Phase-Averaged Reynolds Stress and Turbulent Kinetic Energy

1. Turbulent kinetic energy k field

Flow Reynolds number Re = U_CL/ν = 4.6 x 10⁶
The overall structure of k field exhibit coherence with the phase averaged axial velocity U field; thin layers at the bow, growing along the model length, and becomes very thick at the stern
The core regions with high k values (e.g. k > 0.01) exist typically near at the model surface, at the center of the vortical flow regions, and near the free surface behind the model transom.
The apparent shapes of the k field and the locations of the core region vary in time, i.e. along with the pure yaw motion phase

k_mean is the average k value over the region where k > 0.001, representing the average k value within the turbulent flow.
In general, k_mean = 0.002 ~ 0.005, oscillating with γ, particularly at x = 0.135.
Period-mean value of k_mean is nearly constant along the model length with H₀ ≈ 0.003, with slightly larger H₀ values at x = 0.135 and 1.035.
In a mean sense, the flow overall turbulent intensity I = (2/3·k)^1/2 ≈ 0.045 in the flow, or a turbulent velocity fluctuation ≈ 4.5% of U_C.
k_mean oscillation amplitude is the maximum at x = 0.135 with H₂ = 0.0007 (19% of H₀), and then undulates with x, with a mean H₂ = 0.0002 (7% of H₀)

k_max is defined as the average k value over the turbulent core region where k > 0.01, representing the approximate maximum k value within the turbulent flow
In general, k_max = 0.01 ~ 0.02, oscillating with γ
Period mean k_max value decreases gradually along the model length, from H₀ = 0.014 at x = 0.135 to H₀ = 0.011 at x = 0.935, and then increases sharply to H₀ = 0.14 at x = 1.035, just behind the model.
Turbulent intensity I = 0.086 ~ 0.097 in the core region, or the turbulent fluctuation velocity about 9 ~ 10% of U_C
The oscillation amplitude of k_max is maximum at the bow with H₂ = 0.0015 (11% of H₀), and drops fast along the model length, H₂ ≈ 0.00025 (2% of H₀) at the aft body.

In average, the normal Reynolds stresses (uu^1/2, vv^1/2, ww^1/2) ≈ (5.4 %, 4.4%, 3%) of U_C and the root-mean-square (rms) of shear Reynolds stresses (uv^1/2, uw^1/2, vw^1/2) ≈ (3.5%, 2.2%, 1.7%) of U_C.
Locally, the maximum normal Reynolds stresses (uu^1/2, vv^1/2, ww^1/2) ≈ (12.9 %, 8.5%, 3.3%) of U_C and the root-mean-square (rms) of shear Reynolds stresses (uv^1/2, uw^1/2, vw^1/2) ≈ (8.8%, 3.6%, 2.8%) of U_C
The Reynolds stresses are anisotropic; typically uu and uv are the largest normal and shear Reynolds stresses, respectively. However, flow becomes more isotropic locally at the stern part and in the wake region.

average u_iu_j (k > 0.001 region)	maximum u_iu_j (k > 0.01 region)	Anisotropic tensor, b_ij^* (k > 0.01 region)

^* b_ij = u_iu_j/2k - δ_ij/3 is the deviatoric part of the Reynolds stress tensor. Close to zero b_ij indicates more isotropic stress tensor.