^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

A hybrid RANS-LES approach is used to resolve the Fore-body Side Vortex (FSV) separating from the KVLCC2 hull at 30° drift angle and Reynolds number
Re_{L}_{oa} ≈ 2.56
e6. The performance of the DES approach is evaluated using a proper grid study. Besides, the following aspects of the CFD results are investigated: the resolution of turbulent energy, the prediction of instantaneous and time-averaged vortical structures, local flow features, the limiting streamlines and the evolution of the vortex core flow. New PIV data from wind tunnel experiments is compared to the latter. The results form a basis for future investigations in particular on the vortex interaction further downstream and the applicability of different kinds of turbulence models to trailing vortices like the FSV. Turbulence modelling is realised with the
k-
ω-SST-IDDES model presented in [1], the grids’ cell count is 6.4 M, 10.5 M and 17.5 M. Grid convergence of the time-averaged vortex core flow is observed. OpenFOAM version 1806 is used to carry out the simulations and snappyHexMesh to build the mesh.

Coherent vortices occur for example as wing tip vortices. Several test cases have been investigated including e.g. [

The Fore-body Side Vortex (FSV) analysed in the following, see

Although there are similarities to wing tip vortices the following literature review focuses mainly on research on this vortical structure to consider the findings on turbulence models, etc. The authors analysed the influence of different turbulence modelling approaches and different grid types and sizes on the formation of the complex vortex system in the near wake of the hull.

The KVLCC2 hull and its modified versions like KVLCC2M have been investigated in several workshops: Gothenburg 2000 [

Fureby et al. [

Abdel-Maksoud et al. [

Xing and colleagues [

Ismail et al. [

The present approach is based on scale resolving simulations applying the k-ω-SST-IDDES model presented in [

This paper is structured as follows: After a description of the case including the vortex system, inflow conditions and the hull model, the modelling approach is explained referring to the turbulence model, the solver and discretisation settings and the mesh. Finally, the results are analysed considering first the proper resolution of the near wall flow and the turbulent energy in the LES zone around the FSV. Secondly the vortical structures, local flow features, the development of the vortex core flow and the streamlines are discussed. This includes the comparison to the experimental data.

Vortex system and FSV The flow around the KVLCC2 hull at a drift angle of

30˚ and at the Reynolds number R e L o a ≈ 2.56 e 6 is simulated with special focus

on the coherent leeward vortex.

The ASV separates on the windward side of the hull and develops close to the hull’s bottom in the boundary layer. Near the ship’s stern several small and large vortices (AHPV, SV and ABV) separate and interact further downstream. The hull’s wake is dominated by this interaction which creates a much more complex flow field than upstream where the FSV evolves separately.

Flow parameters The inflow conditions are presented in

KVLCC2 model geometry The KVLCC2 hull introduced in [

A double-body model is placed in the wind tunnel but flow simulations are carried out for a single hull. Both approaches realise a symmetry boundary condition at the imaginary waterline. For CFD this is realised with a slip or zero-gradient boundary condition.

Inflow velocity U ∞ | Air temperature | Viscosity ν | Density ρ | R e L o a | I = 2 3 k / U ∞ |
---|---|---|---|---|---|

25 m/s | ≈23˚C | 1.56e-5 m^{2}/s | 1.19 kg/m^{3} | ≈2.56e6 | 0.3% |

Main particulars | Ship | Model | ||
---|---|---|---|---|

Length over all | L o a | [m] | 333.6 | 1.600 |

Length between perpendiculars | L p p | [m] | 320.0 | 1.535 |

Breadth | B | [m] | 58.0 | *0.278 |

Original draft | T | [m] | 20.8 | *0.100 |

Modified draft | D | [m] | 21.5 | 0.103 |

Scale | λ | [-] | 1.0 | 208.500 |

The Cartesian coordinate system is aligned to the ship model’s longitudinal axis: The x-axis points towards the stern, the y-axis in the portside direction and the z-axis towards the ships bottom (perpendicular to the inflow).

As this investigation succeeds the one presented in [

• Different inflow conditions (speed/Reynolds number);

• Different orientation and location of the measurement planes;

• New coordinate system aligned to the new measurement planes. This is important as the velocity and vorticity component normal to the planes are analysed.

Wind tunnel The TUHH low-speed wind tunnel (

The test section allows manual and optical access from the top and the lateral sides. Positioning the PIV measurement system is supported by a multiple-axes traversing system mounted to the lateral sides of the section, thus allowing the measurement of the flow velocity at various planes and positions.

The orientation of the double-body model in the test section is shown in

Dimensions of the test section | Length 5.5 m Width 3 m Height 2 m |
---|---|

Max. velocity | 35 m/s |

Contraction ratio nozzle | 4.125 |

Fan unit power | 400 kW |

perpendicular (FP) is about 1.93 m and the distance between the wind tunnel bottom and the line which represents the intersection between the double-body symmetry plane and the transom stern is about 0.50 m (values rounded to [cm]). The model is placed in the middle of the test section width, hence the “boundaries” of the open test section are located at z W = z = ± 1.5 m . The model is suspended inside the test section by 8 wires, each 1.0 mm in diameter. The forward and the aft four wires are fixed at x / L p p = 0.085 and at x / L p p = 0.908 , respectively.

At the forward perpendicular the model is equipped with zig-zag strips (see

Measuring planes Within the current measurements, PIV data has been obtained for planes parallel to ship frames (y-z-planes). The reason for this choice of the plane orientation is that the flow in the model wake is mostly parallel to the hull walls, hence is normal to the measurement planes. So e.g. the axis of the large fore-body side vortex (FSV) is (nearly) orthogonal to the planes and following its axial vorticity can be determined. (This is valid until the FSV bends towards the symmetry plane.) The fact that the FSV is almost perpendicular to the ship frames is also the reason for the introduction of the new coordinate system.

x [ m ] = [ 0.361,0.461, ⋯ , 1.561 * ,1.661 ]

this corresponds to (rounded values)

x / L p p = [ 0.235,0.300, ⋯ ,1.017 * ,1.082 ] .

The measurement plane at the transom stern is marked with *.

The following data is available from the experiments for each plane:

• velocity (vector field: U x , U y , U z );

• vorticity normal to the planes (scalar field: ω x ).

PIV system The spatial distribution of the velocity components in different planes is measured by a modular commercial 2D-3C-PIV system of TSI Inc. The stereoscopic PIV system (SPIV) consists of a pulsed laser, light sheet optics, two cameras, a synchronizer and a computer with software to control image generation and processing.

The light sheet is generated by a 200 mJ two-head Nd-YAG-laser (Quantel Big Sky) and the light sheet optics. Scattered light is received by two PowerView 4 M (2048 × 2048 pixel, 12 bit, monochrome) cameras equipped with Nikon 300 mm f/4D AF-S lenses; their baseline is located approximately 1.7 m from the middle of the test section and positioned on both sides of the laser plane. Six measuring planes at each measuring station were investigated. The planes are arranged

in 2 × 3 configuration, where two planes are measured beside each other in y-direction and 3 planes are measured above each other in z-direction. The overlap between neighbouring planes is 50%. The optical axes of the lenses are inclined 27.5˚ and 24.5˚ to the normal of the laser plane for the middle planes and ±0.5˚ for the other planes. At capture frequency of 7.25 Hz, 1000 images were recorded at each measuring plane.

In order to avoid blur caused by the oblique view of the cameras, a rotatable base adjusts the angle between the lens and CCD chip to satisfy Scheimpflug condition. The cameras record two images each with a short time separation ( Δ T = 15 μ s ). In order to reach sufficient signal-to-noise ratio for the subsequent image processing and to minimize the loss of particle pairs, the time separation was selected to meet the condition that a particle would travel more than 25% of the light sheet thickness.

For the PIV measurements, particles of an average diameter of about 1 μm are generated as the seeding. The Laskin type droplet generator uses dioctyl sebacate (DOS). The generator is placed downstream of the test section. The fog generated spreads through the wind tunnel at a closed loop operational mode and leads to a global seeding. Therefore, any influence due to turbulence of the generated fog is negligible.

The images was analysed by means of a FFT-transformation, cross correlation technique and ensemble averaging of the calculated correlation maps. Gaussian curve fitting was applied to estimate the location of the correlation peak with sub-pixel accuracy. No pixel locking effects were recognized. The results were calculated with a 50% overlap of neighbouring vectors for the two-component vector maps of each camera. Reconstruction of three component velocities is based upon the vector maps of both cameras as well as calibration data.

The calibration is executed by capturing a set of images for a calibration target. A black calibration target with a predefined rectangular grid of dots spread on two planes was used to capture the calibration images. The required calibration data was calculated by evaluating these images. The calibration of the PIV system is sensitive to even small changes of the geometrical and optical configuration. Therefore, the whole PIV-components were installed on one crossbar. Then the crossbar can be moved by the traversing mechanism in vertical and horizontal direction.

Data reduction The velocity components are calculated by using the full set of data. In order to calculate the velocity fluctuation, the set of data is divided into 20 smaller packs of 50 images each. For comparison reasons, the same investigation was conducted with 10 packs of 100 images each, 50 packs of 20 images each and 100 packs with 10 images each. The velocity components U i are computed using:

U i = 1 N ∑ k = 1 N U i , k (1)

where the index i = x , y , z represents the velocity components. The index N represents the collected data for every interrogation area. The differentials of the velocity components are used to compute the vorticity vector components according to its definition as the rotation of the velocity field.

Uncertainty assessment The PIV system was mounted on a crossbar, 2D automated traverse system. The uncertainty of the positioning is ±0.1 mm. The estimated uncertainties of the three velocity components according to prior measurements with the same configuration are W = ± 0.08 m / s , V = ± 0.06 m / s and U = ± 0.33 m / s . The relative uncertainties of the velocity components to the free stream velocity are W = ± 0.30 % , V = ± 0.22 % and U = ± 1.22 % .

Within the following section several aspects of the CFD approach are presented in detail dealing with turbulence modelling, the flow solver and the computational mesh.

Turbulence modelling A standard well-established hybrid RANS-LES approach is applied to predict the flow around the hull: k-ω-SST-IDDES. The DES model presented in [

Solver OpenFOAM The simulations of the flow around the ship hull are based on a cell-centred, unstructured finite volume method (FVM). OpenFOAM version 1806, first presented in [

The discretisation setup is chosen to reduce its contribution to the numerical diffusion: Temporal terms are discretised with a second order implicit scheme (backward). The convection terms of the turbulence properties k and ω are discretised with a TVD scheme and the convection of the velocity is discretised using a blending between linear upwind (with gradient limiter) and central differences (linear). The latter blending factor is determined based on the flow (RANS-LES subdivision) and the local mesh quality following [

The simulations are initialized with steady RANS solutions on the respective grid. Probes for velocity and turbulence properties are used to monitor the development of resolved turbulence. After about one hull pass ( L o a / U ∞ ≈ 0.064 s ) turbulence is assumed to be developed and time averaging starts. The simulations are analysed after the time-averaged velocity field was converged which corresponds to a simulation time of about seven hull passes. Considering a maximum Courant number of 0.8 the time steps for the coarse, medium and fine mesh are approximately 3.7e-6 s, 3.4e-6 s and 2.5e-6 s respectively.

Compuational domain: mesh and boundary conditions

The different meshes coarse, medium and fine are created by refining the background block mesh with a factor of exactly 1.25. The cell count along the domain edges in z-direction changes from 16 (coarse mesh) to 20 (medium) to 25 (fine) and in x- and y-direction with the same ratio. As the body mesh and the refinement regions are based on the block mesh, the cell size ratio is also 1.25.

Coarse | Medium | Fine | |
---|---|---|---|

Total cell count | 6.4 M | 10.5 M | 17.5 M |

Mesh refinement | 1 | 1.25 | 1.25^{2} |

Cell size level 0 | 150 mm | 120 mm | 96 mm |

Cell size level 6 | 2.34 mm | 1.88 mm | 1.50 mm |

Cell size level 7 | 1.17 mm | 0.94 mm | 0.75 mm |

Layer count | 19 | 19 | 19 |

Layer extrusion ratio | 1.2 | 1.2 | 1.2 |

1st layer thickness | 3.15e-2 mm | 2.52e-2 mm | 2.02e-2 mm |

/level 6 | 1.34% | 1.34% | 1.34% |

Last layer thickness | 0.84 mm | 0.67 mm | 0.54 mm |

/level 6 | 36% | 36% | 36% |

Total layer thickness | 4.5 mm | 3.9 mm | 3.1 mm |

/level 6 | 210% | 210% | 210% |

/1st layer th. | 155 | 155 | 155 |

Faces non-ortho >70˚ | 13 | 35 | 56 |

The mesh structure The block mesh is considered level 0, the cells at the hull and around the FSV are isotropically refined six times (up to level 6 respectively) and up to level 7 at four distinct zones (see also

• bow: high velocity magnitude due to the flow around the corner;

• stern: small geometry features to capture by the mesh;

• FSV initial separation (until x / L p p ≈ 0.33 ): high mesh density to resolve initial vortex formation and roll up of free shear layer;

• FSV core (until x / L p p ≈ 0.73 ): vortex core where high flow gradients occur. Following the vortex core line of a preliminary result on the medium mesh the cells are refined within a certain distance around it.

Viscous layers are added after the mesh has been snapped to the surface. Their size is preset by the expansion ratio, the number of layers and the first cell size. Following from expansion ratio and layer count the total layer thickness is about 150 times the first layer thickness, details given in

For the coarse and medium mesh the layer coverage on the hull is 100%, for the medium mesh there is a tiny part (below 0.3% of the hull surface) where no or less than 19 layers are extruded by the mesh algorithm. This part is located near the forward shoulder and the waterline at the windward side (starboard). As this is located close the stagnation point where the flow velocity is small, no serious influence on the result is observed.

According to [

At the hull the only no-slip boundary condition is preset, the imaginary watersurface is modelled with a slip (zero-gradient) condition, both inlet patches with a velocity inlet setting the velocity magnitude to U ∞ = 25 m / s and the inflow angle to 30˚ and the outlet is modelled with a pressure outlet condition (with undisturbed static pressure p ∞ = 0 ).

First, the near wall resolution and the resolved turbulent energy are analysed. Afterwards, the investigation deals with the vortex structures, local flow effects, the development at the vortex core line and the wall shear. Most results are obtained with CFD, only the vortex core flow also compared to PIV data.

CFD verification The flow near the ship hull is computed in RANS mode down to the wall, so without wall functions. As required the dimensionless wall distance is near unity, see

Mesh | Min. | Max. | Avg. |
---|---|---|---|

Coarse | 0.05 | 4.01 | 1.16 |

Medium | <0.01 | 2.35 | 0.88 |

Fine | 0.02 | 18.80 | 0.72 |

to fine) arises due to the refinement of the first cell. The high maximum value for the fine mesh originates from the missing layer extrusion mentioned above. Analysing the local flow the impact is considered negligible.

The wall distance and the wall near flow are presented in

Considering the hybrid RANS-LES approach, the RANS-LES interface and the amount of resolved TKE is analysed in the following.

Furthermore the proper resolution of hybrid RANS-LES or LES approaches can be verified with the relation of resolved TKE to total TKE

k r e s / k t o t = 1 2 〈 u ′ i u ′ i 〉 1 2 〈 u ′ i u ′ i 〉 + k m o d (2)

using the trace of the reynolds stress tensor 1 2 〈 u ′ i u ′ i 〉 and TKE from the subgrid

model k m o d . As originally proposed in [

Flow analysis Within the following section different aspects of the flow are analysed starting with the vortical structures in the vicinity of the FSV followed by a local analysis at a plane, the flow at the vortex core line and the limiting streamlines.

Vortical structures The time-averaged flow fields,

resolution becomes visible as the size of the FSV isosurface decreases from coarse to medium. On the medium and the fine mesh the difference is small; this corresponds to the grid convergence of the results discussed below.

Plenty vortices of different size are visible in the instantaneous flow field. The pattern on the medium and fine mesh is similar and the vortices are restricted to the refined mesh zone. On the coarse mesh, larger scales occur. In general the instantaneous vortex pattern consists of circular structures surrounding the FSV, these patterns represent the separated free shear layer that rolls up and induces velocities onto the FSV. This observation was also mentioned in [

Local flow analysis In the following the flow at plane perpendicular to the vortex axis will be analysed, see

Both velocity and vorticity show distinct local maxima inside the vortex core on the finer meshes. The more the shear layer rolls up (gets closer to the vortex centre) the higher the axial velocity becomes. And the velocity overshoot of about 15% at the centre is clearly visible. On the coarse mesh there is no local

velocity maximum and no velocity overshoot. The free shear layer seems to be smeared out. Comparing the results on the different meshes shows the similarity between the medium and fine mesh and the significantly weaker vortex on the coarse mesh.

On the coarse mesh, two effects occur: the lower resolution is a possible explanation for the weaker vortex (increased numerical diffusion) but still the velocity oscillations lead to a high TKE level. So there are many flow instabilities or fluctuations which seem to have a numerical origin, at least they do not contribute to a coherent vortex structure. For example, as seen in

In the vorticity field, the current pattern of the free shear layer can be determined. It consists of zones with different axial vorticity which represents the unsteady nature of the flow separation. The local maximum at the vortex centre is clearly visible; most of the vorticity is concentrated there.

Although it is a snapshot of the unsteady wake flow, the vortex core of the FSV can be determined. This fact can be used to analysed a possible wandering motion of the FSV: Wandering is expected to occur at very low frequencies compared to turbulent fluctuations [

Vortex core properties The algorithm to extract the centre of the FSV is based on the local alignment of the velocity and the vorticity vector (the normalized helicity is the cosine between both vectors), as the velocity is parallel to the vorticity at the vortex centre. It was proposed by Levy and colleagues in [

The axial velocity, vorticity, the pressure coefficient and the resolved TKE at the vortex centre as well as its position are shown in

Considering the pressure coefficient it is necessary to mention that the undisturbed static pressure p ∞ is zero, so the numerator becomes p − p ∞ = p . All subfigures show similar results for the medium and fine mesh, but the vortex on the coarse mesh is significantly weaker (smaller vorticity and higher pressure). Considering that the cell size changes with a factor of 1.25 from mesh to mesh, the time-averaged flow shows grid convergence because the change from coarse to medium is large and from medium to fine is about an order smaller. As the resolution of the coarse mesh is considered not sufficient, the vortex core properties are not analysed in the following.

At the initial vortex evolution the pressure reduction and the velocity overshoot develop on a short distance compared to the decay further downstream. Near x / L p p = 0.3 the extremal value is (nearly) reached and both the velocity and the pressure decrease at a similar rate to approximately 90% of their extremal values at x / L p p = 0.73 . The vorticity shows a small peak near x / L p p = 0.3 with a sudden decrease by about 10%, further downstream the vorticity decreases linearly to about 50% of the extremal value. Near x / L p p = 0.33 the mesh coarsens, the refinement box around the free shear layer and at the hull where the FSV separates ends, see

The resolved TKE is approximately two orders of magnitude larger than the modelled part, this corresponds to the observations in

Comparison of the vortex core flow to PIV data For the experimental results, the vortex core line is extracted by local extrema of the axial vorticity. It is assumed that the influence of the different vortex core line algorithms is negligible as usually the vorticity has a local extremum at the vortex centre for a coherent structure. At least the difference in the algorithms cannot explain the discrepancy in the vorticity level.

The shaded region in

The scattering of the results may be induced by the wandering motion of the FSV in the wind tunnel test section. Due to flow unsteadiness in the inflow or disturbances due to the PIV and camera system the low-frequency oscillation may be induced. During the postprocessing of the PIV data the flow field is averaged on a fixed grid, see Equation (1). Vortex wandering may lead to a smoothing of the high velocity gradients in the vortex core which in turn would lead to a reduced vorticity level.

Limiting streamlines The limiting streamlines shown in

The current results provide a basis for future research on the resolution of coherent vortices with proper RANS models (e.g. with curvature correction) or scale resolving simulations. The hybrid RANS-LES approach based on the k-ω-SST-IDDES model seems to be well suited for the prediction of the FSV at 30˚ drift angle as the LES region around the FSV is properly resolved and the time-averaged flow at the vortex core line shows grid convergence. A proof for the proper resolution inside the LES region is the high value of resolved to total TKE which is mostly higher than 80% and near unity inside vortex core and free shear layer. Grid convergence can be assumed as the time-averaged velocity, vorticity, pressure and the resolved TKE at the vortex centre as well as its location change very little from the medium to the fine mesh compared to a large difference from the coarse to the medium mesh. Besides, the instantaneous vortical

structures on both finer meshes have a similar pattern.

Considering the grid study, the results on the coarse mesh differ significantly from the ones on both finer meshes. Taking into account high-velocity fluctuations whose origin seems to be a numerical issue, the coarse mesh may be too coarse to properly resolve the flow. Hence, the following conclusions on the flow are referred to the medium and fine mesh.

The vortex pattern of the time-averaged and instantaneous flow is quite different: For the first a coherent smooth vortex tube exists and for the latter there are many small vortical structures wrapping around the FSV. Even for the instantaneous flow the nearly circular shape of the FSV core was observed for one time step. This observation supports the dominant nature of a trailing vortex.

The comparison of the PIV and CFD results for the vortex core shows that the position coincides well, the velocity overshoot is about 15% higher for CFD and the vorticity is initially about 100% higher but decreases to the constant experimental value. The reason for the deviation is not certain; a possible explanation is the smoothing of the experimental data due to the analysis on a fixed grid that does not consider a possible wandering motion of the FSV in the wind tunnel.

Outlook Several aspects will be analysed in the future considering validation, different modelling and numerical approaches and a comparison to other flows with coherent trailing vortices:

• As the coarse mesh shows convergence issues and high-velocity fluctuations without proper physical explanation, meshes with a resolution between the coarse and medium ones should be investigated.

• The promising results of the medium and fine mesh lead to the question whether the approach is applicable to smaller drift angles that occur in the everyday operation of ships. A first test case would be the drift angle 12˚.

• As the free shear layer rolls up into the FSV it may induce axial velocity to the vortex core. This can be further investigated by refining the the region where the shear layer is located.

• After the analysis of the initial separation and the separate development of the FSV it would be of interest to analyse its interaction further downstream inside the vortex wake, see

• Another point considering the modelling refers to the comparison of different kinds of turbulence models. Possible models need to consider the strong curvature inside the vortex core, e.g. with curvature correction or like in EARSM framework. These RANS models offer a huge gain in computational efficiency and the question is how accurate the vortex flow can be predicted.

• Wandering: This coherent low-frequency motion of the vortex may be induced by wind tunnel unsteadiness [

This research was partially sponsored by the Office of Naval Research Global under Grant N62909-18-1-2080 under the administration of Drs Salahuddin Ahmed and Wei-Min Lin. The authors would like to express their thanks for the support. Besides, the authors would like to thank their colleagues Klaus Wieczorek, Dr. Volker Müller and Jörg Voigt for their work related to the experiments.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Office of Naval Research.

The authors declare no conflicts of interest regarding the publication of this paper.

Feder, D.-F., Shevchuk, I., Sahab, A., Gerwers, L. and Abdel-Maksoud, M. (2019) Fore-Body Side Vortex of KVLCC2 at 30˚ Drift: A Trailing Vortex Resolved with DES and Compared to PIV Data. Open Journal of Fluid Dynamics, 9, 303-325. https://doi.org/10.4236/ojfd.2019.94020