**3. Numerical Procedure**

We consider an incompressible Newtonian fluid in the plane Poiseuille (channel) geometry, driven by a constant volumetric flux *Q*. The *x*, *y* and *z* coordinates are aligned with the streamwise, wall-normal and spanwise directions, respectively. Periodic boundary conditions are imposed in the *x* and *z* directions with fundamental periods *Lx* and *Lz*, and a no-slip boundary condition is imposed at the walls *y* = ±*h*, where *h* = *Ly*/2 is the half-channel height. The laminar centreline velocity for a given volumetric flux is given as *Ucl*,*lam* = (3/4)*Q*/*h*. Using the half-height *h* of the channel and the laminar centreline velocity *Ucl*,*lam* as the characteristic length and velocity scales, respectively, the non-dimensionalised Navier–Stokes equations are given as

$$
\nabla \cdot \boldsymbol{u} = 0,\tag{2}
$$

$$\frac{\partial \mu}{\partial t} + \boldsymbol{u} \cdot \nabla \boldsymbol{u} = -\nabla p + 1/(\text{Re}\_c)\nabla^2 \boldsymbol{u}.\tag{3}$$

Here, we define the Reynolds number for the given laminar centreline velocity as *Rec* = *Ucl*,*lamh*/*ν*, where *ν* is the kinematic viscosity of the fluid. Characteristic inner scales are the friction velocity *u<sup>τ</sup>* = (*τw*/*ρ*) and the near-wall length scale, or wall unit, *δν* = *ν*/*uτ*, where *ρ* is the fluid density and *τ<sup>w</sup>* is the time- and area-averaged wall shear stress. Quantities non-dimensionalised by the inner scales are denoted with a superscript '+'. The friction Reynolds number is then defined as *Re<sup>τ</sup>* = *uτh*/*ν* = *h*/*δν*. For the current simulations, friction Reynolds numbers of *Re<sup>τ</sup>* = 70 and 85 are considered. Simulations are performed using the open source code ChannelFlow written and maintained by Gibson [40]. We focus on a domain of *Lx* × *Ly* × *Lz* = 13.64 *πh* × 2 *h* × 3.64 *πh*. These dimensions correspond to *L*<sup>+</sup> *<sup>x</sup>* <sup>×</sup> *<sup>L</sup>*<sup>+</sup> *<sup>z</sup>* <sup>≈</sup> <sup>3000</sup> <sup>×</sup> 800 for *Re<sup>τ</sup>* <sup>=</sup> 70, and *<sup>L</sup>*<sup>+</sup> *<sup>x</sup>* <sup>×</sup> *<sup>L</sup>*<sup>+</sup> *<sup>z</sup>* ≈ 3640 × 970 for *Re<sup>τ</sup>* = 85. A numerical grid system is generated on *Nx* × *Ny* × *Nz* (in *x*, *y*, and *z*) meshes, where a Fourier–Chebyshev–Fourier spectral spatial discretisation is applied to all variables. A resolution of (*Nx*, *Ny*, *Nz*) = (196, 73, 164) is used for both Reynolds numbers. The numerical grid spacing in the streamwise and spanwise direction are Δ*x*<sup>+</sup> *min* <sup>≈</sup> 15.3 (18.6) and <sup>Δ</sup>*z*<sup>+</sup> *min* ≈ 4.9(5.9) for *Reτ* = 70 and (*Reτ* = 85) cases. The nonuniform Chebyshev spacing used in the wall-normal direction results in Δ*y*<sup>+</sup> *min* <sup>≈</sup> 0.07 (0.08) at the wall and <sup>Δ</sup>*y*<sup>+</sup> *max* ≈3.0 (3.7) at the channel centre for *Re<sup>τ</sup>* = 70 and (*Re<sup>τ</sup>* = 85) cases. For the computation time, 50 × 103 strain times (> <sup>25</sup>*Rec*) is chosen to attain meaningful statistics.

The present experiment provides temporal information for the flow, and therefore for a comparison of the DNS and experimental data, temporal information from the DNS data is extracted. To obtain reliable statistics, nine wall locations are chosen at the wall on the top and on the bottom walls of the computational domain. These locations are selected in such a way that each spatial location is not correlated with the others [22]. The streamwise/spanwise spatial locations correspond to the combinations of three *<sup>x</sup>*<sup>+</sup> locations and three *<sup>z</sup>*<sup>+</sup> locations: *<sup>x</sup>*<sup>+</sup> <sup>≈</sup> 505, 1500 and 2495; *<sup>z</sup>*<sup>+</sup> <sup>≈</sup> 151, 400 and 649 for *Re<sup>τ</sup>* = 70, and *<sup>x</sup>*<sup>+</sup> <sup>≈</sup> 613, 1820 and 3027; *<sup>z</sup>*<sup>+</sup> <sup>≈</sup> 183, 485 and 787 for *Re<sup>τ</sup>* = 85. The instantaneous wall shear stress is obtained by using the streamwise velocity gradient information at *<sup>y</sup>*<sup>+</sup> <sup>≈</sup> 1, although no difference in its value was observed between *<sup>y</sup>*<sup>+</sup> <sup>≈</sup> 1 and lower *<sup>y</sup>*<sup>+</sup> locations.
