Overview

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:

Tutorial case:

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:

Numerical domain (not in scale)

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:
Mesh (Front part) Mesh

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:

Temporal derivatives and variables:

Numerical boundary conditions:

  • Velocity, \(\mathbf{U}\)
Patch Condition Value [m⋅s-1]
Inlet turbulentDFSEMInlet (17.55, 0.00, 0.00)
Outlet inletOutlet (0.0, 0.0, 0.0)
Sides (\(z\)-dir) cyclic -
Walls (\(y\)-dir) fixedValue (0.0, 0.0, 0.0)
  • Pressure, p
Patch Condition Value [m2⋅s-2]
Inlet zeroGradient -
Outlet fixedValue 0.0
Sides (\(z\)-dir) cyclic -
Walls (\(y\)-dir) zeroGradient -
  • Turbulent kinematic viscosity, nut (i.e. \(\nu_t\))
Patch Condition Value [m2⋅s-1]
Inlet calculated -
Outlet calculated -
Sides (\(z\)-dir) cyclic -
Walls (\(y\)-dir) zeroGradient -

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:
Field Linear Solver Smoother Relative Tolerance
U smooth GaussSeidel 0.0
p GAMG GaussSeidel 0.0
nuTilda smooth GaussSeidel 0.0

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
Prescribed vs. reproduced Reynolds stress tensor at inlet patch (Poletto et al., Fig. 4)  
<u'u'>-component of Reynolds stress tensor - Downstream development (Poletto et al., Fig. 14)
<v'v'>-component of Reynolds stress tensor - Downstream development (Poletto et al., Fig. 15)
<u'v'>-component of Reynolds stress tensor - Downstream development (Poletto et al., Fig. 13)
Longitudinal skin friction coefficient at top patch - Downstream development (Poletto et al., Fig. 9)
Longitudinal skin friction coefficient at bottom patch - Downstream development (Poletto et al., Fig. 9)
Streamwise vorticity component at y/δ=0.05 (Poletto et al., Fig. 11)
Streamwise vorticity component at y/δ=1.0 (Poletto et al., Fig. 12)

Resources

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

Datasets for verifications (plain text)

Reynolds stress tensor profiles:

Mean velocity profiles:

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