Where can I find out more about the relationship between solid-body and Bessel-function
swirl?
In kiva, initial swirling velocities can be specified as solid-body rotation, where the
azimuthal (swirling) velocity steadily increases with distance from the axis. Initial
swirling velocities can also be specified with a Bessel-function, where again the
azimuthal velocities increase with distance from the axis, but at some point the azimuthal
velocities begin to decrease. This velocity decrease is caused by the flow interacting with
the cylinder liner.
Kiva has coding to ensure that the angular momentum imparted to the flow field is the
same regardless of solid-body, or the Bessel-function specification, when using the same
swirl ratio input value.
Experiments usually assume solid-body rotation when arriving at the measured swirl ratio
value. A common swirl measurement device is an impulse torque meter, which essential
measures torque imparted to it from the flow field angular momentum via a set of steady
flow tests and converts those torque measurements into a swirl ratio.
This document uses definitions used in swirl ratio measurements and in model
implementation of swirl in initiating angular momentum to the flow field to allow the
reader to confirm that assigning swirl via solid-body, or Bessel-function assumptions are
equivalent.
 sb   bessel
(1)
where:
 sb  torque if flow is solid body, subscript sb = solid body
 bessel  torque if flow follows a Bessel function, subscript bessel = Bessel function
Below is a definition for either solid-body, or Bessel-function torque. The equation states
that torque = (r x V) * massFlowRate.

2 cylrad
 
0
r
v

rv  vz rdrd
0
= radial distance from axis
=
=
azimuthal velocity
density
Randy Hessel, Engine Research Center, University of Wisconsin-Madison
(2)
vz
=
=
cylrad
axial velocity
cylinder radius
The torque measurements assume that the flow is axisymmetric, therefore equation (2)
becomes,
cylrad
  2

rv  vz rdr
0
(3)
Substituting (3) into (1) gives,
cylrad
2

cylrad
r (rsb )  vz rdr  2
0

r (effvel )  vz rdr
0
(4)
where:
 sb = solid body angular velocity
effvel = effective velocity at distance r via a Bessel function calculation
Torque meter analysis also assumes that the axial flow and density are uniform, so
equation (4) becomes,
cylrad

cylrad
r (r sb )rdr 
0

r (effvel )rdr
0
(5)
Simplifying and noting that by definition the solid-body swirl is also constant, (5)
becomes,
cylrad
 sb

cylrad
r dr 
0
3

(effvel )r 2 dr
0
Further evaluation gives,
Randy Hessel, Engine Research Center, University of Wisconsin-Madison
(6)
cylrad 4
sb

4
cylrad

(effvel )r 2 dr
0
(7)
Modelers might recognize effvel is a kiva variable. Expanding its definition using other
kiva variables (see subroutine setup) yields,
effvel  bessel * angtrm
(8)
 swipro * r  bessel * cylrad * swipro
bessel * angtrm  besj1
*
cylrad
4.0* bessel 2


(9)
where
bessel
= angvel in kiva terminology. Substituting (9) into (7) gives
cylrad 4
sb

4
cylrad

0
 swipro * r  bessel * cylrad * swipro
r 2 * besj1
dr
*
4.0* bessel 2
 cylrad 
(10)
Removing constants from the integral and taking the ratios of angular velocities gives,
 sb / bessel
swipro

cylrad 3 * bessel 2
cylrad

0
 swipro * r 
r 2 * besj1
dr
cylrad


(11)
Thus, writing a program to solve equation (11) and entering different values of swipro
should prove to the reader that the ratio of angular velocities assuming solid-body, or
Bessel-function swirl results in a representation of the same angular momentum. That is,
the value of equation (11) should equal something close to 1.0.
ERC staff can access source code for solving this equation on the ERC Toolbox page.
Search for angVelSolidToBessel.
Randy Hessel, Engine Research Center, University of Wisconsin-Madison