Turbulent plane channel flow with smooth walls

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:

Keywords: Detached eddy simulation (DES), Large eddy simulation (LES), Synthetic turbulence generation Smagorinsky sub-filter scale model, pimpleFoam, pisoFoam

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: \( \text{Re}_{u_\tau} = |\mathbf{U}_\tau| \delta / \nu_\text{fluid} = 395\) [-]
    • (Estimated) Friction velocity: \( \mathbf{U}_\tau = (1.0, 0.0, 0.0)\), and \( | \mathbf{U}_\tau | = u_\tau = 1.0 \) [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 -
    • 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 algorithm
  • 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 solvers Gauss Seidel Smoother 0.0
p GAMG Solver Gauss Seidel Smoother 0.0
nuTilda Smooth solvers Gauss Seidel Smoother 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 = \( \approx 2.7 \) [60]
  • Sampling pass-throughs = \( \approx 24.5 \) [60]

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):