Inviscid flow around bodies moving in weak density gradients without

buoyancy effects
Authors: EAMES I. , HUNT J. C. R. 
Published: July 2002

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.