Vortex Pair Simulations
Introduction
Numerical simulations of two-dimensional vortex pairs in conjunction with radially-dependent eddy viscosity distributions were performed in order to test the turbulent diffusion hypothesis and to study the effects of buoyancy. The simulations make use of the analytically-determined eddy viscosity distribution described in the vortex diffusion write-up. The analytic eddy viscosity distribution required some modifications since it was was derived for a single vortex and it diverges like r^2 for large distances from the vortex center, and like 1/(1-t/tau) for t-->tau (with tau being ~10t*). The fix was remove the time dependence and to apply a radially-dependent damping function, so that the eddy viscosity would actually decrease at large distances from the vortex center. The damping function is derived from a hyperbolic tangent function as followsD(r) = 0.5*[ 1 - tanh(2.5*(r-r_c)/r_w) ]
where r_c=b, r_w=b/5, where b is the instantaneous vortex spacing. The damped eddy viscosity distributions from the two vortices was superimposed using a special non-overlapping summation where the eddy viscosity at any field point is simply equal to the value due to the closest vortex. The end result is shown in the image below
While the basic shape of the eddy viscosity distribution is fixed in time,
it translates with the vortex pair as it descends. To do this, the vortex
cores are tracked in time by finding the two locations of minimum pressure.
The simulations target a B747 wake with initial vortex spacing of b0=46 m, an initial altitude of 279 m, an initial circulation of gamma0=550 m^2/s, and an initial vortex core radius of R=1 m. The initial velocity field is simply the superposition of two Burnham-Hallock vortices.
The simulations were performed on a computational domain extending from the surface to 560 m in altitude (z direction), and spanning 375 m in the horizontal (x) direction. A grid with Nx=1875, Nz=2800 was used so that the mesh spacing is 0.2 m in both directions. Slip walls were used at the top and bottom, whereas periodic boundary conditions were used in the horizontal directions. The slip wall condition at the top and the periodic conditions in the horizontal are slightly problematic in that they imply the presence of non-physical image vortices. However, the effect of the image vortices was minimized by making the domain much larger than the minimal dimensions required to resolve the primary vortices. In order to study the effects of buoyancy, two simulations were performed, one with an isothermal atmosphere, and one with a neutral atmosphere.
Results
The simulations target a B747 wake with initial vortex spacing of b0=46 m, an initial altitude of 279 m, an initial circulation of gamma0=550 m^2/s, and an initial vortex core radius of R=1 m. The initial velocity field is simply the superposition of two Burnham-Hallock vortices.
The simulations were performed on a computational domain extending from the surface to 560 m in altitude (z direction), and spanning 375 m in the horizontal (x) direction. A grid with Nx=1875, Nz=2800 was used so that the mesh spacing is 0.2 m in both directions. Slip walls were used at the top and bottom, whereas periodic boundary conditions were used in the horizontal directions. The slip wall condition at the top and the periodic conditions in the horizontal are slightly problematic in that they imply the presence of non-physical image vortices. However, the effect of the image vortices was minimized by making the domain much larger than the minimal dimensions required to resolve the primary vortices. In order to study the effects of buoyancy, two simulations were performed, one with an isothermal atmosphere, and one with a neutral atmosphere.
Results
Case 1 uses an isothermal atmosphere with temperature T=300k. This gives a
buoyancy period of 352 seconds, or equivalently 5.9 minutes. Case 2 uses
a temperature lapse rate of -9.77 k/km, which produces a neutral atmosphere.
Circulation time histories from the two simulations, (computed on a circle
of radius 15 m), are shown in the following plot.
The circulation time histories from the two simulations are rather similar
and both are in reasonable agreement with the expected linear decay.
The gradual flattening of the simulation results at later times is due
to the lack of inclusion of an eddy viscosity time dependence. The
analytic vortex diffusion model for strict linear decay requires a time
dependence that is inversely proportional to the instantaneous circulation.
This feature does not seem physically correct (at least later in the decay
process) and thus it was not included in the simulations. The analytic
diffusion model can also be solved without the eddy viscosity time dependence
and this result is included as the dashed curve on the plot. The
stably-stratified simulation is seen to agree almost perfectly with the
static eddy viscosity diffusion model results. Although it might be
fortuitous, this result would seem to suggest that the simple axisymmetric
diffusion model for a single vortex is suprisingly accurate when applied to
the non axisymmetric flow in a vortex pair.
The effectiveness of the eddy viscosity at diffusing vorticity away from the vortex centers can be seen through a time sequence of vorticity images, thresholded at a very small value. Such a time sequence taken from the unstratified simulation, starting at t=0 and sampled every 20 seconds is show below.
The effectiveness of the eddy viscosity at diffusing vorticity away from the vortex centers can be seen through a time sequence of vorticity images, thresholded at a very small value. Such a time sequence taken from the unstratified simulation, starting at t=0 and sampled every 20 seconds is show below.
The min and max values for the color scale are set to extremely small values
in these plots in order to visualize the weak vorticity away from the vortex
cores. For reference, the maximum vorticity value at t=0 is 170 1/sec and
thus the contour level at 0.2 /sec is at the 0.1% level.
The present choice of vorticity color scale makes the cores look much, much
larger than their actual size. From these images it is clear that vorticity
is rather effectively diffused away from the cores, but then is deposited
weakly in rings near the radius where the damping function takes over
to decrease the eddy viscosity. Analytic solutions generated with the
exact eddy viscosity distribution do not show this effect since the r^2
growth in eddy viscosity continues to diffuse vorticity away from the cores
at all radii. Due to the addition of the damping function in the simulations,
we are left with a very slight accumulation of vorticity far from the vortex
cores. The accumulation occurs outside the 15 m circle where circulation
is computed and thus does not affect the measured circulation decay.
Similar to the circulation decay, the simulations produces velocity profiles that look very much like what is found in experimental data. The following plot shows a comparison of the initial Burnham-Hallock profile and the profile taken from the unstratified simulation at t=100 s.
Similar to the circulation decay, the simulations produces velocity profiles that look very much like what is found in experimental data. The following plot shows a comparison of the initial Burnham-Hallock profile and the profile taken from the unstratified simulation at t=100 s.
The vortex core size is seen to remain fixed and the overall profile shape
remains rather close to the Burnham-Hallock initial condition.
One of the main objectives of this simulation is to aid in the understanding of how buoyancy affects the evolution of the vortex pair. The following image sequence of the perturbation potential temperature, taken from the stably-stratified simulation, is rather useful in this regard. These images start at t=0 and are sampled at 20 second intervals.
One of the main objectives of this simulation is to aid in the understanding of how buoyancy affects the evolution of the vortex pair. The following image sequence of the perturbation potential temperature, taken from the stably-stratified simulation, is rather useful in this regard. These images start at t=0 and are sampled at 20 second intervals.
The simulations were initialized with no potential temperature perturbations and
thus the image at t=0 is blank. As soon as the simulation begins, a strong
downwash is produced between the vortices and weaker upwashes are produced
on the outer edges of the pair. These vertical flows serve to transport
lighter fluid (higher potential temperature) to the center and below the
vortices, and heavier (lower potential temperature) along the edges and on
top of the vortices. The vortices then wind in alternating layer of heavy
and light fluid from the top and bottom to form a spiral pattern. This
mechanism appears to be rather different from the previous view incorporated
in viper where the wake oval would retain its initial value of potential
temperature in the absence of turbulent entrainment or detrainment across
the oval boundary. In reality it appears that the potential temperature in
the wake oval is largely controlled by ingestion of lighter fluid from above
the vortex pair.
The vortex cores achieve a reasonably constant value of potential temperature that is different from that in wake oval. Part of this effect is due to the heat generated by viscous dissipation near the core (this effect can be measured in the neutrally-stable simulation). However, the bulk of this effect seems to be due to trapping of fluid within the core, as we have postulated earlier.
Vortex trajectories from the unstratified and stratified simulations are shown in the following two plots.
The vortex cores achieve a reasonably constant value of potential temperature that is different from that in wake oval. Part of this effect is due to the heat generated by viscous dissipation near the core (this effect can be measured in the neutrally-stable simulation). However, the bulk of this effect seems to be due to trapping of fluid within the core, as we have postulated earlier.
Vortex trajectories from the unstratified and stratified simulations are shown in the following two plots.
There is considerable difference between the trajectories for the two cases.
In the unstratified case the vortices gradually separate as they descend. This
behavior is caused by ground effect and it becomes more pronounced as the
vortices move closer to the ground. The stably stratified case shows the
opposite trend where the vortices move together as they descend. This
behavior appears to be the result of baroclinic generation of secondary
vorticity, which induced a net flow inward. It also appears that the simulation
overestimates this effect. Evidence for this can be seen in the following plot
which compares the simulated trajectories with lab measurements.
Here we see that the unstratified case agrees reasonably well with the
laboratory data. The stably-stratified case, on the other hand, shows a
premature decrease in the descent rate followed by an increase after t*=5.
The increase at later times appears to be the result of enhanced vertical
induced velocity due to the fact that the cores have moved closer together
during this time period. Note that the data shows the opposite trend, where
the descent rate should be slowing down (and even reversing) as buoyancy
forces act against the downward motion.
The secondary vorticity can be seen in the following image sequence.
The secondary vorticity can be seen in the following image sequence.