The rapid-distortion model of Hunt & Graham (1978) for the initial distortion of
turbulence by a flat boundary is extended to account fully for viscous processes. Two
types of boundary are considered: a solid wall and a free surface. The model is shown
to be formally valid provided two conditions are satisfied. The first condition is that
time is short compared with the decorrelation time of the energy-containing eddies,
so that nonlinear processes can be neglected. The second condition is that the viscous
layer near the boundary, where tangential motions adjust to the boundary condition,
is thin compared with the scales of the smallest eddies. The viscous layer can then
be treated using thin-boundary-layer methods. Given these conditions, the distorted
turbulence near the boundary is related to the undistorted turbulence, and thence
profiles of turbulence dissipation rate near the two types of boundary are calculated
and shown to agree extremely well with profiles obtained by Perot & Moin (1993) by
direct numerical simulation. The dissipation rates are higher near a solid wall than
in the bulk of the flow because the no-slip boundary condition leads to large velocity
gradients across the viscous layer. In contrast, the weaker constraint of no stress at a
free surface leads to the dissipation rate close to a free surface actually being smaller
than in the bulk of the flow. This explains why tangential velocity fluctuations parallel
to a free surface are so large. In addition we show that it is the adjustment of the
large energy-containing eddies across the viscous layer that controls the dissipation
rate, which explains why rapid-distortion theory can give quantitatively accurate
values for the dissipation rate. We also find that the dissipation rate obtained from
the model evaluated at the time when the model is expected to fail actually yields
useful estimates of the dissipation obtained from the direct numerical simulation at
times when the nonlinear processes are significant. We conclude that the main role of
nonlinear processes is to arrest growth by linear processes of the viscous layer after
about one large-eddy turnover time.
Dissipation of shear-free turbulence near boundaries
