Original Article
Estimating the mean flow field in
combustion chambers
International J of Engine Research
2014, Vol. 15(3) 338–345
Ó IMechE 2013
Reprints and permissions:
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DOI: 10.1177/1468087413485576
jer.sagepub.com
Thad S Morton
Abstract
Swirling flow fields in combustion chambers can be determined based on swirl ratio and a velocity profile specified along
some path to the vortex center. A method is presented whereby flow fields can be constructed by applying the continuity equation in a streamline coordinate system and imposing irrotationality about the symmetry axis of the vortex ring.
The swirl ratio may be specified at the vortex core, along with a velocity profile along any semi-axis of the vortex cross
section.
Keywords
Intake swirl, in-cylinder flow, vortex, flow structure, swirl ratio
Date received: 6 December 2012; accepted: 2 March 2013
Introduction
Combustion engines are usually designed to generate
coherent vortex structures in the cylinder during intake
in order to enhance mixing. These motions are typically
categorized as either swirl or tumble motion, with the
term ‘‘swirl’’ referring to rotation about an axis aligned
with the cylinder axis (toroidal rotation) and ‘‘tumble’’
referring to rotational motion about an axis perpendicular to the cylinder axis (poloidal rotation). The combination of swirl and tumble is a vortex ring with swirl.
Swirl motion provides lower variability than tumble
motion1 and more stability due to its lower decay rate.2
In fact, swirl is amplified when the flow structure is
squished as the piston approaches top-dead center.3
This is due to conservation of angular momentum.
Swirl in spark-ignition engines enhances atomization4
and is a key factor in engine performance,5–7 but its
cyclic features have been difficult to describe.1 Pipitone
and Mancuso2 showed that the swirl ratio in a cylinder
could be increased from 0.3 to around 0.65 at the
expense of 35% of breathing capacity. Kampanis et al.5
reported that through the use of a swirl-inducing device
in a steady-flow test cylinder, a swirl ratio of 5.5 was
achieved. Figure 1 shows streaklines of vortices in an
internal combustion engine with negligible swirl.8,9
Gas turbine combustors rely on some type of stationary toroidal vortex to facilitate stable combustion.
Understanding the primary characteristics of the mean
flow in such a vortex can guide conceptual designs
and computational fluid dynamics (CFD) studies. Two
state-of-the-art burner concepts, namely, swirlstabilized combustors10,11 and trapped-vortex combustors,12,13 are shown in Figure 2.
Vortex rings are found in a range of other phenomena, including tornado and hurricane structures,
interstellar media,14 dilute atomic Bose–Einstein condensates,15 tokomaks,16 and the growth of single-wall
carbon nanotubes.17 Moffatt18 gave an asymptotic
solution for flow in a vortex ring without swirl and
showed that a solution must exist for a vortex ring with
swirl.19 Etnyre and Ghrist20,21 showed that smooth
steady Euler flows in a solid torus possess closed trajectories. No exact solution has yet been found for a largecore swirling vortex ring.
Currently, an assumption usually made when treating piston-engine swirl is that the flow acts as a solidbody vortex with uniform axial velocity.2 However, the
actual flow in the cylinder of a piston engine is unsteady
and affected by factors such as motion of the piston
and the action of spray. A CFD study cannot provide,
for example, a functional relationship between swirl
ratio and complex geometrical changes in the flow
region in the cylinder, nor can it be easily tuned to
Morton
339
Figure 1. (a) Vortex structure with negligible swirl setup in an engine cylinder by the intake valve.8,9 (b) Swirling vortex ring
computed by the present method.
Figure 2. Gas turbine combustors utilizing stationary vortex rings: (a) swirl-stabilized combustor and (b) trapped-vortex
combustor.
match experimental data. Therefore, the engine
designer could be benefited if, with few measurements
through the combustion chamber, he or she could
obtain a good understanding of what the actual flow is
doing throughout the chamber. Presently no analytical
method exists to accomplish this.
While not all of the above complexities neglected by
the usual solid-body treatment will be addressed herein,
the purpose of the present work is to provide a means
of estimating, with a minimal set of velocity measurements through the vortex, complete steady-flow fields
in gas turbine combustors, as well as instantaneous flow
fields in port-injected piston engines during intake. The
work extends a method given previously for finding the
velocity field in a torus with an elliptical cross
section22,23 to one for which an irrotational toroidal
velocity (irrotational swirl) component is present.
Depending upon how the swirl is generated, this irrotationality assumption may not be out of place in studies
of engine swirl. Uzkan et al., for example, reported very
low net torque in some high-swirl intake ports.24,25 The
present method predicts, for example, a toroidal-topoloidal frequency ratio of f3 = f2 = 7:3 : 1 for the outer
streamline of the flow reported by Kampanis et al.5 to
have achieved a swirl ratio of 5.5 (see Figure 5(d) and
(e)). In other words, a fluid particle on the outer streamline travels 7.3 times around in the swirl direction in the
time it makes one loop through the central core. This
prediction made by equation (28) below takes as its
input, the empirical velocity profile of equation (27).
Similarly, the swirl ratios reported by Pipitone and
Mancuso of 0.3 to 0.65 correspond to a toroidal-topoloidal frequency ratio range of 0:44f3 =f2 40:87.
Using the present method, characteristics such as this
may be checked by hand for many scenarios before
detailed and lengthy CFD simulations are run.
Outline of approach
The present method does not attempt to solve the
momentum equation throughout the domain. Rather,
an experimental velocity profile along some path to the
vortex center is incorporated, and the continuity equation and an irrotationality condition are used to determine the instantaneous flow throughout the remainder
of the domain. The steady continuity equation for general coordinate systems22,23
∂ pffiffiffi i r gv = 0
∂xi
ð1Þ
can be used for this purpose. Here, the raised index on
the velocity v denotes components of a contravariant
pffiffiffi
tensor, and g is the determinant of the Jacobian of the
transformation from any suitable coordinate system to
a rectangular coordinate system.
340
International J of Engine Research 15(3)
representing different streamlines, all of which are nested
on the manifold M2 containing the motion. Different
values of x3 correspond to different manifolds similar to
M2 . Holding only x1 fixed, and allowing x2 and x3 to
vary, represents a specific toric submanifold in T 3 .
Equation (2) is satisfied identically by choosing a
stream function c, such that
Jab ∂c va = pffiffiffi b x 2 M2
x
r g ∂
ð4Þ
where Jab is the symplectic matrix
0 1
Jab =
1 0
Since the only nonzero velocity component in the
streamlined coordinate system is to be v2 , we can set
a = 2 in equation (2) and integrate, giving
h(
x1 , x3 )
pffiffiffi
r g
v2 =
Figure 3. Vortex ring.
In order to integrate equation (1), one of the coordinates in the coordinate system should coincide with the
direction of the fluid velocity. These streamlined coordinates xi will be denoted by overbars. Not only will
curves of one of the coordinates be designated as
streamlines, but these streamlines will also be considered to lie entirely on a two-dimensional manifold M2 ,
defined by the quotient M2 = T 3 = V. (The core circle
lies in the one-dimensional subspace V.) Therefore, all
quantities will be independent of the coordinate direction in the subspace V of the torus T 3 (see Figure 3).
For such a dissection of the flow field, equation (1) can
be rewritten as two separate statements
∂ pffiffiffi a r g v = 0 xa 2 M2 for a = 1, 2
a
∂
x
ð2Þ
and
∂ pffiffiffi 3 r g v = 0
∂
x3
x3 2 V
ð3Þ
For the vortex ring studied herein, equation (3) will
be satisfied by letting v3 = 0. This does not mean, however, that the swirl velocity is assumed to be 0. Rather,
swirl will be accomplished by letting the manifold M2 ,
on which the xk frame is embedded, deform
appropriately.
The closed streamlines lying on any given manifold
M2 never leave that manifold. Flow along any streamline on that manifold will be represented by increasing
x2 , and the remaining coordinate, x1 , will be a constant
of the motion. Therefore, v1 = 0, and the set of all points
determined by holding x1 fixed will be invariant sets
ð5Þ
The ‘‘constant’’ of integration h is a function of at most
x1 and x3 , and if a given set of streamlines is confined to
the manifold M2 , we can reduce h(
x1 , x3 ) to simply
1
h(
x ). Comparing equations (4) and (5) reveals that
x1 ). The advantage of this method lies in
∂c = ∂
x1 = h(
the fact that any function h(
x1 ) satisfies the continuity
equation, even if h is a discontinuous function of x1 .
Coordinate system
The cross section of the core of the vortex ring is
formed by the following coordinate system
x2 (t))
x^1 = x1 a cos (
x2 (t)) + RC
x^2 = x1 b sin (
3
3
1
2
3
x , x (t), x )
x^ = x^ (
ðZÞ
ð6Þ
(R)
ð7Þ
(u)
ð8Þ
Here, a and b are, respectively, the semi-major and
semi-minor axes of the elliptical cross section of the
vortex, and RC is the distance from the central axis to
the core circle of the vortex ring. Notice that in the
barred coordinate system, only x2 varies with time, and
x1 and x3 remain constant as the fluid particle travels.
This coordinate system yields a vortex ring with an
elliptical cross section defined by the following bounds
x1 4
x1max = 1)
on the dimensionless variables xk : (04
2
3
and ( p \ x , x 4p).
The core circle of the torus is given by x1 = 0, and
the outer surface by x1 = 1. Streamtubes, described by
curves of constant x1 , are concentric ellipses.
The transformation is completed by relating the
cylindrical coordinates x^k to those in a rectangular
coordinate system, as follows
9
x1 = x^2 (t) cos (^
x3 (t)) =
ð9Þ
x2 = x^2 (t) sin (^
x3 (t))
;
1
3
x = x^
Morton
341
The manifold on which the velocity field in equation
(5) is defined will be allowed to deform so as to introduce a toroidal velocity component. A toroidal velocity
will arise as soon as the shape of the manifold M2 in
Figure 3 is distorted from planar, and it will also be a
function of the velocity v2 on that manifold. A description of this manifold will be sought by letting all toroidal motions arise by a deformation of the manifold.
Fortunately, in spite of the deformation p
introduced
ffiffiffi
into the manifold, the Jacobian determinant g appearing in equation (5) simplifies to
pffiffiffi
ð10Þ
g = x1 abR
where F(
x1 , x3 ) = x3 . Therefore, by incorporating equation (10) and the assumption of constant density along
streamlines, we arrive at the following for equation (8)
Z
1
c1 rx1 ab 1 2
d
x + x3 ,
v
^ =0
ð15Þ
x^3 =
h
R
due to significant cancellation of terms. It is, in fact,
independent of both ∂^
x3 = ∂
x1 and ∂^
x3 = ∂
x2 . For the
cylindrical coordinate
system, the Jacobian of the transpffiffiffi pffiffiffiffiffiffi
formation is g^ = g^33 = R.
The remaining two coordinates x^1 and x^2 are given
by equations (6) and (7). The resulting manifold defined
x2 is deformed according to equation (15). That
by x1 3
is, the coordinate x^3 (of the cylindrical system) of any
fluid particle is given by equation (15).
where the integral is given by
Z
0
1
2 1
1 2
2
BRC tan x =2 + x bC
d
x = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi atan@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A
R
2
2
RC 2 (
RC 2 (
x1 b)
x1 b)
ð16Þ
Fluid particle trajectories
A condition can be placed on the velocity field in the
vortex ring by requiring that the vorticity about the central symmetry axis be 0. The component of the vorticity
tensor corresponding to the direction of the symmetry
axis (the z direction in Figure 3) is
1
∂ ∂ p
p
1
v
^ ¼ pffiffiffi
g^3p v^ 3 g^2p v^
ð11Þ
g^ ∂^
x2
∂^
x
where v^3 = d^
x3 = dt, and eijk is the permutation symbol,
equal to 1 when i, j, and k form a cyclic permutation,
1 when anticyclic, and 0 otherwise. Since the cylindrical coordinate system is orthogonal, only diagonal components of the metric tensor g^in remain. If the flow is
irrotational about the symmetry axis, (11) simplifies to
1 ∂ v
^ 1 ¼ pffiffiffi 2 g^33 v^3 ¼ 0
g^ ∂^
x
Velocity field
The velocity distribution along streamlines lying on the
manifold will satisfy continuity and therefore be given
by equation (5). The function h(
x1 ) appearing in equations (5) and (15) can be determined by specifying the
magnitude of the physical velocity profile along a semiaxis of the vortex ring core. This physical velocity is
first related to the streamline component v2 of the velocity tensor, as follows
pffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi
v = vk vk = gki vi vk = g22 v2
ð17Þ
The component g22 of the metric tensor must be
computed from
ds2 =
c1
R2
ð12Þ
Notepffiffiffiffiffiffi
that the physical velocity component is
v^(3) = g^33 v^3 = R^
v3 . Therefore
v^(3) =
c1
R
ð13Þ
Using the fact that v^i = vk ∂^
xi =∂
xk , we may write
v^3 = v2 ∂^
x3 =∂
x2 , so that equation (12) becomes
Another integration yields
Z pffiffiffi
c1
r g 2
d
x + F(
x1 , x3 )
x^3 =
1
3
R2
h(
x , x )
=
3
X
∂xk ∂^
xi
k
xp q
j ∂x ∂^
dx
dx = gjq dxj dxq
p
i ∂
j
q
x
∂^
x
∂
x
∂^
x
k=1
where
!
3
X
∂xk ∂xk ∂^
xi ∂^
xp
∂^
xi ∂^
xp
^
=
g
gjq =
ip
xp ∂
xj ∂
xq
∂
xj ∂
xq
xi ∂^
k = 1 ∂^
ð18Þ
Therefore, by taking the indicated derivatives of
equations (6), (7), (15), and (9), one arrives at the following relation for g22
2
2 2 2
c1 r x1 ab
2
2 2
x ) a sin (
x ) + b cos (
x) +
g22 = (
h(
x1 )
ð19Þ
1 2
3
∂^
x
c1
= 2 2
∂
x2
v R
dxk dxk
k=1
Integrating once gives the component v^3 of the velocity tensor
v^3 =
3
X
ð14Þ
Using equation (19), the magnitude v of the physical
velocity may now be evaluated on the semi-axis given
by, say, x2 = p=2, as follows
342
International J of Engine Research 15(3)
pffiffiffiffiffiffi h g22 pffiffiffi 2
r g ðx2 = p=2Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c1 rab 2 h(
x1 )
= a2 +
1
h(
x ) rabRjp=2
So by equation (13), on the outermost streamline
where x1 = 1;
v jp =
ð20Þ
so that
1
h(
x )=rb
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
vjp=2 (RC + x1 b)
2
(c1 )2
ð21Þ
Since h is only a function of x1 , it will satisfy the
continuity equation. The constant c1 , according to
v(3), where v^(3) is the physical
equation (13), is c1 = R^
swirl (toroidal) velocity. The velocity given by equation
(20) however, is the magnitude of the physical velocity,
which includes the swirl component.
The swirl ratio SI is defined as the ratio of the swirl
(toroidal) velocity v^(3) to the poloidal velocity component v^(2) at the minimum radius of the vortex ring
v^(3)
v^I (3)
ð22Þ
SI =
[
v^I (2)
v^(2)(x1 = 1, x2 = p)
2
(Note that at the minimum radius, the poloidal velocity
component v^(2) is in the axial direction, so this definition of swirl ratio coincides with the usual definition at
the minimum radius.) And at the minimum radius
vI 2 = v^I (2)2 + v^I (3)2
1
RI
vO = pffiffiffi = vI
RO
2
The velocity components throughout the vortex
remain such that irrotationality is maintained about
the central symmetry axis.
Figures 4 and 5 show vortex rings that are irrotational about their symmetry axis. Their trajectories are
computed from equations (6), (7), (15), and (9). (The
vortices in Figure 2 were also generated by the present
method.) Low-velocity regions in the figures can be
identified by a general increase in the spacing between
streamlines. From this observation, it is clear that the
velocity decreases with the increasing radius. For
clarity, the toroidal-to-poloidal frequency ratios, f3 = f2 ,
of the vortex rings shown in the figures are near small
rational numbers.
For plotting purposes, the trajectories are transformed to a rectangular coordinate system using equation (9). The inverse tangent function appearing in
equation (16) is continuous in the range
(p \ x2 \ p), so for plotting purposes, when this
range is exceeded, x2 can be reset to p, and x3 can be
set to whatever value is necessary in order for equation
(15) to return the same value of x^3 that it had before
the reset. This will ensure continuity of the streamline
plots.
so using equation (22), the constant c1 can be written as
c1 = v^I (3) RI = SI v^I (2) RI
ð23Þ
that is, as the product of the inner radius RI with the swirl
component of the physical velocity there. If vI is easier to
measure in practice, then the constant may be written as
c1
SI vI RI
= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 + SI 2
Frequency
A relation for the poloidal frequency f2 can be found by
integrating equation (5) over one period T2 , as follows
2p
ð
ð24Þ
pffiffiffi 2
g d
x =
T2ð
(
x1 )
h(
x1 )
dt
r
0
0
The velocity magnitude is obtained by substituting The period will, in general, depend upon x1 , which
equation (24) into equation (21), then equation (21) remains fixed throughout the integration above.
into equation (5), and finally equations (5) and (19) Therefore, by equations (10) and (7)
into equation (17). This gives
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
ih
i2
2
b2 i
1
(SI vI RI )2 h
x2 ) + ba cos2 (
x2 ) vjp=2 (RC + x1 b) +
1
x2 )
ð25Þ
sin2 (
cos2 (
v(
xi ) =
a
R
1 + SI 2
The function vjp=2 in equations (21) and (25) can be any
function of x1 , determined empirically. Specifying this function determines only the velocity magnitude from the eye of
the vortex ring to the outer radius. The components of the
velocity at the minimum radius are dictated by the swirl ratio
SI . In the figures to follow, the following function was used
for the velocity magnitude from the core circle outward
v jp=2 =
1
2
(1 + x1 )
ð26Þ
1
Z2p
x ab
2
h(
x1 )
T2 (
x =
x2 ) + RC d
x1 )
x1 b sin (
r
0
The integration gives
h(
x1 )
2p RC =
x1 )
T2 (
r x1 ab
Morton
343
Figure 4. Each vortex ring contains the trace of two streamlines: x1 = 0:2 and x1 = 1:0. The frequency ratio f3 =f2 refers to that of
the streamline x1 = 1:0. RC = 0:1, RI = 0:02, a=b = 1:0, RO = (2RC RI ) = 0:18. (a) SI = 0:1074, f3 =f2 ’ 1=7; (b) SI = 0:1504, f3 = f2 ’ 1=5;
(c) SI = 0:1876, f3 = f2 ’ 1=4; (d) SI = 0:2509, f3 = f2 ’ 1=3; (e) SI = 0:50, f3 =f2 ’ 1=3; and (f) SI = 0:7508, f3 =f2 ’ 1.
) f2 (
x1 ) =
1 h(
x1 )
1
2p r x abRC
ð27Þ
If the ratio of the frequency of toroidal rotation f3 to
that of poloidal rotation f2 is rational, the streamline
will be a closed loop. Doubling the aspect ratio of the
vortex doubles the ratio f3 = f2 , all else being equal. By
setting the frequency ratio near, but not equal to, small
rational numbers, a spreading of neighboring streamlines is achieved. This provides a qualitative depiction
of the acceleration of the fluid near the core.
The most prominent set of frequency harmonics in
the vortices can be found with the relation
SI ’
3 b f3
4 a f2
ð28Þ
344
International J of Engine Research 15(3)
Figure 5. Each vortex ring contains the trace of two streamlines: x1 = 0:2 and x1 = 1:0. The frequency ratio f3 =f2 refers to that of
the streamline x1 = 1:0. RC = 0:1, RI = 0:02, a=b = 1:0, RO = (2RC RI ) = 0:18. (a) SI = 1:1253, f3 =f2 ’ 3=2; (b) SI = 1:5026, f3 =f2 2;
(c) SI = 2:9942, f3 = f2 ’ 4; (d) SI = 4:521, f3 = f2 ’ 6; (e) SI = 6:3859, f3 =f2 ’ 17 =2; and (f) SI = 10:505, f3 =f2 ’ 14.
where f3 = f2 is a simple rational number. This relation
is valid for x1 = 1 (the outer surface of the vortex) with
RC , RI , and RO as given in the figure captions.
However, other frequency harmonics that do not conform to equation (28) also exist, as evidenced by the
vortex in Figure 4(e).
Conclusion
Flows throughout vortex rings with swirl can be estimated by specifying the velocity magnitude along some
semi-axis of the vortex cross section, applying the
continuity equation throughout, and imposing irrotationality about the symmetry axis of the torus. The
flows found may be useful in understanding and modeling practical flows in piston-engine cylinders, gas turbine combustors, and other industrial settings.
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit
sectors.
Morton
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