Quantification of resolution

Mesh resolution

The grid resolution corresponds to uniform $\Delta x^{+}=12.2,\ \Delta z^{+}=6.3$ and ranges from $\Delta y^{+}=0.16$ near the wall (first solution point) to $\Delta y^{+}=4.6$ at the centre of the channel. The near-wall Kolmogorov length scale, $\eta =\left({\bar {\rho }}\nu ^{3}/\epsilon \right)^{1/4}$ , in wall units is $\eta ^{+}=1.54$ .

Solution verification

The solution accurately captures the near wall and log-law behaviour of averaged streamwise velocity profile as shown in Fig. 2.

Figure 2: Average streamwise velocity,

Re_{\tau }=180

, vs data from Moser et al. (1999)

The turbulent stress profiles agree well with the data from Moser et al.

Figure 3: Turbulent stress profiles

The expected near wall behaviour of turbulent stresses are ${\widetilde {u^{\prime \prime 2}}},{\widetilde {w^{\prime \prime 2}}}\sim y^{2},\ {\widetilde {v^{\prime \prime 2}}}\sim y^{4}$ and ${\widetilde {u^{\prime }v^{\prime }}}\sim y^{3}$ . These stresses near the wall from the current set of simulations are shown below.

Figure 4: Near wall turbulent stress
${\widetilde {u^{\prime \prime }u^{\prime \prime }}}$
${\widetilde {v^{\prime \prime }v^{\prime \prime }}}$
${\widetilde {w^{\prime \prime }w^{\prime \prime }}}$
${\widetilde {u^{\prime \prime }v^{\prime \prime }}}$

The turbulent kinetic energy budget across the channel is also compared with data from Moser et al. (Fig. 5). The dashed black line denotes the residual of the budget.