Overview🔗

• Replicates Poletto et al.’s [58] setup and settings for Direct numerical simulation of turbulent channel flow up to Ret=590 [51]
• References: Van Driest (1956) [84], Smagorinsky (1963) [64], Kim et al. (1987) [28], Moser et al. (1999) [51], and Poletto et al. (2013) [58] (Synthetic turbulence inflow generator)
• See the resources section for additional data files

Flow physics:

• Internal flow
• Moderate Reynolds number
• Spanwise direction: Statistically homogeneous
• Streamwise and channel-height directions: Statistically developing
• Newtonian, single-phase, incompressible, non-reacting

Solver:

• pimpleFoam

Tutorial case:

• $FOAM_TUTORIALS/incompressible/pimpleFoam/LES/planeChannel

Physics and Numerics🔗

Physical domain:

• The case is a statistically-developing internal flow through parallel smooth walls which are two characteristic-length apart.
  • \(x\): Longitudinal direction (Mean-flow direction)
  • \(y\): Vertical direction (Wall-normal direction)
  • \(z\): Spanwise direction (Statistically homogeneous direction)
  • \(O\): Origin at the left-bottom corner of the numerical domain

Physical modelling:

• Reynolds number based on friction velocity: \(\text{Re}_{u_\tau} = \vert \mathbf{U}_\tau \vert \delta / \nu_\text{fluid} = 395\) [-]
  • (Estimated) Friction velocity: \(\mathbf{U}_\tau = (1.0, 0.0, 0.0)\), and \(\vert \mathbf{U}_\tau \vert = u_\tau = 1.0\) [m⋅s^-1]
  • Characteristic length (Channel half-height): \(\delta = 1.0\) [m]
  • Kinematic viscosity of fluid: \(\nu_{\text{fluid}} \approx 0.002532\) [m²⋅s^-1]
  • Bulk velocity of flow: \(\mathbf{U}_b = (17.55, 0.00, 0.00)\) [m⋅s^-1]
• Turbulence model: Large eddy simulation with the Smagorinsky sub-filter scale model utilising the van Driest wall-damping function. The sub-filter scale model constants:
  • \(C_k \approx 0.02655\)
  • \(C_e = 1.048\)
  • \(C_s \approx 0.065\)
    • \(C_s = (C_k \{C_k/C_e\}^{0.5} )^{0.5}\)

Numerical domain modelling:

• Shape: Rectangular prism
• Dimensions: \((x, y, z) = (20.0\pi, 2.0, \pi)\) [m]
• Sketch:

Spatial domain discretisation:

• Mesh type: Rectangular cuboid mesh
• Mesher: blockMesh
• Number of nodes, \(N\): \((N_x, N_y, N_z) = (500, 46, 82)\) [nodes]
• Spatial resolution \((\Delta)\) distribution:
  • Uniform in \((x, z)\)-directions
  • Stretched in \((y)\)-direction; clustered nearby walls
• Uniform mesh particulars:
  • \(\Delta_x^+ = (\Delta_x u_\tau )/\nu_{\text{fluid}} \approx 49.6\) [-]
  • \(\Delta_z^+ \approx 15.1\) [-]
• Wall-normal mesh particulars:
  • Simple grading expansion ratio: 25.0 [-] (From top to bottom wall, the ratio is 0.04)
  • First wall-normal node height: \(\Delta_y^+ \approx 1.1\)
  • Mesh details:

Temporal domain discretisation:

• Time-step size: \(\Delta_t = 0.004\) [s]
• Estimated Courant-Friedrichs-Lewy (CFL) number based on \(\{ \overline{u_x} \}_{y^+ = 392} = 20.133\)[m⋅s^-1]: CFL \(\approx 0.64\)

Equation discretisation:

Spatial derivatives and variables:

• Gradient: Gauss linear
• Divergence: Gauss linear
• Laplacian: Gauss linear orthogonal
• Surface-normal gradient: orthogonal

Temporal derivatives and variables:

• ddtSchemes: backward

Numerical boundary conditions:

• Velocity, \(\mathbf{U}\)

• Pressure, p

• Turbulent kinematic viscosity, nut (i.e. \(\nu_t\))

Solution algorithms and solvers:

• Pressure-velocity: PISO
• Parallel decomposition of spatial domain and fields: scotch
• The bandwidth of the coefficient matrix is minimised by renumberMesh
• Linear solvers:

Initialisation and sampling:

• Computation time for a single domain pass-through based on \({ \overline{U_x} }_{y^+ = 392} = 20.133\)) [m²⋅s^-1] \(\approx 3.121\) [s]
• Initialisation pass-throughs = \(\approx 2.7\) [58]
• Sampling pass-throughs = \(\approx 24.5\) [58]

Results🔗

List of metrics:

• Prescribed vs. reproduced Reynolds stress tensor components at inlet patch
• \(\overline{u^\prime u^\prime}\) downstream development vs. \(x/ \delta\)
• \(\overline{v^\prime v^\prime}\) downstream development vs. \(x/ \delta\)
• \(\overline{u^\prime v^\prime}\) downstream development vs. \(x/ \delta\)
• Surface skin friction coefficient \(\mathrm{C}_f\) vs. \(x/ \delta\)
• Streamwise mean flow speed and Reynolds stress tensor components at uniform-interval downstream profiles
• Streamwise vorticity \(\omega_x\) at \(x/ \delta = 0.1\)
• Streamwise vorticity \(\omega_x\) at \(x/ \delta = 1.0\)
• Metrics are time and spanwise averaged
• \(< \cdot >\) is the time-averaging operator

Resources🔗

Note: Links will take you to the University of Texas at Austin website

Datasets for verifications (plain text)🔗

Reynolds stress tensor profiles:

• Numerical (DNS) [51]

Mean velocity profiles:

• Numerical (DNS) [51]

Two-point velocity correlations (i.e. Auto- and cross-correlation functions):

• Numerical (DNS) [51]