Page 49 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

32 SLENDER STRUCTURES AND AXIAL FLOW

is the turbulence kinetic energy per unit mass. In view of the foregoing, this may also be
written as
co
K = E(k)dk = i@ii(k), (2.94)

in which E(k) is the energy spectrim function, i.e. the density of contributions to the
kinetic energy on the wavenumber magnitude axis (Batchelor 1960).
Some progress has been made in understanding the changing scales of turbulence,
as measured by its spectra and expressed in terms of the scalar wavenumber k. The
spectra at low k (large eddies) often retain something of the original unsteady laminar
flow; but, with increasing k, there is a continual stretching of the eddies by the medium
scales, which causes a transfer of turbulence energy to large k (small eddies) and also
randomizes the orientation of the eddies so that turbulence becomes locally isotropic. If
the Reynolds number is very large, the intermediate spectrum is inertial (Le. it sensibly
does not depend on viscosity), and it may be shown by dimensional analysis that the
spectrum is proportional to k-5/3. For the smallest eddies, where k > ~(E/V~)'/~, the
Kolmogoroff wavenumber, viscosity takes over and causes a decay of the cascading energy
with dissipation rate E to heat. This structure, as described in the foregoing, enables a
dramatic assumption to be made, namely that away from walls, the Reynolds stresses are
independent of urn. In this one respect, turbulent flow may often be easier to analyse than
laminar flow.
In analysing the boundary layer near walls, the so-called law ofthe wall is often used. In
this discussion, 2-D or axisymmetric boundary layers only are considered. Let UI be the
streamwise flow velocity in the boundary layer and x2 = y the distance perpendicularly
away from the wall. Then, near enough to the wall, UI = Ul(p, p, U,, y), where U, =
(~,,,/p)l/~ the skin-friction velocity and r,, is the shear stress at the wall; thus, U1
is
is independent of outer parameters, such as the overall boundary-layer thickness, the
free-stream velocity U, and the pressure gradient when not too large. Thus,

(2.95)

which is the law of the wall. Rotta (1962) predicts the functional form of B by noting that
changes in U1 in most of the region outside the viscous sublayer are independent of p,
because the shear stress is almost entirely due to -pw there. Dimensional analysis then
leads to (y/U,)(aUl/ay) = 1/K 2: 0.41, a universal constant named after von KBrmBn.
After integration, this gives

(2.96)

where B = 5.5 for a smooth wall. This proof applies to rough walls, 'fully rough walls'
(where p is unimportant even near the wall), and ribletted walls for which there is a drag
reduction. The only thing that changes is the value of B, which is lower for rough walls,
increasingly with the roughness, and slightly higher for ribletted walls.
The law of the wall has been accepted for the purposes of CFD (Computational Fluid
Dynamics), where it often becomes the inner boundary condition, but it must be noted
that the corresponding law for turbulence intensity is not exactly true when comparing,
say, boundary-layer flow and pipe flow; i.e. O/Ur # %(yUr/u).