This first application investigates a submerged generic DARPA (Defense Advanced Re-search Projects Agency) SUBOFF geometry as described in Groves et al. [1989] without appendages. The test case is frequently used during studies that focus on the propulsion and manoeuvring of deeply submerged submarines or in the vicinity of the free surface, e.g. Wang et al. [2020], Daum et al. [2017], Chase and Carrica [2013]. The present study is focused on the near surface operation. When operating close to the free surface, the wave field induced by the interaction of the dynamic pressure field with the free surface is unfavorable in terms of wave resistance as well as the signature of the submarine.
As illustrated in Fig. 5.1, the hull consists of three sections: A bow section (0 ≤ x∗1 ≤ 2D∗), a middle body (2D∗ ≤ x∗1 ≤ 0.745L∗S) as well as an aft section including cap (0.745L∗S ≤ x∗1 ≤ L∗S). The origin of the coordinate system is located at the front tip.
Here,L∗Sand D∗ represent the length of the submarine and the maximum body diameter.
The present study is performed for a turbulent flow at ReL=V1∗L∗S/ν∗b = 8.54·106 and Fn = V1∗/√
G∗L∗S = 0.3, based on the gravitational acceleration G∗, the inflow velocity V1∗ and the kinematic viscosity of the water ν∗b. The non-dimensional submergence reads L∗/D∗ = 1.1.
The three-dimensional domain has a length, height and width of 20L∗S, 10L∗Sas well as 5L∗S, where the inlet, bottom and outer boundaries are located five geometry-lengths away from the origin. A dimensionless wave length ofλ =λ∗/L∗ = 2πFn2 = 4.4 is expected and the wave elevation w.r.t. still water should be minimized, viz. Ω∗O = [−L∗S, L∗S/10,0]× [5L∗S,1.5L∗S,5L∗S], by modifying only the middle body of the underwater vehicle while conserving its displacement. The utilized unstructured numerical grid is displayed in Fig.
5.1 (b) and consists of approximately 4·106 control volumes. Due to symmetry, only half of the geometry is modeled in the lateral direction.
The fully turbulent simulations employ a wall function-basedk∗−ω∗SST model and con-vective terms for momentum and turbulence are approximated using the QUICK scheme.
x∗1 x∗2
20L∗S
5L∗S 10L∗S
5L∗S
L∗S D∗ 1.1D∗
gi∗ Ω∗O
(a) (b)
Figure 5.1: Submerged DARPA SUBOFF case (ReL = 8.54·106, Fn = 0.3): (a) Schematic drawing of the initial configuration and (b) unstructured numerical grid around the generic underwater vehicle and the free surface.
The CICSAM scheme is used for the compressive concentration transport. In contrast, the adjoint equations employ the QDICK and adCICSAM procedure. At the inlet, velocity and concentration values are prescribed, slip walls are used along the top, bottom as well as outer boundaries and a hydrostatic pressure boundary is employed along the outlet.
Along the mid ship plane a symmetry condition is declared. The dimensionless wall nor-mal distance of the first grid layer reads x+2 = y+ ≈ 30 and the free surface refinement employs approximately δx∗1/λ∗ =δx∗3/λ∗ = 1/50 cells in the longitudinal as well as lateral and δx∗2/λ∗ = 1/500 cells in the normal direction. The tangential resolution of the free surface region is refined within a Kelvin-wedge to capture the wave field generated by the submerged geometry, cf. Fig. 5.2. According to Alg. (2) the integration in pseudo-time applies a time step size based on Cotar = 0.4 embedded in five sub-cycles. The employed steepest descent approach uses a step size based on a prescribed maximum deformation for the initial design, viz. α∗di = L∗S/1000 based on one per mil of the reference length as maximum initial deformation, cf. Alg. 3. The step size is kept constant over the optimization, leading to a smooth convergence of the objective functional. Five optimiza-tions are performed: Four of them carefully increase the adjoint apparent viscosity from νˆcρ∗b/µ∗b = 1 to νˆcρ∗b/µ∗b = 1000. The relative decrease of the cost functional is de-picted over the number of gradient steps (performance evaluations) in Fig. 5.3. The fifth optimization employs the smallest apparent viscosity but neglects all three source terms scaling with the density and viscosity difference within the adjoint concentration equation and thus resembles a frozen material property approach. These source/sink terms are a strong motivation for the introduction of an apparent viscosity, as they drastically increase the coupling of the adjoint equation system and by that decrease the numerical stability.
Except for the optimization with frozen fluid properties, all optimizations achieve a similar order of magnitude for the minimization of the cost functional. The adjoint coupling terms resulting from a variation of the fluid properties seem to have a much stronger influence on the shape derivative than the adjoint apparent viscosity. An increase of the latter re-sults in a deviating convergence behavior and a somewhat smaller reduction of the cost
Figure 5.2: Submerged DARPA SUBOFF case (ReL = 8.54·106, Fn = 0.3): Numerical grid in the still water plane.
0 20 40 60 80 100
−40
−30
−20
−10 0
nopt [-]
(J∗ −J∗ini )/J∗ini ·102 [%] νˆc= 1
νˆc= 10 νˆc = 100 νˆc= 1000 νˆc= 1, frozen C
Figure 5.3: Submerged DARPA SUBOFF at ReL = 8.54·106, Fn = 0.3: Inverse concen-tration objective decrease over the number of performed shape updates during a steepest descent procedure. Four optimizations differ in the amplitude of the apparent viscosity and one optimization neglects all adjoint coupling terms that originate from the derivative of material properties.
functional. Interestingly, the fastest convergence is achieved with the highest apparent viscosity. The inferior influence of the apparent viscosity compared to the coupling terms is also demonstrated by the wave patterns displayed in Fig. 5.4. The three graphs are ex-tracted at three different lateral positions, viz. x∗3/D∗ = 0 (left), x∗3/D∗ = 2 (middle) and x∗3/D∗ = 4 (right) as indicated in Fig. 5.5. This observation is confirmed by the resulting hull geometries. Figures 5.6 and 5.7 present water lines and buttocks of the initial and the optimized geometry with νˆc = 1 against the optimization with frozen material properties (top) and the final shape resulting fromνˆc = 1000 (bottom). The slices underline the local character of the optimization approach as well as the strength of the proposed method wr.t. large shape and mesh deformations.
−1 0 1 2 3 -1
0 1 2
x∗1/L∗S [-]
x∗ FS,2/L∗S ·100[-]
initial
−1 0 1 2 3 -1
0 1
x∗1/L∗S [-]
x∗ FS,2/L∗S ·100[-]
νˆc= 1 νˆc= 1000
−1 0 1 2 3 -1
0 1
x∗1/L∗S [-]
x∗ FS,2/L∗S ·100[-]
νˆc= 1, frozen C.
Figure 5.4: Submerged DARPA SUBOFF case (ReL= 8.54·106, Fn = 0.3): Wave elevation for the initial and for three optimized shapes with νˆc = 1 (with and without adjoint two-phase coupling terms) as well as νˆc = 1000 along the main flow direction (x∗1) at three different lateral positions, viz. (left)x∗3/D∗ = 0, (center) x∗3/D∗ = 2 and (right) x∗3/D∗ = 4.
x∗FS,2/L∗S·2/100
-1 -0.5 0 0.5 1
Initial
Optimized
Figure 5.5: Submerged DARPA SUBOFF case (ReL = 8.54·106, Fn = 0.3): Normalized wave elevation for top) the initial geometry and bottom) the optimized hull resulting from an optimization with νˆc= 1.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.600.55
0.500.45 0.400.35 0.300.25 0.200.15 0.100.05 0.00
x∗1/L∗S [-]
x∗ 3/D∗ [-]
initial
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.600.55 0.500.45 0.400.35 0.300.25 0.200.15 0.100.05 0.00
x∗1/L∗S [-]
x∗ 3/D∗ [-]
νˆc= 1
Figure 5.6: Submerged DARPA SUBOFF case (ReL = 8.54·106, Fn = 0.3): Water lines for the initial geometry (black), two optimized geometries based onνˆc= 1 (orange) and νˆc = 1000 (blue) as well as the resulting slices for an optimization that employsνˆc = 1 with a frozen material property approach (purple).
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -0.60
-0.50 -0.40 -0.30 -0.20 -0.100.000.100.200.300.400.500.60
x∗1/L∗S [-]
x∗ 2/D∗ [-]
νˆc= 1, frozen mat.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-0.60 -0.50 -0.40 -0.30 -0.20 -0.100.000.100.200.300.400.500.60
x∗1/L∗S [-]
x∗ 2/D∗ [-]
νˆc= 1000
Figure 5.7: Submerged DARPA SUBOFF case (ReL = 8.54·106, Fn = 0.3): Buttocks for the initial geometry (black), two optimized geometries based onνˆc= 1 (orange) and νˆc = 1000 (blue) as well as the resulting slices for an optimization that employs νˆc= 1 with a frozen material property approach (purple).