Turbulent flat plate

• Skin friction. Downstream of the leading-edge region the curve follows the reference of the resource, computed with the same model on a grid of 545 × 385 points.
• Law of the wall. In wall units the velocity profile leaves the sublayer line $u^+ = y^+$ at the first nodes and follows the logarithmic law with the model’s von Kármán constant of 0.41 further out. Where two elements meet, the chart shows both nodal values: the small jumps between them are the discontinuous approximation, not an error of the plot.
• Eddy viscosity. It is zero at the wall, several hundred times the molecular viscosity in the outer layer, and returns to the small free-stream value outside the boundary layer. At that edge $\tilde\nu$ undershoots below zero in a few nodes, which is what the negative branch of the model is for.

The model is integrated to the wall, so the first element must reach into the viscous sublayer: the table gives $y^+$ of the first solution node off the wall. With a high-order element the requirement is on the nodes inside the element, not on the element height, which is why 10 elements across the layer are enough here. To refine:

gyre run flat_plate_sa --set mesh.params.nx=20 --set mesh.params.ny=12 --set mesh.params.h_wall=3e-5
gyre run flat_plate_sa --set discretization.order=3

The turbulence model is part of the differentiated residual, and so is the wall distance:

gyre run flat_plate_sa --gradients --of cd --wrt mach,re,chi_inf

returns the total derivatives of the drag from one adjoint solve. They are verified against finite differences, including a displacement of the wall, under Turbulence closure.

Turbulent cases start from a uniform free stream with a small pseudo-time step and a moderate growth (cfl0: 1, cfl_growth: 3), and they keep the piecewise-constant multigrid level (levels: [2, 1, 0]): on a wall-resolved mesh the linear solves need it (why).

The line smoother (solver.smoother: line) cuts the Krylov iterations of linear solves to a tight tolerance to less than half on this mesh; the flow solve from the free stream does not get faster with it, and a single run with one adjoint solve is not where it pays (measured). It is meant for a case that is kept, in a session or a design loop, for many derivative evaluations at one flow state after another:

solver:
  levels: [2, 1, 0]
  smoother: line
  precond_lag: 2      # assemble the blocks every second step of a flow solve

gyre run prints, under the line of the case, how many lines the smoother found on the mesh, and gyre mesh info reports the same for a mesh file before anything is run.