232
STtVEN A. ORSLAG A N D YIH-HO P A 0
incompressibility constraint. This is done by writing
(13)
v‘(x) = V x A’(x)
where the vector potential A‘(x) is chosen to achieve the desired local energy
spectrum and turbulence intensity. It suffices to choose A’(x) to be of the
form
(14)
A‘(x) = [I(r)]”’B(x)
where the turbulence intensity function f(r) is a nonrandom function of only
the distance from the wake axis x I , and the fluctuation component B(x) is
chosen as a realization of a homogeneous, isotropic random field with
specified isotropic energy spectrum, as done by Orszag and Patterson
(1972). If the intensity function is chosen to vanish outside a radius ro from
the wake axis, the resulting turbulent velocity field is, by (13). nonturbulent
outside the cylinder of radius ro centered on the wake axis.
In summary. our technique of imposing the initial conditions allows arbitrary mean velocity profile, turbulence intensity profile, and local turbulence
energy spectrum.
4. MOMENT~JMLESS WAKE
The wake model of Section 2 and the numerical methods of Section 3
permit simulation of the turbulent wake of a self-propelled body. Free-slip
houndary conditions are applied al x j = 0, L3 and the spectral cutoffs
K l = K 2 = 16, K 3 = 32 are used. The’ following choice of initial parameters
was made: L~ = L~ = L~ = 2, ird(r) = vo sin(r/rl)/(r/rI) where r l = $.
I(r) = max[ 1 - r2/4.5, 01, and turbulence energy spectrum E(k) =
Ak4 exp( - BkZ) (cb Orszag and Patterson, 19721. The initial conditions are
chosen to match as closely as possible the results of Naudaschcr (1965) and
Wang (1965) at a location four body diameters behind a self-propelled disk
in a wind tunnel. In the numerical simulations, the viscosity is chosen to be
v = 0.005, so that the Reynolds number of the simulation is roughly 13,000
(run 2a). and v = 0.003 with Reynolds number roughly 22,000 (run 3). In
comparison, the laboratory experiments of Naudascher and Wang were run
at Reynolds numbers of roughly 55.000.
Some measure of the degree of complication of the flow that is simulated
is given by Figs. 3 and 4. I n Fig. 3, contours of the run 3 axial mean velocity
arc plotted at t = 0.8, corresponding to about seven diameters downstream
from thc body upon making identification of body velocity by the ratio
max[rld(r)J/llo at the station at x / D = 4 with the experimental results. The
axial mean is determined by averaging over the s1 direction. In Fig. 4,
contours of the run 3 x, component of vorticity are plotted at t = 0.8.
STtVEN A. ORSLAG A N D YIH-HO P A 0
incompressibility constraint. This is done by writing
(13)
v‘(x) = V x A’(x)
where the vector potential A‘(x) is chosen to achieve the desired local energy
spectrum and turbulence intensity. It suffices to choose A’(x) to be of the
form
(14)
A‘(x) = [I(r)]”’B(x)
where the turbulence intensity function f(r) is a nonrandom function of only
the distance from the wake axis x I , and the fluctuation component B(x) is
chosen as a realization of a homogeneous, isotropic random field with
specified isotropic energy spectrum, as done by Orszag and Patterson
(1972). If the intensity function is chosen to vanish outside a radius ro from
the wake axis, the resulting turbulent velocity field is, by (13). nonturbulent
outside the cylinder of radius ro centered on the wake axis.
In summary. our technique of imposing the initial conditions allows arbitrary mean velocity profile, turbulence intensity profile, and local turbulence
energy spectrum.
4. MOMENT~JMLESS WAKE
The wake model of Section 2 and the numerical methods of Section 3
permit simulation of the turbulent wake of a self-propelled body. Free-slip
houndary conditions are applied al x j = 0, L3 and the spectral cutoffs
K l = K 2 = 16, K 3 = 32 are used. The’ following choice of initial parameters
was made: L~ = L~ = L~ = 2, ird(r) = vo sin(r/rl)/(r/rI) where r l = $.
I(r) = max[ 1 - r2/4.5, 01, and turbulence energy spectrum E(k) =
Ak4 exp( - BkZ) (cb Orszag and Patterson, 19721. The initial conditions are
chosen to match as closely as possible the results of Naudaschcr (1965) and
Wang (1965) at a location four body diameters behind a self-propelled disk
in a wind tunnel. In the numerical simulations, the viscosity is chosen to be
v = 0.005, so that the Reynolds number of the simulation is roughly 13,000
(run 2a). and v = 0.003 with Reynolds number roughly 22,000 (run 3). In
comparison, the laboratory experiments of Naudascher and Wang were run
at Reynolds numbers of roughly 55.000.
Some measure of the degree of complication of the flow that is simulated
is given by Figs. 3 and 4. I n Fig. 3, contours of the run 3 axial mean velocity
arc plotted at t = 0.8, corresponding to about seven diameters downstream
from thc body upon making identification of body velocity by the ratio
max[rld(r)J/llo at the station at x / D = 4 with the experimental results. The
axial mean is determined by averaging over the s1 direction. In Fig. 4,
contours of the run 3 x, component of vorticity are plotted at t = 0.8.