CYLINDERS A DETAILED NUMERICAL STUDY WITH VALIDATION. 1 2 ρv 2 C D. 2 ρv in

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1 Proceedings of Proceedings of OMAE7 th International Conference on Offshore Mechanics and Arctic Engineering June -5, 7, San Diego, California, USA OMAE7-975 VISCOUS FLOW COMPUTATIONS ON SMOOTH CYLINDERS A DETAILED NUMERICAL STUDY WITH VALIDATION Guilherme Vaz MARIN R&D Department, CFD Projects 7 AA Wageningen The Netherlands G.Vaz@Marin.nl Christophe Mabilat ATKINS Woodcote Grove, Ashley Road Epsom, Surrey, KT8 5BW United Kingdom Christophe.Mabilat@AtkinsGlobal.com Remmelt van der Wal MARIN Offshore Department 7 AA Wageningen The Netherlands R.vdWal@Marin.nl Paul Gallagher ATKINS Woodcote Grove, Ashley Road Epsom, Surrey, KT8 5BW United Kingdom Paul.Gallagher@AtkinsGlobal.com ABSTRACT The objective of this paper is to investigate several numerical and modelling features that the CFD community is currently using to compute the flow around a fixed smooth circular cylinder. Two high Reynolds numbers, 9 and 5 5, are chosen which are in the so called drag-crisis region. Using a viscous flow solver, these features are assessed in terms of quality by comparing the numerical results with experimental data. The study involves grid sensitivity, time step sensitivity, the use of different turbulence models, three-dimensional effects, and a RANS/DES (Reynolds Averaged Navier Stokes, Detached Eddy Simulation) comparison. The resulting drag forces and Strouhal numbers are compared with experimental data of different sources. Major flow features such as velocity and vorticity fields are presented. One of the main conclusions of the present study is that all models predict forces which are far from the experimental values, particularly for the higher Reynolds numbers in the drag-crisis region. Three-dimensional and unsteadiness effects are present, but are only fully captured by sophisticated turbulence models or by DES. DES seems to be the key to better solve the flow problem and obtain better agreement with experimental data. However, its considerable computational demands still do not allow to use it for engineering design purposes. NOMENCLATURE V in Reference inflow velocity [m/s] V = (V x,v y,v z ) Velocity field [m/s] ω = (ω x,ω y,ω z ) Vorticity field [s ] p Pressure [N/m ] F = (F x,f y,f z ) Forces [N] ρ Water density [kg/m 3 ] µ Water viscosity [N.s/m ] d Cylinder diameter [m] l Cylinder length [m] A = d l Projected area [m ] = ρv ind µ Reynolds number [ ] C D = F x Drag coefficient [ ] ρv in A C D Time-averaged drag coefficient [ ] C L = F y Lift coefficient [ ] ρv C p = p ina Pressure coefficient [ ] ρv in f s Vortex-shedding frequency [s ] St = f sd V in Strouhal number [ ] t Time [s] t Time step [s] INTRODUCTION One of the major challenges in the offshore industry is to assess the loads and motions of risers in currents. The fatigue life of these structures is often dominated by Vortex Induced Vibrations (VIV) and Vortex Induced Motions (VIM). Experimental model tests are carried out extensively to better understand these flow phenomena. As a result of the availability and user-friendliness of commercial CFD codes in combination with high computational power the number of numerical calculations is also rapidly Copyright c 7 by ASME

2 increasing, see for instance [ 3]. The flow around a bare cylinder is the starting point to understand the VIV/VIM phenomenon and it is perhaps one of the most challenging problems of fluid dynamics due to: the large influence of the Reynolds number; large sensitiveness to surface roughness; different types of shear layers; and different time and spatial scales of vorticity and turbulence. This poses high demands on the models to be used, on the choice of numerical settings, and on the computational resources. The flow around the bare smooth cylinder is dominated by several instabilities associated with the cylinder wake, separated shear layer and boundary layer. These instabilities are explained in detail by Williamson []. A detailed analysis of their formation mechanisms is highly complex and has been studied in the past by means of experiments [5, ], and is outside the scope of the present work. However, the physical understanding of these instabilities, their influence on the flow regime and properties, and relation with the Reynolds number, is essential for any reasonable numerical computation. The flow phenomena dependence on the diameter-based Reynolds number can be summarized as follows [,, 7]: - Up to 7, the flow is steady and two-dimensional with two symmetric vortices on each side of the wake center line. The first wake instability, a Hopf bifurcation, occurs at 7. - For > 7, the flow, still laminar, becomes unsteady and asymmetric, with Von Karman vortex shedding appearing for slightly larger. - At 9 up to, three-dimensional instabilities, such as formation of vortex loops, deformation of primary vortices, streamwise and spanwise vortices, appear in the wake. Up to these three-dimensional instabilities increase and finer scales arise. - Between and 5 the wake flow transits to turbulent, while the flow on the cylinder is still laminar. Beyond a certain critical within this range, the shear layer separating from the upper and lower surface of the cylinder becomes unstable via the Kelvin-Helmholtz mode of instability. This instability transitional point moves upstream with increasing. - At 5, this instability reaches the body and the boundary layer on the surface of the cylinder transits from laminar to turbulent (critical regime). This transition leads to a delay of the separation of the flow on the cylinder, causing a substantial reduction of the drag force. This region, also called drag-crisis, reaches up to around Between 5 5 up to (super-critical regime) the flow is symmetric with two separationreattachment bubbles and a thin wake. The high Reynolds stresses of the boundary layer following the separation bubble allows the boundary layer to survive greater adverse pressure gradients than in the following regime (postcritical) and the drag forces remain fairly constant. - Above > a post-critical regime appears where transition to turbulence occurs before separation. With increasing the transition point moves upstream and most of the cylinder boundary layer is turbulent. Separation occurs also further upstream, yielding higher drag values and a wider downstream wake which is fully turbulent. Average C D Figure. D Steady Laminar D Unsteady Laminar Wieselsberger Experiments (from Schlichting) 3D Unsteady Laminar 3D Unsteady Laminar + Turbulent Wake 3D Unsteady Laminar + Turbulent + Turbulent Wake 3D Unsteady Turbulent + Turbulent Wake from Schlichting [5]. C D variation with. Experiments by Wieselsberger taken This global picture of the flow is important for the application of the numerical models: steady or unsteady, twodimensional or three-dimensional, which turbulence models, with or without transition models. Also the existence of different spatial and temporal scales requires a choice of which simulation technique to be applied, and a detailed study on the spatial and temporal discretization errors. Bearing in mind the selected numbers for the present work, 9 and 5 5, it could be expected beforehand that three-dimensional unsteady turbulent viscous flow computations had to be done, and, due to the complex and various scales of turbulence, a D RANS, Reynolds Averaged Navier- Stokes, approach together with simple eddy-viscosity based turbulence models would not be sufficient to capture quantitatively all the previously referred instabilities. Therefore both D and 3D RANS computations have been done, not only with eddy-viscosity based turbulence models but with more complex Reynolds-Stress transport models,, and also using a transition model. 3D DES, Detached-Eddy Simulations, have been done in order to check the effect of larger scale turbulence on the solution. This paper tries to show that even with these more complex models accurate results for the forces on the bare smooth cylinder for the drag-crisis region are extremely difficult to obtain, even if the flow physics is better captured. It should be emphasized that not only the magnitude of the calculated forces is of interest but also the overall trend with. Additionally, it is shown that the computational costs increase rapidly with increasing complexity of the models used, leading to highly unfavourable quality vs. costs ratios for the 3D complex models. THEORETICAL FORMULATION Problem Formulation Consider the one-phase viscous turbulent flow around a fixed circular cylinder with diameter D and length L fully im- Copyright c 7 by ASME

3 mersed in water (Newtonian and incompressible fluid) and where free-surface (Froude) effects are negligible. Body forces are not taken into account. An absolute inertial space-fixed reference frame (x,y,z) is adopted and in this system the fluid velocity V components are denoted by V i with i =,,3 or x,y,z. The fluid static pressure, density and dynamic viscosity are defined respectively by p,ρ and µ. A constant velocity profile V in aligned with the x direction is the onset inflow. The equations ruling the flow are the incompressible unsteady Navier-Stokes equations: V =, V i t + (V iv) = ρ ( p τ i), where τ τ i j = µ( Vi x j + V j is the viscous stress tensor, being i = j =,,3 running indices. With the force vector F on the cylinder walls, the drag and the lift coefficients C D,C L are calculated. Important dimensionless numbers that rule the flow are the Reynolds number based on the diameter and the Strouhal number S t, f s being the vortex-shedding frequency calculated from the lift force history with time. x i ) Modelling Techniques Direct Numerical Simulation, DNS, of turbulence is the most exact approach to the solution of turbulent flows. In DNS the Navier-Stokes equations, Eq.(), are discretized directly and solved numerically. If the mesh is fine enough to resolve the smallest scales of motion, one obtains an accurate threedimensional, time-dependent solution of the governing equations completely free of modelling assumptions. DNS is a very useful tool for the study of transitional and turbulent flow physics, but has some limitations. To resolve all flow scales, one requires a number of grid points proportional to Re 9/, and the cost of the computation scales like Re 3. For this reason, DNS is still largely limited to simple geometries at low Reynolds numbers, and its application to engineering-type problems within the next decade appears unlikely [8]. When only some quantitative properties of the turbulent flow are of interest, such as average forces, a statistical approach is followed by averaging in time the flow equations and obtaining the RANS, Reynolds Averaged Navier-Stokes, equations. This averages out all intrinsic unsteadiness of the flow, which is then attributed to the turbulence. On averaging, the non-linearity of the Navier-Stokes equations gives rise to extra terms that have to be modeled by closure models, so-called turbulence models. This approach, being the mostly widely used in CFD, suffers from one principal shortcoming: it cannot capture all range of turbulent scales. While the small scales tend to depend mainly on viscosity, which effect can be modeled by turbulence models and RANS, the large scales do not. They depend strongly on the boundary conditions, and are difficult, if not impossible, to be modeled in a general sense for all types of flows with RANS [9]. In a Large-Eddy Simulation, LES, the contribution of the large energy carrying scales and structures is computed exactly and only the effect of the small scales is modeled. The Navier- Stokes equations are filtered from the small scales. Since the small scales tend to be more homogenous and universal, and less () affected by the boundary conditions than the large ones, there is hope that their modelling (in this context called subgrid-scale models) can be simpler and require fewer adjustments when applied to different flows than the turbulence models for the RANS approach. LES provides three-dimensional time dependent solutions of the Navier-Stokes equations, still requires fairly fine meshes, but can be used at higher Re than DNS [8]. However, LES of boundary layer type of flows have shown poor results when compared with RANS together with highly calibrated turbulence models [8]. The Detached-Eddy Simulation methodology, DES, is a hybrid method in which the attached boundary layers are solved using a RANS approach, while the wake region is solved using a LES approach. This allows a smooth RANS-like flow solution in flow-attached regions together with accurate capturing of detached eddies dynamics in separated-flow regions []. URANS Model In a statistically steady flow, every variable can be written as the sum of a time-average component plus a fluctuation. If the flow is unsteady ensemble averaging has to be used []. While any linear term in Eq.() gives the identical term, from quadratic nonlinear terms extra terms called correlations (such as v i v j ) are obtained. Based on this and on Eq.(), the URANS equations are written as V i t + x j (V i V j + v i v j V i =, x i ) = ρ ( p + τ ) i j, x i x j where the τ i j = τ ji = µ( V i x j + V j is the mean viscous stress tensor and τ i j = v i v j the Reynolds stress tensor. In order to compute all mean-flow properties of the turbulent flow, Eq.(), we need a prescription for computing the Reynolds stress tensor, i.e. to close the system of equations we have to introduce equations for the Reynolds stress components. Noticing that τ i j is a symmetric tensor we have additional unknowns. The function of the turbulence models is to find these extra equations. x i ) Turbulence Modelling Eddy-Viscosity Based Turbulence Models. In laminar flows, energy dissipation, transport of mass and momentum are mediated by the viscosity, so it is natural to assume that the effect of turbulence can be represented as an increased viscosity. This leads to the eddy-viscosity model (also called Boussinesq approximation) for the Reynolds stress tensor, v i v j = ν t () ( V i + V ) j x j x i 3 ρδ i jk, (3) where the k term is the turbulence kinetic energy given by k = v i v i and ν t the turbulence kinematic viscosity. This model assumes that the ν t is approximately isotropic, an important simplification which may not be realistic for several flows. 3 Copyright c 7 by ASME

4 For the eddy-viscosity -equation models two dimensionless parameters are important, a turbulence velocity scale v and a turbulence length scale l. While k can be the basis for the velocity scale, extra considerations have to be taken into account for the length scale, but dimensional arguments dictate ν t k l. Based on the available literature on these models, it is widely accepted that in spite of its enormous popularity, the k ε turbulence model of Launder [] (ε k 3 /l is the so-called turbulence dissipation) has significant drawbacks to capture boundary layer flows in adverse pressure gradients and separated flows. The k ω SST two-equation model by Menter [3] (ω ε k ) is here preferred. It blends the good behaviour of k ε models for free shear flows and the good behaviour of the standard k ω models for boundary layer flows. The equations for this model are given in detail in [3]. For the cylinder case Constatinescu et al [] showed for a sub-critical Reynolds number, that this model could predict reasonably well both the separation point and the drag value when compared with the experiments, while the k ε model did not predict any shear-layer development, with disastrous consequences for any quantitative analysis. It has to be emphasized that all turbulence models, including k ω SST, involve so-called closure coefficients which were empirically tuned based on simple flow experiments. Transition Model. As mentioned in the Introduction, the onset and extent of transition is of relevant importance for the smooth cylinder test case. Turbulence model equations can be solved through transition from laminar to turbulent flow, although it is well known that most models predict transition at Reynolds numbers that are at least an order of magnitude too low. This is because the calibration of its coefficients is based on reproducing the boundary layer behaviour and not on predicting transition. Other methods are used in the literature to predict transition: based on stability analysis of the boundary layer equations; purely based on experimental correlations; or based on the vorticity Reynolds number concept. Most of these models are difficult to be implemented in general-purpose massively parallel and unstructured grids based CFD codes. The so-called Local Correlation-based Transition Model, LCTM, purposed by Langtry and Menter [5], combines the vorticity Reynolds number concept with experimental correlations, using standard transport equations which are easily implemented in a general CFD code. The model adds two extra transport equations which are coupled with the k ω SST turbulence model: one for the boundary layer intermittency, and another for the momentum-thickness Reynolds number. The full details of this model are described in [5], but as the authors of the model emphasize, and as the name suggests, the method does not describe fully the physics of the transition process but rather forms a framework for the implementation of correlation-based models into general-purpose CFD codes. Therefore, and since the version of the method used in the present work was calibrated for aerodynamics applications and not for offshore flows, conclusions about the behaviour of this model must be drawn with special care. Reynolds-Stress Transport Model. While models based on the eddy-viscosity approximation provide excellent predictions for many flows of engineering interest, there are some applications where agreement with experiments seems to fail. Generally speaking, such models are inaccurate ( for flows ) with sudden changes in the mean strain rate (term V i x j + V j x i of Eq.()) or for flows with extra rates of strain. Some examples of these flows are flows over highly curved surfaces, flows in rotating fluids, highly three-dimensional flows and flows with larger zones of boundary-layer separation. The flow around the smooth cylinder, for the two considered, has some of these features. The starting point for any Reynolds-stress transport model,, is the exact differential transport equation describing the behaviour of the Reynolds-stress tensor τ i j = v i v j, see for instance Wilcox []. All independent components of this tensor (3 components for D flows) are computed rather than modeled. However, several terms of this equation have to be also modeled based on simplifications and assumptions. In fact, the turbulence dissipation term ε i j appearing in this equation, is usually modeled by solving an equation for the scalar ε like in the -equation models previously explained. It is often stated in the literature, [9,], that Reynolds-stress models based on ε retain the inaccuracies of the k ε model for boundary layers. In order to avoid these issues, a Reynolds Stress model has been used in this work which uses the ω equation for the turbulence dissipation. The complete description of the model can be found in []. DES Model As briefly explained above, the DES model tries to combine the accuracy of the RANS models for describing boundarylayers, and the better description of shear-layer zones provided by LES. The starting point for the DES model is therefore the filtered version of Eq.() used for LES, being the connection between RANS and LES made by the subgrid-scale model. Most of the popular subgrid-scale models are still based on the eddyviscosity ν t concept. The most widely used subgrid-scale model is the Smagorinsky model [], where ν t is function of the filter width used to derive the LES equations, which is proportional to the grid size. DES relies on the comparison of the turbulent length scale l computed from the subgrid-scale model and (or local grid spacing). In case the grid spacing is sufficiently lower than l, the model switches to the grid spacing as the defining l. The argument being that under those conditions, the grid is sufficiently fine for resolving the turbulent structures. From a practical standpoint, the use of the grid spacing as reduces ν t and allows the formation of a three-dimensional turbulent spectrum in detached flow regions. This makes it possible for the user to adjust the solution to include more complex physics by refining the grid (driving the solution towards LES) in regions of interest. Notice also that, if the mesh is sufficient to resolved all eddies, DES will result in a standard LES. Further grid refinement results in DES approaching the DNS limit, disregarding other numerical constraints. This may be seen as a very attractive characteristic of the model but in fact is one of its drawbacks []: ) it requires strict rules for the grid generation to obtain a consistent solution; ) DES solutions Copyright c 7 by ASME

5 seem not to converge with systematic grid refinement, which is an unsatisfactory situation in numerical analysis. The original DES model proposed by Spalart et al. [] has the -equation Spalart-Allmaras turbulence model [7] as subgrid-scale model. However, the DES approach may be used with any turbulence model that has an appropriately defined l. The version here used is based on the work of Strelets [8] and uses a modified version of the k ω SST model as subgrid-scale model. Y Pressure Outlet Symmetry X Z Inflow Wall Symmetry NUMERICAL DETAILS The commercial flow solver ANSYS-CFX was used for the computations here presented. It is a general-purpose code based on a unstructured grid based finite volume discretization of any flow equation. A collocated grid arrangement is used. For solving the continuity equation, Eq.(), a weakly-coupled solver is used, where a fourth-order spatial derivative of the pressure is added to the equation. Following the standard finite element approach, local-element shape functions are used to evaluate the derivatives for all diffusion terms. The solver of the linear system of equations resulting from the discretization of the flow equations is based on a multi-grid technique. For the discretization of time and convective terms of the momentum transport equations of Eq.(), and of any turbulent quantities, several schemes are available. In the present work, for the time dependent terms a second order backward Euler scheme is chosen for all equations. For the discretization of the convective terms a local blending between upwind and central discretization is used. For vector quantities, like velocity, this blending is independently done for each vector component. For the DES, in the RANS regime the high-resolution scheme is used for the convection terms, while for the LES regime a central scheme is applied. Figure. Computational domain and boundary conditions used. Quadrant Figure 3. Overview of grid layout grid is coarser, again to save computational resources. The grids are identified by the number of cells in the circumferential direction per quadrant of the cylinder. The grid presented in Fig.(3) is a D grid with 9 cells per quadrant and 5 cells along the body surface normal direction, with a total number of 75 cells. For 3D grids the lengthwise direction of the cylinder is also discretized with several planes which have the same type of grids as the ones for D. For both D and 3D computations, parallel processing is used, to shorten the computation times. TEST SETUP Physical and Numerical Conditions The cylinder tested is a fixed smooth bare cylinder with diameter d =.m and length l = 3.8m. The inflow conditions are Vin =.5m/s and Vin =.95m/s resulting respectively in Red = 9.3 and Red = 5.5 5, using standard environmental conditions. Both free-surface and bottom effects are neglected. Fig.() illustrates the computational domain as well as the boundary conditions considered. Four symmetry boundary conditions are considered for the side, top and bottom walls, together with an inlet inflow and a outlet pressure boundary condition. The top and bottom walls are located approximately 3.5 diameters from the cylinder, in order to save computer resources (memory and CPU time). The outlet boundary is located approximately diameters away from the cylinder. Fig.(3) shows the grid layout. An O-type grid is used around the cylinder, covering the zones close to the body (boundarylayer) and close to the body wake (shear-layer zones) with extra refined mesh. No wall-functions are used (the cylinder is smooth and therefore no roughness, nor rough-wall-functions are considered), and a very fine grid is used close to the body surface in order to maintain (for all grids and Reynolds numbers) a y+ for wall-adjacent grid cells. Far downstream of the cylinder the Experimental Data Most of the computations on smooth cylinders are validated against experiments dating back to the mid th century, [5, ]. Recently at MARIN, several experimental campaigns have been performed to assess the VIV phenomenon for fixed and freely vibrating, smooth and rough cylinders, the first measurements being made by de Wilde and Huijsmans [9]. These experiments were carried out in MARIN s High Speed towing tank and provided time traces of the forces on the cylinder, CD and St. With respect to the lift coefficient CL, a maximum value is difficult to be defined due to the irregular behaviour of the measured signals. This is caused by the three-dimensional effects observed in the basin. Fig.() shows both the original data by Wieselsberger taken from Schlichting [5] and MARIN average drag data resulting from different experiments carried out between and, for the fixed smooth cylinder (average roughness k/d < ). For the lower Reynolds number, the agreement is acceptable, and for CD an experimental value of. is considered. For St, MARIN experiments gave a value of.. For the higher Red the experimental values have to be considered with care since, at this region, the values are very sensitive to Red and surface roughness, a situation that is also corroborated by the different 5 Copyright c 7 by ASME

6 Average C D = 9.3E+ Wieselsberger Experiments (from Schlichting) MARIN Experiments = 5.5E+ 5 rms (res) p -3 V x V y k ω Figure. C D variation with. Experiments by Wieselsberger taken from Schlichting [5] and by MARIN. Conditions C D (e CD ) C Lmax St(e St ) 9. BaseCase ( 8%).7 (+%) BaseCase 9 (+%).35.9 (+%) Table. Numerical results for base case conditions. values found in the available literature [ 7,, ]. Values of C D =.3 and of St =.5 determined by MARIN experiments are considered. RESULTS Two-Dimensional Results Base Case Results. For the base case computations, the D grid previously shown in Fig.(3) and the SST k ω turbulence model are used. A time step of t =.s together with T s = s of simulation time, and t =.3s with T s = 5.s, are considered, respectively for the lower and higher. These values were chosen in order to achieve time convergence (steadystate cyclic solution) on the drag and lift forces. For the numerical convergence for each time step, a criterion of rms values of normalized residuals for the equations for p, V, k and ω lower than was considered. Fig.(5) shows the numerical convergence behaviour of the main quantities for the lower case. The convergence criterium is attained for each time step and the residuals evolution mimics the cyclic behaviour of the flow. Fig.() shows the evolution of the drag and lift coefficients during the simulation for both, indicating that cyclic steady-state convergence is reached. Tab.() shows the values obtained and the relative deviations from the experimental values, which are considerable for the C D estimate for the higher test case. The flow solution can be compared for approximately the same lift-force regime instant, within the last cycle of the computation. We consider a cycle to start at θ = corresponding to a zero lift force and θ = π/ corresponding to a maximum positive lift force. Fig.(7) shows the axial dimensionless velocity component V x /V in, and the out-of plane dimensionless vorticity component ω zd V in for the two and the same instant θ =. Fig.(8) -7 3 t [s] Figure 5. Residuals convergence history. Base case conditions, = 9.3. C D = 9.3E+ = 5.5E Figure. C L = 9.3E+ = 5.5E t/t s t/t s C L and C D time history. Base case conditions. shows the pressure coefficient and the dimensionless eddy viscosity distribution. The eddy-viscosity distribution shows only a very limited area of turbulent wake for the lower, but an almost total turbulent flow in the wake for the higher. Fig.(9) and Fig.() illustrate the shedding mechanism for several θ values. It is visible that positive and negative lift occurs when the maximum vorticity values on the wake are also positive and negative respectively. The higher values of vorticity for the lower test case explain the higher values of the forces for this condition. However, from all the previous pictures presented it is evident that the wake of the cylinder is only marginally narrower for the higher. Sensitivity Studies. Since the agreement between the calculations and the experiments was not satisfactory, a number Copyright c 7 by ASME

7 Red=9.3E+ Vx/Vin: Red=5.5E+5 Vx/Vin: θ= Figure 7. Red=9.3E+ C p: Red=5.5E+5 C p: θ=π θ=3π/ ωz d Vin for several µt/µ: θ. Base case conditions, Red = θ= Figure θ =. Base case conditions θ=3π/ Figure 9. µt/µ: θ=π/ Vx /Vin and ωvzind for θ=π/ θ=π/ Cp and µt /µ for θ =. Base case conditions. θ=π/ of additional numerical tests for the base case was done: - y+ : Small grid changes were done in order to obtain better y+ distributions. For the previous and following computations all the cells close to the cylinder surface have a y+ <.. - Background turbulence levels: Different background turbulence levels were studied, %, 5% and %. These prescribe the initial conditions for the turbulence model equations. The effect on the solution was however minimal. The medium level of 5% was chosen for the following computations. θ=3π/ θ=π Figure. 7 θ=3π/ ωz d Vin for several θ. Base case conditions, Red = Copyright c 7 by ASME

8 Cells/quad Cells C D (e CD ) C Lmax St(e St ) 9 75 ( 8%).7 (+%) ( %) (+9%) ( 5%) (+9%) ( 3%) 9.5 (+8%) Table. Numerical results for grid refinement study. = 9.3. t C D (e CD ) C Lmax St(e St ). 9 ( %) (+9%)..9 ( 3%) 9.57 (+9%) ( %) 8.57 (+9%) Table 3. Numerical results for time step refinement study. = Convergence tolerance: The rms convergence tolerance was lowered to for the higher. Both for C D and C Lmax the effect is almost negligible, C D = 5 instead of C D = 9 and C Lmax =.93 instead of C Lmax =.9. Since the number of iterations per time step is increased, with a consequent augment of computational resources, the value of was retained for the following computations. - Grid refinement: The influence of the spatial discretization error on the solution was studied by grid refinement. For the lowest the grid was refined by increasing the number of cells per quadrant. Tab.() and Fig.() shows the results obtained for the different grids. The trends are clear: with an increase of the grid level the average drag coefficient increases and tends to the experimental value and the maximum lift coefficient increases substantially. With respect to the Strouhal number the tendency is negative since the values tend to deviate, even if slightly, from the experimental value. This last trend requires additional investigation. - Time step refinement: The influence of the time step on the forces is illustrated in Tab.(3) and Fig.(). They show that more temporal resolution gives better results for C D, however the St number being insensitive to the time step. C D cells/quad 8cells/quad cells/quad 3cells/quad.5.. C D. t=. t=.. t= t/t s t/t s Figure. C D time history vs. grid level and time step. = k-ω =9.3E+ k-ω+ LCTM.3 C D Figure. t/t s. k-ω =5.5E+5 k-ω+ LCTM.3 C D t/t s C D time history vs. turbulence models. Turbulence Model Studies. Given the above results, and considering a balance of quality and computational resources, further computations were done using a grid with 8 cells per quadrant, and a time step of.s for the lower and.3s for the higher. With these numerical settings, the following turbulence models were applied: a) the k ω SST model; b) the k ω SST model using the LCTM transition model; c) the ω-based. Fig.() illustrates the evolution of C D with time, and shows that the model presents a steady-state convergence behaviour but with large variations with respect to the average value. For the higher higher frequency components are visible. The other two turbulence models apart from presenting different average values, show a smooth sinusoidal type convergence behaviour. Tab.() presents the numerical results for C D, C Lmax and St, and the following trends: - The C D value is more difficult to capture for the higher, this being independent of the turbulence model used. On the other hand, the St numbers are more difficult to calculate for the lower condition. - Both C D and C L values follow the same trends with the different turbulence models, i.e. if the drag values increase also the maximum lift values do so. - The results obtained with the different turbulence models show different trends for the two Reynolds numbers. For instance, while for the lower case the application of improve the results, by increasing the average drag value, for the higher case the results are worst since the average drag is also increased. 8 Copyright c 7 by ASME

9 Turb. Model C D (e CD ) C Lmax St (e St ) k ω 9 ( %) (+9%) 9.3 k ω + LCTM.733 ( 39%) 3.78 (+39%).8 ( %) (+7%) k ω 7 (+%).95 (+8%) k ω + LCTM (+%).5.33 (+3%) 8 (+8%) 55.3 (+%) Table. Numerical results for turbulence model study. Average C D Wieselsberger Experiments (from Schlichting) MARIN Experiments.8 k-ω k-ω + LCTM... = 5.5E+ 5 =9.3E+ =5.5E+5. = 9.3E+ V x /V in: -... V x /V in: Figure. 5 C D results vs. experimental data for turbulence model study. V x /V in: -... V x /V in: -... Figure 3. k-ω k-ω+lctm k-ω+tran V x /V in: -... V x /V in: -... V x /V in for θ =. Turbulence model study. Finally, Fig.(3) shows the influence of the different turbulence models on the velocity field. While the effect of the LCTM transition model on the wake field is not large for both, the computations with the show stronger velocity components on the wake which means strong vorticity values. It is again visible that the flow structure on the wake computed by all models is not very different for the two numbers, while it was expected for the higher number a stronger narrowing of the wake due to less flow separation. Remarks. From all the D computations presented before, one can conclude that both discretization errors, spatial and temporal, and the models applied, have a large influence on the results of the forces on the cylinder. With respect to the discretization error, the average drag values obtained for the lower test case (see Tab.() and Tab.(3)) show that with increasing spatial and temporal resolution the results are closer to the experimental value of.. However, when compared with Tab.() one can see that the effect of the discretization error is considerably smaller than the effect of the turbulence models. When analyzing the results of Fig.(), it is apparent that the gives larger drag values than the k ω model, and that the k ω model together with the LCTM transition model gives always values lower than the k ω model. This implies that for the lower the gives the best results and for the higher the k ω model with transition model provides results closer to the experiments. The fact that the computations with the transition model present the lowest errors for the higher, may be explained by the fact that here the boundary layer on the cylinder is partially laminar and partially turbulent, a situation for which the model was designed. Fig.() shows that the trends with the different turbulence models are consistent, but that the results are not satisfactory for the higher. This is partially explained by the fact that in all computations no significant changes in the wake flow structure have been seen, i.e. no less flow separation and therefore not so steep decrease of the drag forces. A comparison with experimental data for the velocity field on the wake region, obtained for instance by means of PIV, is of extreme importance in order to capture, identify and possibly correct the imperfections of the models. Also, due to the steepness of the drag curve in this region, a small change in can induce large variations in the drag values obtained. In a short-term, further calculations with the same models should be done for respectively smaller and larger in order to draw more definite conclusions. Three-Dimensional Results Considerations. Following the trends from the D calculations, the use of different turbulence models has been seen to have a larger effect on the values of the forces than the discretization error. Also, as explained in the Introduction, for the 9 Copyright c 7 by ASME

10 tested Red 3D effects play a role on the flow physics. Therefore, for the 3D studies more efforts will be spent on analyzing the effect of the three-dimensionality of the flow and its connection with the different turbulence models. DES computations were also performed to see if the three-dimensionality effects are better captured. However, without performing exhaustive discretization studies (grid and time step refinement) as have been done for the D test cases absolute conclusions are difficult to be drawn. We consider therefore the 3D results here presented as preliminary results. The grids used for the 3D calculations consist of 3 cells/quad and cells to model the height of the cylinder z (two diameters considered), reaching a total number of -. million cells. This will permit to obtain a deeper insight in the threedimensionality of the flow, and by reducing the number of cells on the xy plane, still keeping the computational effort within admissible research-based limits. The time step used for the k ω and computations was.5s. For the DES a lower value of 5 3 s for the low Red and 5 s for the high Red was considered, in order to obtain an rms Courant number of.5. The convergence tolerance for each time step was decreased to 5. In order to facilitate the convergence procedure in time, the converged solutions of the easier models were used as initial solutions for the more complex models. For instance, for the DES simulations the initial solution is the solution obtained with the, which has also been started with the solution of the k ω calculation... k-ω DES.3.3 Red=9.3E+.. CD CD Red=5.5E k-ω DES t/ts Figure 5. Red 9.3 Results. 3D computations with the k ω model, with the and using a DES approach have been performed. The convergence behaviour of the computations can be seen in Fig.(5). It is visible that a sinusoidal cyclic behaviour is more difficult to be obtained for the DES. The computations were nevertheless stopped when a steady-state cyclic behaviour was achieved for the lift force. The convergence trend for the DES is very similar to what have be shown in the literature [], where more frequency content and unsteadiness are found, which is also found in experiments. Tab.(5) shows the values obtained for the forces and Strouhal number. For the lower Red number the DES is the one that gives the most accurate results for the drag values. For the higher Red number it is remarkable that all computations estimate drag values which are very similar, and give good predictions of the St number. Nevertheless, and once more, for the higher Red test case the accuracy of the predictions for the average drag is far from satisfactory. A better insight in three-dimensional effects is obtained by analyzing the velocity distribution predicted by the 3D computations. Fig.() shows for Red = the velocity iso-surface Vx /Vin =.75. One can see that the 3D k ω results show very few velocity fluctuations in the cross-flow direction, while for the and the DES results these are evident. Furthermore, the DES typical velocity fluctuation pattern is notably different from the other two computations t/ts CD time history vs. turbulence models. 3D computations. Calc. CD (ecd ) CLmax St (est ) k ω ( 8%).5.35 (+8%).79 ( 3%) (+33%) DES.95 ±.5 ( ± 5%).9.3 (+8%) k ω (+7%).3.7 (+%) 79 (+%) (+%) DES 5 (+7%).9 5 (+%) Table 5. Numerical results for 3D study. Y X Z k-ω Y Y X X Z Z Figure. Vx /Vin =.75 Red = D computations with a grid of 3 cells/quad have been performed and the differences on the results with respect with the grid of 9 cells/quad observed to be smaller than the effect of the turbulence models. DES iso-surface for θ =. 3D computations. Copyright c 7 by ASME

11 Average C D... Wieselsberger Experiments (from Schlichting) MARIN Experiments D k-ω D 3D k-ω 3D DES 3D = 5.5E+ 5 Calc. C D (e CD ) C D (e CD ) CPU Time [hours] = 9.3 = processor D k ω 9 ( %) 7 (+%) D.8 ( %) 8 (+8%) 75 3D k ω ( 8%) (+7%) 8 3D.79 ( 3%) 79 (+%) 3D DES.95 ( %) 5 (+7%) 9 Table. Error for C D and CPU time vs. D and 3D computations... = 9.3E E+ Figure 7. C D numerical results vs. experimental data for D and 3D computations. Remarks. Fig.(7) summarizes the drag forces results of the 3D calculations and compares them with their D analogous. The forces predicted by the 3D computations are in general smaller than the ones by the D computations, which is due to the three-dimensional effects. This has also been observed before in the literature []. The almost negligible difference between the D and 3D k ω results are also due to the same reason. For the lower case the three-dimensional effects seem to play a less significant role than for the higher case, since the D computations present better results than the 3D ones. The drastic decrease of drag force is not captured by any of the computations here shown. The better physical model offered by the DES approach, as seen in Fig.(), includes more three-dimensional effects and different structure scales that are not modeled by the RANS approach with eddy-viscosity based turbulence models. For the difficult this may be hypothetically the key to perform better estimates, since as it can be seen in Fig.(7) the steepness of the drag variation with the is the largest for the DES. Tab.() shows the error for C D vs. the computational times (total clock time if using one processor) for the several D and 3D computations. With increasing complexity of the models applied the computational costs increase rapidly. The step from D to 3D computations increases the costs even further, even if the grid has been coarsened in the xy plane. The DES have computational costs that are perhaps acceptable for research purposes but definitely not adequate for daily design or even weekly analysis of offshore structures. Considering a performance ratio based on quality of the computations vs. the computational costs, it seems that the optimum choice is in fact the use of D URANS together with a k ω model. It is emphasized once more that, for the 3D computations, no conclusions about quality can be drawn without further grid and time step convergence studies. However, for finer grids and finer time-steps, the computational demands will increase significantly, which further penalizes any quality vs. cost performance ratio. CONCLUSIONS Several numerical and modelling features, currently used by the CFD community, have been used to compute the flow around a fixed smooth cylinder for different diameter based Reynolds numbers, = 9.3 and = 5.5 5, close to the socalled drag-crisis region. The present study involves grid sensitivity, time step sensitivity, the use of different turbulence models, D/3D effects and a RANS/DES comparison. The results of the computations have been compared with experimental data available in the literature and at MARIN. Based on the presented data and results, the following conclusions may be drawn: - For the lower case, a general under-prediction of the drag loads has been found, while for the higher case, a general over-prediction has been observed. The Strouhal numbers estimates presented values closer to the experiments than the estimates for the drag coefficient. - The numerical studies done with the D models for the lower showed that, with increasing grid discretization level and with decreasing time step, the results presented a slightly better agreement with the experimental values. However, this effect was seen to be much smaller than the effect of using different turbulence models. - For the D computations, the effect of using more complex turbulence models was large, but consistent: for both the results for the drag coefficient were the largest and the k ω model together with the LCTM transition model results the lowest. - The flow solution on the cylinder wake did not present substantial differences for the two different. The expected strong narrowing and weakening of the wake for the higher was not visible, which explains the least agreement with the experiments. The predictions showed also a total turbulent flow on the cylinder wake for the higher. - The three-dimensional effects visible in the experiments were only modeled by the use of or DES. The eddyviscosity based k ω computations did not show any crossflow velocity fluctuations. - The DES results seem to be the most realistic: not only three-dimensional effects were visible, but also showed high frequency content on the time history of the forces, which are also visible on the experiments. The drag decrease between the lower and the higher was the steepest. Nevertheless, the drag coefficient obtained with DES did not present the best agreement with the experimental data. - The computational costs of 3D RANS computations, and specially of 3D DES computations are considerable large. Copyright c 7 by ASME

12 They are maybe adequate for research purposes but still do not permit a daily design or analysis of offshore structures. Having in mind a quality vs. cost performance ratio, and considering that good DES results are only obtained using RANS solutions as initial guesses, it is preferred to start analyzing any cylinder flow using a RANS approach together with eddy-viscosity based turbulence models. Before making final conclusions of which model produces the most accurate results for this flow problem, additional work has to be carried out. More D/3D calculations have to be done for different within and close to the drag-crisis region, always with detailed grid and time step refinement studies. This will permit to make a complete assessment of the capability of the models to predict the steep drag decrease, and to check the sensibility of the computations with the. The sensitiveness with the surface roughness should be also assessed. Additionally, not only global integral values such as forces, should be compared with experiments, but also velocity and vorticity fields, in order to identify the weakness of the models. If possible, the models should be calibrated based on these comparisons. All these topics will be considered in a near future. It has been proven again that the bare smooth cylinder is a complicated flow problem around a simple geometry. In practical applications the riser geometry is even more complex: the surface is not smooth, presents roughness, and usually has strakes or other devices designed to reduce motions. Being geometrically more complex, the flow around some of these geometries is in fact easier to be modeled, whence the separation points are very well defined by the sharp strakes. This is corroborated by the several published cases of good agreement between CFD computations and experiments. However, real risers typically have strakes on only a portion of their length, and thus much of the cylinder will still be bare with flow past it over much of the water column. Also, when dealing with fairings, accurate computation of the separation points can be as difficult as that for a bare cylinder. The flow around the bare smooth cylinder is therefore of extreme practical relevance for real life risers analysis. ACKNOWLEDGMENT The authors are thankful for the effort spent by Martin Hoekstra on the revision of this paper and on the discussions about the results here presented. The authors also acknowledge the help of René Huijsmans and Jaap de Wilde regarding the start-up of this project and the supply of the experimental data. REFERENCES [] Constantinides, Y., and Oakley, O.,. Numerical Prediction of Bare and Straked Cylinder VIV. OMAE Proceedings, Hamburg, Germany. [] Constantinides, Y., Oakley, O., and Holmes, S.,. Analysis of Turbulent Flows and VIV of Truss Spar Risers. 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La Recherce Aerospatiale, (5). [8] Strelets, M.,. Detached Eddy Simulation of Massively Separated Flows. AIAA Journal(879). [9] Wilde, J. J., and Huijsmans, R.,. Experiments for High Reynolds Numbers VIV on Risers. ISOPE Proceedings, Stavanger, Norway, June. [] Güven, O. et al., 975. Surface Roughness Effects on the Mean Flow Past Circular Cylinders. Tech. Rep. 75, IOWA Institute. Hydraulic Research Dpt. [] Shih, W.C.L. et al., 99. Experiments on Flow Past Rough Circular Cylinders at Large Reynolds Numbers. nd International Colloquium on Bluff Body Aerodynamics and Applications. Melbourne, Australia, December. Copyright c 7 by ASME