This document presents a temporal stability analysis of a swirling gas jet discharging into an ambient gas. The analysis considers the inviscid limit of the compressible swirling jet flow. The dispersion relation parameters studied include swirl number, Mach number, coflow velocity, and molar weight ratio. The base flow is solved using a self-similar solution, and a pseudospectral method is used to discretize the governing equations. The stability analysis derives the eigenvalue problem from the linearized Navier-Stokes equations to determine how perturbations of the base flow will grow or decay over time.

1. Physics of Fluids
Temporal stability analysis of a swirling gas jet discharging into an ambient
gas
D. Moreno Boza,1
C. del Pino Pe˜nas,1
and R. Fern´andez Feria1
1
Departamento de Ingenier´ıa Mec´anica y Mec´anica de Fluidos. ´Area de Mec´anica de Fluidos. Universidad de M´alaga.
(Dated: 8 September 2013)
Jet flows dynamics is one of the most fundamental parts of Fluid Mechanics due to its practical application
on industrial processing. Indeed, the structure of this flows must be taken into account when designing the
injection system of internal combustion engines, swirling combustion in gas turbines, fuel atomization and
atomizers technology, air handling units, vortex formation and breakdown, etc. To that end, it is precise
to solve de Navier Stokes equations with its corresponding boundary and initial conditions to give the so
called base or basic flow. This solution may be affected by small perturbations, temporal or spatial, so it will
likely lead to the formation of the formation of vortex and other kind of instabilities. Thus, a linear temporal
stability analysis is performed in this report considering the inviscid (Re−1
→ 0) swirling jet flow of a fuel
species discharging into a different ambient gas (or liquid). The main aim of this paper resides in describing
the influence of the dispersion relation parameters, such as swirl number, Mach number (wich models the
compressibility effects), coflow velocity of the ambient gas and the molar weight ratio. Precisely, the explicit
dependence of the governing equations on the molar weight ratio, enables us not to be constrained regarding
whether the discharging is made gas to gas, liquid to gas or liquid to liquid. In the same way, the formulation
for the viscous case (Re−1
̸= 0) is also presented and discussed for future work. Thus, the basic flow is solved
in terms of a self-similar solution and a pseudospectral collocation method is used in the discretization of the
equations.
PACS numbers: Valid PACS appear here
Keywords: Stability analysis - Jet flows - Swirling jets - Inviscid limit - Spectral methods
I. INTRODUCTION
This reports aims to carry out a hydrodynamic linear-
inviscid-temporal stability analysis of a non-reactive
compressible swirling jet discharging into a different am-
bient gas. By agreement, the ambient gas will be denoted
as the oxidant species and the gas flowing inside the tube
will be the fuel. To perform this stability analysis, the
governing equations must be rewritten in terms of the
following decomposition:
M = m + m′
,
where m is any magnitude of the base flow and m′
is a perturbation. Assuming that |m′
/m| ≪ 1, the
Navier-Stokes equations lead to an eigenvalue problem,
known as the dispersion relation, once the base solution
is known and a normal modes decomposition is made for
the perturbations. In this study, a self-similar solution
(see ref.10
) is considered in order to study the stability of
the flow near the nozzle exit, i.e., the solution for z ≪ 1.
The derivation of the eigenvalue problem, resulting
after the azimuthal and axial modes are fixed, is de-
scribed in section II, and its solution for the inviscid
limit (Re → ∞) is also shown in the same section. In
the section III, the main results obtained are explained
and finally some conclusions are given in section IV.
II. FORMULATION AND NUMERICAL SCHEME
Consider the discharge of a swirling gas jet through a
pipe of radius RI into an ambient gas. In figure 1 a sketch
of the geometry can be seen. As it has been commented
before, the ambient gas will be the oxidant whilst the fuel
will be the gas flowing through the pipe. Therefore, the
equations governing the problem are:
1. Continuity equation:
∂ρ
∂t
+ ∇ · ρ⃗v = 0, (1)
where ρ is the mixture density, ⃗v its velocity field,
and t is time.
2. Conservation of species: only necessary for one of
the species. For example, for the fuel it yields:
ρ
∂y
∂t
+ ρ⃗v · ∇y = ∇ · ρD∇y, (2)
where y is the mass fraction of the fuel (then 1−y is
the mass fraction for the oxidant), D is the binary
diffusion coefficient between both species, and ther-
mal diffusion and chemical reaction are neglected.
3. Equation of state for a binary gas mixture:

2. Physics of Fluids 2
RI
z
r
θ
FIG. 1: Sketch of the geometry and the coordinate system.
p
ρ
= RT
(
y
MF
+
1 − y
MO
)
= RT (y + ϵ(1 − y)) /MF , (3)
where ϵ = MF /MO is the molar weight ratio, p
is the pressure of the mixture and T its tempera-
ture, R ≈ 8.314 J/mol K is the universal molar gas
constant, M denotes molecular weight and the sub-
scripts F and O stand for fuel and oxidant species,
respectively.
4. Momentum equation:
ρ
∂⃗v
∂t
+ ρ⃗v · ∇⃗v = −∇p − ρg⃗ez + ∇ · µ∇v, (4)
where g is the acceleration due to gravity and µ is
the kinematic viscosity of the gas mixture.
5. Energy equation:
ρcp
DT
Dt
=
∂p
∂t
+⃗v·∇p+∇·K∇T +ρD(cpF −cpO)∇T ·∇y,
(5)
where D/Dt is the substantial derivative operator,
cp denotes specific heat capacity, K is the thermal
conductivity of the mixture.
These equations allows to solve the base flow in terms of
the dimensional variables, x = (r, θ, z) (from now, cylin-
drical polar coordinates will be used) and t, once the
transport coefficients are modelled in terms of empirical
expressions. By scaling the position vector in terms of
the transformation, thus we have:
x = RI(r, θ, z/δ), δ = RI/zc, (6)
where zc is a characteristic axial length. If the Reynolds
number is defined as it follows:
Re =
WcRIρF c
µF c
, (7)
where Wc is a characteristic axial velocity of the incoming
jet at z = 0, and ρF c, µF c, are characteristic value for
the density and the kinematic viscosity, respecitively, of
the fuel species at the (inlet) reference temperature then,
for convinience, zc may be choosen as:
zc = ReRI , (8)
so it yields Re = δ−1
. Once the base flow non-
dimensional magnitudes are solved in terms of the self-
similar solution, that is, (U, V, W, Y, ϱ, P, Θ) (i.e., the
non-dimensional radial, azimuthal and axial velocity
fields, mass fraction, density, pressure, and temperature)
are known (see figure 2, where the velocity field is shown
for several values of the swirl parameter and the coflow),
then a linear stability analysis can be performed by mak-
ing a decomposition of these magnitudes as follows:
u = Wc(δU + u′
), (9)
v = Wc(V + v′
), (10)
w = Wc(W + w′
), (11)
ρ = ρF c(ϱ + ρ′
), (12)
p = pc(1 + γMa2
(P + p′
)), (13)
T = Tc(Θ + T′
), (14)
where the magnitudes with primes denotes perturba-
tions, Ma = Wc/
√
γpc/ρF c is the Mach number, γ is
the specific heat ratio, P = (P − 1)/γMa2
, and pc and
Tc are characteristic values for the pressure and tempera-
ture, respectively, related by pc = ρF cRTc/MF . Finally,
a non-dimensional time, τ, can be written by using the
following relation:
τ =
Wc
RI
t. (15)
These decomposed variables when applied in equations
1-5 lead to a set of linear partial differential equations
(PDEs) for the perturbations if the non-linear terms are
neglected and if it is assumed that the magnitude of the
perturbations is much smaller than the non-perturbed
variables, that is, |m′
|/|m| ≪ 1, for every magnitude, m.

3. Physics of Fluids 3
0 0.5 1 1.5 2
−7
−5
−3
−1
1
3
r
U(r)
W(r)
V (r)
(a)
0 0.5 1 1.5 2
−7
−5
−3
−1
1
3
r
V (r)
W(r)
U(r)
(b)
FIG. 2: Velocity field profiles, U, V, W, as functions of r and (a)
swirl parameter, S (wich varies between 0 and 0.75) and (b)
coflow magnitude, WO (wich ranges from 0 and 1). These figures
correspond to the following flow parameters:
ϵ = 1, Eu = 1, Pr = 0.7, Sc = 1.39.
Assuming a normal modes decomposition whose ampli-
tude depends exclusively on the radial coordinate (that
is, parallel flow approximation), we have:
s′
= ˆs(r)ei(a/δ z+nθ−ωτ)
, (16)
where s′
= (ρ′
, y′
, p′
, u′
, v′
, w′
, T′
) is the perturbation vec-
tor, and ˆs(r) its amplitude. In equation 16, a is the
non-dimensional axial wavenumber, n is the azimuthal
wave number, and ω = ωr + iωi is the non-dimensional
frequency of the disturbances. Therefore, a and ω are
defined as:
a ≡ RI
˜k, ω ≡ ˜ωRI/Wc, (17)
where ˜k and ˜ω are the dimensional axial wavenumber
and frequency, respectively. In a hydrodinamic stability
analysis, wich is the present case, one fixes a real axial
wavenumber, a, and looks for complex values of ω, where
the real part ωr is the frequency of the perturbations, and
the imaginary part, ωi, is the temporal growth rate. The
flow is said to be convectively unstable when ωi > 0. For
the azimuthal wavenumber, n, axisymmetric perturba-
tions are given for n = 0, while helical perturbations are
those for n ̸= 0 (normally, n ∈ Z. To simplify this analy-
sis, the flow stability will be considered locally for a given
axial location z ≪ 1, neglecting buoyancy effects and,
in first approximation, considering constant values for
Sc, µ, K, cp = cpF = cpO, αF = αO = 1. Thus, the set of
resulting PDEs when bearing in mind the normal modes
decomposition 16 can be re-arranged in matrix form since
the problem is linear. Notice that the definition of the
density perturbations leads to the non-dimensional state
equation:
ϱ =
(1 + γMa2
)P
Θ
1
Y + ϵ(1 − Y )
. (18)
This being the case, what we have is the following homo-
geneus problem:
D · ˜s = 0, D = D1 + δD2, (19)
where the terms of O(δ2
) or higher are neglected. In the
other hand, the matrix operators D1,2 are given by:
D1 =











iΩ 0 0 · · ·
0
(
iΩ + a2
Sc
)
ϱ 0 · · ·
1 −(1 − ϵ)ϱΘ − ϱ
1+γMa2P
· · ·
−V 2
r 0 d
dr · · ·
0 0 in
r · · ·
0 0 ia · · ·
0 0 −iΩ(γ − 1)Ma2
· · ·
· · · α1 inϱ
r iaϱ 0
· · · ϱ∂Y
∂r 0 0 0
· · · 0 0 0 ϱ
· · · iΩϱ −2ϱV
r 0 0
· · · α2 iΩϱ 0 0
· · · ϱ∂W
∂r 0 iΩϱ 0
· · · α3 0 0 iΩϱ









(20)
and:

4. Physics of Fluids 4
D2 =









ψ1 0 0 · · ·
β1 ψ2 0 · · ·
0 0 0 · · ·
0 0 0 · · ·
U ∂Y
∂r + UV
r + W ∂V
∂z 0 0 · · ·
U ∂W
∂r + W ∂W
∂z 0 0 · · ·
U ∂Θ
∂r + W ∂Θ
∂z 0 (γ − 1)Ma2
U d
dr · · ·
· · · 0 0 ∂ϱ
∂z 0
· · · 0 0 ϱ∂Y
∂z 0
· · · 0 0 0 0
· · · ϕ1
in
r
(3
r − d
dr
)
−ia d
dr 0
· · · −in
r
( d
dr + 2
r
)
ϕ2 ϱ∂V
∂z + na
r 0
· · · −iaDr
na
r ϕ3 0
· · · 0 0 β2 ϕ4










(21)
where the differential operators Dr and Drr are written:
Dr ≡
d
dr
+
1
r
, Drr ≡
d2
dr2
+
1
r
d
dr
, (22)
and the following definitions have been made:
α1 =ρDr +
∂ϱ
∂r
, (23)
α2 =
(
V
r
+
∂V
∂r
)
ϱ, (24)
α3 =ϱ
∂Θ
∂r
− (γ − 1)Ma2 ∂P
∂r
, (25)
ψ1 =UDr +
∂U
∂r
+
∂W
∂z
, (26)
ψ2 =
(
ϱU −
1
Sc
∂ϱ
∂r
)
d
dr
−
ϱ
Sc
Drr +
n2
r2
ϱ
Sc
−
ia
Sc
∂ϱ
∂z
,
(27)
β1 = −
1
Sc
∂Y
∂r
Dr −
1
Sc
∂2
Y
∂r2
+ U
∂Y
∂r
+
(
W −
ia
Sc
)
∂Y
∂z
,
(28)
β2 =(γ − 1)Ma2 ∂P
∂z
+ ϱ
∂Θ
∂z
, (29)
ϕ1 = − 2Drr +
n2
+ 2
r2
+ ϱU
d
dr
+ ϱ
∂U
∂r
+ a2
, (30)
ϕ2 = − Drr +
2n2
+ 1
r2
+ ϱUDr + a2
, (31)
ϕ3 = − Drr + ϱU
d
dr
+
n2
r2
+ ϱ
∂W
∂z
+ 2a2
, (32)
ϕ4 = −
D2
Pr
+
1
Pr
(
n2
r2
+ a2
)
+ ϱU
d
dr
(33)
and finally:
Ω = −ω + aW + n
V
r
. (34)
The homogeneous equation 19 is to be solved with the
following boundary conditions:
(˜y, ˜u, ˜v, ˜w, ˜T) = 0, r → ∞, (35)
and (see ref.3
):



˜u = ˜v = ∂ ˜w
∂r = ∂˜y
∂r = ∂ ˜T
∂r = 0 n = 0
∂˜u
∂r = ˜w = ˜u + in ˜v = ˜ρ = ˜y = ˜p = ˜T = 0 |n| = 1
˜u = ˜v = ˜w = ˜ρ = ˜y = ˜p = ˜T = 0 |n| > 1
(36)
at r = 0. The pressure and the density must remain
bounded in the case of axisymmetric perturbations, i.e.,
n = 0, so |˜ρ(0)| < ∞ and |˜p(0)| < ∞. Thus, the equation
19 in conjunction with the boundary conditions written
above constitute an homogeneous eigenvalue problem for
ω, once a and n are fixed (which is the main purpose
of this study), whose solution is known as the disper-
sion relation. This relation depends only on the non-
dimensional numbers that govern the flow and, of course,
the axial and azimuthal wavenumbers and may be writ-
ten as follows:
D(ω; a, n, Sc, Pr, Ma, γ, ϵ, S, WO, z) = 0. (37)
In equation 19, the parameter δ determines whether the
fluid is mainly dominated by viscosity or it can be con-
sidered inviscid (δ → 0). If the inviscid limit is con-
sider, then the problem given by equation 19 can be easily
rewritten into the form:
(A − ωB)˜s = 0, (38)
which is then a new non-homogeneous eigenvalue prob-
lem for determining ω. Furthermore, the order of the
system of equations have been reduced in one by elimi-
nating the density from equation 19 thanks to the state
equation, so now:
˜s = (˜y, ˜p, ˜u, ˜v, ˜w, ˜T), (39)
Thus, the matrices A and B can be readily found by
splitting the operator Ω as follows:
Ω = −ω + λ, λ ≡ aW + n
V
r
, (40)
so then it yields:
A =










iλ(1 − ϵ)ϱΘ iλ γMa2
ϱ
1+γMa2P
ϱDr + ∂ϱ
∂r · · ·
(
iλ + a2
Sc
)
ϱ 0 ϱ∂Y
∂r · · ·
−V 2
r (1 − ϵ)ϱΘ ϕ5 iλϱ · · ·
0 in
r
(V
r + ∂V
∂r
)
ϱ · · ·
0 ia ϱ∂W
∂r · · ·
0 −iλ(γ − 1)Ma2
ϕ6 · · ·

5. Physics of Fluids 5
· · · inϱ
r iaϱ −iλϱ
· · · 0 0 0
· · · −2ϱV
r 0 V 2
r ϱ
· · · iλϱ 0 0
· · · 0 iλϱ 0
· · · 0 0 iλϱ








(41)
and:
B = i








(1 − ϵ)ϱΘ γMa2
ϱ
1+γMa2P
0 0 0 −ϱ
ϱ 0 0 0 0 0
0 0 ϱ 0 0 0
0 0 0 ϱ 0 0
0 0 0 0 ϱ 0
0 −(γ − 1)Ma2
0 0 0 ϱ








(42)
while the following definitions are made:
ϕ5 = −
V 2
r
γMa2
ϱ
1 + γMa2P
+
d
dr
, (43)
ϕ6 =ϱ
∂Θ
∂r
− (γ − 1)Ma2 ∂P
∂r
, (44)
To check the results of the temporal stability analysis,
the case of a single species, incompressible and swirless
jet flow is first considered, so we are able to compare with
previous results (see ref.3
). For ϵ = 1, Y = 1 everywhere,
the stability equation coming from the species conserva-
tion becomes irrelevant. On the other hand, for S = 0,
the basic flow is such that V = P = 0, and Θ = ϱ = 1
because Ma = 0. Consequently, ˜rho = ˜T = 0, and the
resulting stability equations (continuity, and the three
componentes of the momentum equation) can be written
as:
Dr ˜u +
in
r
˜v + ia ˜w = 0, (45)
d˜p
r
+ i(aW − ω)˜u = 0, (46)
in
r
˜p + i(aW − ω)˜v = 0, (47)
ia˜p +
∂W
∂r
˜u + i(aW − ω) ˜w = 0, (48)
which coincide with those first given by Batchelor and
Gill (see ref.3
).
A. Numerical scheme
The problem 38, whith its corresponding boundary
conditions, constitutes a linear eigenvalue problem for the
complex frequency, ω, given the (real) axial wavenum-
ber, a (i.e., temporal stability analysis), the azimuthal
wave number, n, and the parameters characterizing the
basic flow (ϵ, Sc, Ma, WO, Pr, γ and S). Since B is non-
hermitian, a QZ algorithm is used to solve numerically
the eigenvalue problem. For the discretization in the
radial direction, we have used the same pseudospectral
Chebyshev collocation method in the radial direction de-
scribed in10
. The boundary conditions at infinity are
applied at a truncated radial distance rmax, chosen large
enough to ensure that the results do not depend on it.
To map the interval 0 ≤ r ≤ rmax into the Chebyshev
polynomials domain −1 ≤ s ≤ 1, we use the transforma-
tion:
r = c1
1 + s
c2 − s
, c2 = 1 + 2
c1
rmax
, (49)
where c1 is a constant such that approximately half of
the nodes are concentrated in the interval 0 ≤ r ≤ c1.
This transformation allows large values of r to be taken
into account with relatively few bassis functions. The
Chebyshev variable s is then discretized in the Gauss-
Lobatto points (see ref.2
), si = cos π i/N, i = 0, . . . , N.
Spurious eigenvalues were discarded by comparing
the computed spectra for an increasing number N of
collocation points. A first selection of physical modes
is made by discarding all the eigenvalues corresponding
to eigenfuncionts that do not decrease conveniently as
r → ∞; that is, we consider only thos eigenfunctions
satisfying the following criterion:
N/10
∑
i=1
|˜s(ri)|2
/
N∑
i=1
|˜s(ri)|2
< T, (50)
where ri are the radial nodes and T is a given tolerance.
B. Convergence tests
To check the efficiency and accuracy of the numerical
method, we firs perform a convergence analysis to se-
lect the numerical parameters that optimize the compu-
tations in accuracy versus computation speed. To that,
end table I shows the obtained eigenvalues ω for the base
flow given by the self-similar solution at z = 0.01 for
perturbations with a = 1 and n = 0 (i.e., axisymet-
ric perturbations), computed with different values of the
number N of radial Chebyshev nodes, rmax and c1. A
faster convergence is noted as N increases with c1 = 1
than with c1 = 2, i.e., concentrating the spectral nodes
in 0 ≤ r ≤ 1 rather than 0 ≤ r ≤ 2. Taking into account
the results in table I, we will consider N = 100, c1 = 1
and rmax = 100.

6. Physics of Fluids 6
rmax c1 N ω
100 1 50 0.656818 + 0.259825 i
100 1 75 0.667424 + 0.255890 i
100 1 100 0.674201 + 0.250054 i
100 1 125 0.678140 + 0.246863 i
100 1 150 0.686830 + 0.239294 i
100 1 175 0.689040 + 0.237239 i
100 1 200 0.691972 + 0.234453 i
100 1 225 0.686822 + 0.239313 i
100 2 100 0.665740 + 0.256851 i
100 2 150 0.676293 + 0.248445 i
100 2 200 0.682786 + 0.242931 i
100 2 250 0.685342 + 0.240653 i
100 2 300 0.688173 + 0.238061 i
100 2 350 0.694199 + 0.232259 i
50 1 175 0.693266 + 0.233181 i
75 1 175 0.687731 + 0.238476 i
100 1 175 0.689040 + 0.237239 i
200 1 175 0.693197 + 0.233265 i
TABLE I: Eigenvalues ω for the self-similar solution at z = 0.01
corresponding to an incompressible, non-swirling flow
(ϵ = 1, S = 0) for axisymmetric perturbations (n = 0) with axial
wavenumber a = 1 and Eu = 1, computed with differents values
of rmax, c1 and N. The tolerance value is T = 10−6
III. RESULTS
IV. CONCLUSIONS
Ma
a
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.5
1
1.5
2
2.5
3
3.5
0.05
0.1
0.15
0.2
0.25
0.3
FIG. 3: Contour lines for the exponential growth rate, ωi, as
function of the Mach number, Ma, and the axial wavenumber, a,
i.e., ωi = ωi(Ma, a). This figure corresponds to the following flow
parameters: ϵ = 1, Eu = 1, Pr = 0.7, Sc = 1.39.

7. Physics of Fluids 7
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