DARPA SUBOFF: using code_saturne for naval hydrodynamics

In brief

Developed by EDF for its own engineering needs, code_saturne is historically associated with internal flows: circuits, heat exchangers, nuclear components. This article takes it to a very different arena: external naval hydrodynamics, with the DARPA SUBOFF, the reference submarine geometry of the 1990s, in its fully appended AFF-8 configuration (hull + sail + four stern fins).

The simulation is run in RANS k-ω SST on a deliberately lightweight mesh of 459,000 cells, representative of a pre-design study. The resistance is compared against tow-tank measurements over the full speed range, and the local results (pressure coefficient, skin-friction coefficient and wake velocity profiles) are confronted with wind-tunnel measurements. The main takeaways:

• Resistance: over the six test speeds (5.93 to 17.79 knots, $Re_L$ from $1.3$ to $4.0 \times 10^7$), the deviation from the measurements lies between $-0.1$ % and $-3.7$ %, and the decrease of the drag coefficient with Reynolds number is correctly captured.
• Local comparisons: the wall-pressure distribution along the hull and the velocity profile in the propeller plane fall within the published experimental uncertainties.
• Identified limitation: the skin friction on the afterbody departs from the measurements. This deviation is consistent with the use of wall functions on a relatively coarse mesh.
• 100 % open-source chain: hexa-dominant snappyHexMesh mesh converted to MED format with meshlane, parametric study over the six speeds set up, launched and monitored with csauto, forces integrated by a user routine compiled at run time.

Why the DARPA SUBOFF?

At the end of the 1980s, DARPA's SUBOFF program had a generic submarine model built and instrumented at the David Taylor Research Center, precisely to give the CFD community a public validation case for naval hydrodynamics. The geometry is defined analytically by Groves et al. (1989) - bow, afterbody, sail and fins are all written in closed-form equations - which makes it exactly reproducible, with no CAD ambiguity. Thirty-five years later, the SUBOFF remains the benchmark of the field: it is the case on which RANS, hybrid and LES codes for deeply submerged hulls are evaluated.

The associated experimental database has two complementary parts. In the tow tank, Liu & Huang (1998) published the resistance of the AFF-8 configuration for six towing speeds. In the wind tunnel, Huang et al. (1992) measured, at $Re_L = 1.2 \times 10^7$, the pressure and skin-friction distributions along the hull as well as the velocity profiles in the propeller plane - together with, most valuably, published experimental uncertainties. All the reference data used here are extracted from these two primary reports.

The goal of the study is twofold: to assess the ability of code_saturne to reproduce the quantities that matter to a naval architect - resistance first, the local flow structure next - and to do so under conditions representative of a pre-design study, that is with a light mesh and wall functions, where the literature on this case often mobilises several tens of millions of cells.

Case set-up

Geometry and flow: the AFF-8 configuration combines the $4.356$ m long axisymmetric hull with the sail and four cross-shaped stern fins. Since the geometry is symmetric about the vertical plane, only half of it is simulated, with a symmetry condition on that plane. The model is computed deeply submerged (no free surface), in fresh water, over the exact range of the test speeds: from $3.05$ to $9.15$ m/s, i.e. $Re_L$ from $1.3$ to $4.0 \times 10^7$.

Pre-design mesh: the hexa-dominant mesh counts 459,378 cells for the half-model. It is refined by octree around the body and in the wake. Generated with snappyHexMesh, it is converted from the polyMesh format to the MED format by meshlane and imported as-is into code_saturne. The wall treatment relies on wall functions (average $y^+$ around 30 at low speed, 100 at full speed): a deliberate choice, representative of the coarse meshes used in early design stages.

Numerical modelling: the computations are run as steady incompressible RANS with the k-ω SST turbulence model and a two-scale wall function. Pressure-velocity coupling relies on the SIMPLEC algorithm with a local pseudo-time step. Each simulation converges in 2,000 iterations, the total resistance being stabilised to within $0.1$ % over the last 500. The parametric study in speed is orchestrated with csauto, the orchestration tool for code_saturne developed by Simvia: starting from a single configuration, csauto derives the set-up of the six test speeds, chains the runs and monitors them (convergence, forces) while they execute.

Post-processing: a user routine (cs_user_extra_operations.cpp) integrates at every iteration the pressure and friction forces per zone (hull, sail, fins). Profiles and fields are extracted with pvpython from the EnSight outputs of code_saturne.

Resistance: comparison with the tow-tank tests

The evolution of the resistance is correctly reproduced over the whole speed range, with a slight systematic under-estimation - between $-0.1$ and $-3.7$ % - and a drag coefficient decreasing from $3.47$ to $3.04 \times 10^{-3}$ with Reynolds number, in line with the tests. The force decomposition helps explain the observed deviations: friction accounts for about 80 % of the total resistance and is also the component most sensitive to the wall-function treatment. With a 459,000-cell engineering mesh, the deviations obtained therefore remain compatible with the objective of a pre-design study.

Velocity and pressure fields

The fields reproduce the physics expected of a streamlined hull: stagnation point and overpressure at the bow, acceleration and suction over the forebody, progressive thickening of the boundary layer along the parallel mid-body, then pressure recovery over the stern where the flow decelerates. The sail and the fins imprint their pressure gradients at the leading edges and leave velocity deficits in the wake that are still visible downstream of the propeller plane.

Wall pressure along the hull

The computed $C_p$ distribution passes within the uncertainty bars for the majority of the pressure taps: the parallel mid-body plateau, the pressure recovery over the stern and the local disturbances induced upstream of the appendages are reproduced without visible bias. Only the first three taps of the forebody ($x/L$ from $0.03$ to $0.11$) escape the bars: the computation predicts a slightly too strong suction there, with a deviation that nevertheless remains modest (less than $0.06$ in $C_p$). This is the quantity that drives the pressure component of the resistance and the longitudinal balance of the hull - and the agreement is good over the whole body, including at the sail and fin stations, whose footprint on the pressure line is correctly positioned.

Skin friction along the hull

The friction shows larger deviations. Over the parallel mid-body, the $C_f$ plateau stays within the experimental uncertainties: the level of turbulent friction is correctly reproduced, consistent with the good prediction of the global resistance. On the afterbody ($x/L$ from $0.85$ to $0.97$), however, the computation departs from the measurements. In this region the boundary layer thickens rapidly under the adverse pressure gradient associated with the stern. The assumptions on which wall functions rest, in particular local equilibrium and the logarithmic profile, become less adequate there. This deviation must therefore be read against the mesh resolution chosen for the study. Resolving the boundary layer down to the wall ($y^+ \approx 1$) with a low-Reynolds treatment is the object of ongoing work.

Velocity profile in the propeller plane

The velocity profile in the propeller plane is the headline quantity of the case: it is what drives the sizing of the propulsor, its efficiency and its signature. The computed azimuthal average of the axial velocity falls within the experimental uncertainties over most of the radius: the wake deficit near the hull, its radial decay and the recovery towards the outer flow are correctly reproduced. The residual deviation at the outer edge of the profile is explained by the wakes of the struts that supported the model in the wind tunnel, absent from the numerical model.

Concluding remarks

Three lessons emerge from this confrontation of code_saturne with the reference naval benchmark.

First, on substance: code_saturne faithfully reproduces the hydrodynamic quantities of a deeply submerged appended hull. The resistance is predicted within $-0.1$ to $-3.7$ % of the tests over the whole speed range, and the wall pressure and the propeller-plane wake fall within the measurement uncertainties. code_saturne can be used to carry out naval hydrodynamics studies.

Second, on method: these results are obtained with a 459 k-cell pre-design mesh and a fully open-source chain - snappyHexMesh mesh converted to MED format by meshlane, parametric study in speed set up and driven by csauto, forces integrated by a user routine. The entry cost of a naval study with code_saturne is modest.

Finally, on the limits: the afterbody friction marks the boundary of what wall functions can deliver on this type of configuration. The adverse-pressure-gradient region that develops over the stern calls for a low-Reynolds integration down to the wall to predict the friction accurately - the natural next step, before opening onto free-surface and self-propelled configurations.

References

• N. C. Groves, T. T. Huang, M. S. Chang, Geometric Characteristics of DARPA SUBOFF Models, DTRC/SHD-1298-01, 1989. DTIC ADA210642
• T. T. Huang, H.-L. Liu, N. C. Groves, T. J. Forlini, J. N. Blanton, S. Gowing, Measurements of Flows Over an Axisymmetric Body with Various Appendages in a Wind Tunnel: the DARPA SUBOFF Experimental Program, 19th Symposium on Naval Hydrodynamics, 1992. DTIC ADA329405
• H.-L. Liu, T. T. Huang, Summary of DARPA SUBOFF Experimental Program Data, CRDKNSWC/HD-1298-11, 1998. DTIC ADA359226
• code_saturne v9.1, EDF. code-saturne.org
• meshlane, Simvia. github.com/simvia-tech/meshlane
• csauto, Simvia. github.com/simvia-tech/csauto