We examine the inviscid flow generated around a body moving impulsively
from rest with a constant velocity U in a constant density gradient,
∇ρ0, which is assumed to be weak in the sense
ε=a[mid ]∇ρ0[mid ]
/ρ0[Lt ]1,
where a is the length scale of the body.
In the absence of a density gradient (ε=0), the flow is irrotational
and no force acts on the body. When 0<ε[Lt ]1, vorticity is generated
by a baroclinic torque and
vortex stretching, which introduce a rotational component into the flow.
The aim is
to calculate both the flow around the body and the force acting on it.When a two-dimensional body moves perpendicularly to the density gradient
U·∇ρ0=0, the density and
velocity field
are both steady in the body's frame of
reference and the vorticity field decays with distance from the body. When
a
three-dimensional body moves perpendicularly to the density gradient, the
vorticity field is regular in the main flow region,
[Dscr ]M, but is singular in a thin inner region
[Dscr ]I
located adjacent to the body and to the downstream-attached streamline,
and the
flow is characterized by trailing horseshoe vortices. When the body moves
parallel
to the density gradient U×∇ρ0=0,
the density field is unsteady in the body's
frame of reference; however to leading order the flow is steady in the
region
[Dscr ]M moving with the body for
Ut/a[Gt ]1. In the thin region
[Dscr ]I of thickness O(aε),
the
density gradient and vorticity are singular. When
U×∇ρ0=0 this singularity
leads
to a downstream ‘jet’ with velocities of
O(−(U·∇ρ0)
Ua/(ρ0U)) on the downstream attached
streamline(s). In the far field the flow is characterized by a sink of
strength
CM[Vscr ]
(U·∇ρ0)
/2ρ0, located at the origin, where
CM is the added-mass coefficient of the body
and
[Vscr ] is the body's volume.The forces acting on a body moving steadily in a weak density gradient
are
calculated by considering the steady relative velocity field in region
[Dscr ]M and evaluating the momentum flux far from the
body. When
U·∇ρ0=0, a lift force,
CL[Vscr ]
(U·∇ρ0)×U,
pushes the body towards the denser fluid, where the lift coefficient is
CL=CM/2 for
a
three-dimensional body, that is axisymmetric about U, and is
CL=(CM+1)/2
for a
two-dimensional body. The direction of the lift force is unchanged when
U is
reversed. A general expression for the forces on bodies moving in a weak
shear and
perpendicularly to a density gradient is calculated. When
U×∇ρ0=0, a drag force
−CD[Vscr ]
(U·∇ρ0)U retards
the body
as it moves into denser fluid, where the drag coefficient is
CD=CM/2,
for both two-
and three-dimensional axisymmetric bodies. The direction of the drag force
changes
sign when U is reversed. There are two contributions to the drag
calculation from the far field; the first is from the wake ‘jet’
on the
attached streamline(s) caused by the rotational component of the flow and
this
leads to an accelerating force. The second and larger contribution arises
from a
downstream density variation, caused by the distortion of the isopycnal
surfaces by
the primary irrotational flow, and this leads to a drag force.When cylinders or spheres move with a velocity U at arbitrary
orientation to the
density gradient, it is shown that they are acted on by a linear combination
of lift
and drag forces. Calculations of their trajectories show that they
initially slow down or accelerate on a length scale of order
ρ0/[mid ]∇ρ0[mid ] (independent
of
[Vscr ] and U) as they
move into regions of increasing or decreasing density, but in general they
turn and
ultimately move parallel to the density gradient in the direction
of increasing density gradient.
Prandtl–Batchelor flow past a flat plate with a forward-facing flap