Introduction | Tank – How to | Tank – Examples | Atmosphere – Examples | **Theory** | For Teachers | Wiki

We develop here some background theory for curved flow, appropriate to the radial inflow tank experiment and to the atmospheric example of hurricane flow.

**Tank experiment
**

__Force balances in the momentum equations__

In the limit in which the tank is rotated rapidly, parcels of fluid circulate around many times before falling out through the drain hole (see, examples section); the pressure gradient force directed radially inwards (set up by the free surface tilt) is in large part balanced by a centrifugal force directed radially outwards.

If V

_{θ}is the azimuthal velocity in the absolute frame of reference and v

_{θ}is the azimuthal speed relative to the tank (measured using the camera co-rotating with the apparatus) then:

_{θ}= v

_{θ}+ Ωr (1)

where Ω is the rate of rotation of the tank in radians per second. Note that Ωr is the azimuthal speed of a particle stationary relative to the tank at radius r from the axis of rotation.

We now consider the balance of forces in the vertical and radial directions, expressed first in terms of the absolute velocity V_{θ} and then in terms of the relative velocity v_{θ}.

__Vertical force balance__

We suppose that hydrostatic balance pertains in the vertical. Integrating in the vertical and noting that the pressure vanishes at the free surface (actually p = atmospheric pressure at the surface, which here can be taken as zero), and with ρ and g assumed constant, we find that:

where H(r) is the height of the free surface (where p = 0) and we suppose that

z = 0 (increasing upwards) on the base of the tank.

**Fig.1** The outer radius of the bucket is r_{1} and the height of the free surface H.

__Radial force balance in the non-rotating frame__

If the pitch of the spiral traced out by fluid particles is tight (i.e. in the limit that v_{r}/v_{θ} << 1 (where v_{r} is the radial speed and v_{θ} is the azimuthal speed) appropriate when Ω is sufficiently large) then the centrifugal force directed radially outwards acting on a particle of fluid is balanced by the pressure gradient force directed inwards associated with the tilt of the free surface. This radial force balance can be written in the non-rotating frame thus:

_{θ}

^{2}/r = (1/ρ)(∂p/∂r) (3)

Using Eq.2, the radial pressure gradient can be directly related to the gradient of free surface height enabling the force balance to be written

_{θ}

^{2}/r = g∂H/∂r (4)

__Radial force balance in the rotating frame __

Using Eq.1, we can express the centrifugal acceleration in Eq.4 in terms of velocities in the rotating frame thus:

_{θ}

^{2}/r = (v

_{θ}+Ωr)

^{2}/r = v

_{θ}

^{2}/r + 2Ω v

_{θ}+ Ω

^{2}r

Hence Eq.4 becomes

_{θ}

^{2}/r + 2Ωv

_{θ}+ Ω

^{2}r = g∂H/∂r (5)

The above can be simplified by measuring the height of the free surface relative to that of a reference parabolic surface Ω^{2}r^{2}/(2g) as follows

^{2}r

^{2}/(2g)

Then, since ∂η/∂r = ∂H/∂r – Ω^{2}r/g, Eq.5 can be written in terms of η thus:

Eq.4 (non-rotating) and Eq.6 (rotating) are completely equivalent statements of the balance of forces. The distinction between them is that the former is expressed in terms of V_{θ}, the latter in terms of v_{θ}. Note that Eq.6 has the same form as Eq.4 except:

(i) η (measured relative to the reference parabola) appears rather than H (measured relative to a flat surface) and

(ii) an extra term, -2Ω v_{θ}, appears on the rhs of Eq.6 — this is called the `Coriolis acceleration’. It has appeared because we have chosen to express our force balance in terms of relative, rather than absolute velocities.

**Atmosphere – Example
**

__Radial force balances__

**Consider parcels of air trapped in closed, axi-symmetric motion, moving with constant tangential velocity v**

_{θ }along a path of curvature r:

**Fig.2 **The velocity of a fluid parcel viewed in the rotating frame of reference: v_{rot} = (v_{θ} , v_{r}) in polar coordinates

The radial balance of forces away from frictional boundary layers and in the steady state is:

_{θ}

^{2}/r + f v

_{θ}= g∂h/∂r (7)

where g is the gravitational acceleration, f=2Ωsinφ is the Coriolis parameter and h is the height of a constant pressure surface.

Eq.7 is known as the `gradient wind’ relation. Note: Eq.7 has a similar form to Eq.6 in the tank experiment.

The Rossby number, Ro=v_{θ}/fr measures the ratio of v_{θ}²/r to fv_{θ}. There are two limit cases:

If Ro<<1, then we have `geostrophic balance’:

_{θ}= g∂h/∂r (8)

If Ro>>1 then the `cyclostrophic’ relationship can be used:

_{θ}

^{2}/r = g∂h/∂r (9)

The definition of the geostrophic wind from Eq.8:

_{θ}= g∂h/∂r (10)

can be used to rewrite the radial momentum equation, Eq.7, in the form

_{θ}

^{2}/r + fv

_{θ}– fv

_{g}= 0

Dividing by fv_{θ} , the ratio of the geostrophic wind to the actual wind can be written:

_{g}/v

_{θ}= 1 + Ro (11)