We consider two-dimensional free surface flow caused by a pressure
wavemaker in a
viscous incompressible fluid of finite depth and infinite horizontal extent.
The
governing equations are expressed in dimensionless form, and attention
is
restricted to
the case δ[Lt ]ε[Lt ]1, where δ is the characteristic
dimensionless thickness of a Stokes
boundary layer and ε is the Strouhal number. Our aim is to provide
a
global picture of the flow, with emphasis on the steady streaming velocity.The asymptotic flow structure near the wavenumber is found to consist
of five
distinct vertical regions: bottom and surface Stokes layers of
dimensionless thickness
O(δ), bottom and surface Stuart layers of dimensionless
thickness O(δ/ε) lying outside
the Stokes layers, and an irrotational outer region of dimensionless
thickness O(1).
Equations describing the flow in all regions are derived, and the lowest-order
steady
streaming velocity in the near-field outer region is computed analytically.It is shown that the flow far from the wavemaker is affected by thickening
of the
Stuart layers on the horizontal length scale
O[(ε/δ)2], by viscous wave
decay on the
scale O(1/δ), and by nonlinear interactions on the scale
O(1/ε2). The analysis of the
flow in this region is simplified by imposing the restriction
δ=O(ε2), so that all three
processes take place on the same scale. The far-field flow structure is
found
to consist
of a viscous outer core bounded by Stokes layers at the bottom boundary
and water
surface. An evolution equation governing the wave amplitude is derived
and solved
analytically. This solution and near-field matching conditions are employed
to
calculate the steady flow in the core numerically, and the results are
compared with
other theories and with observations.
Flow structure and near-field dispersion in arrays of building-like obstacles
