Numerical and asymptotic methods are used to describe the structure of low-temperature laminar gas jets discharging into a hot atmosphere of the same gas in the limit of small jet-to-ambient temperature ratios " = Tj/To 1. In the limit " ! 0, heat conduction cannot modify significantly the temperature in the cold gas, leading to a two-region flow structure consisting of a neatly defined unperturbed cold jet for r < rf (x), where T = T0/To = " and u = U0/Uj = 1, surrounded by a hot gas. These two region are separated by a transition layer where T − " " and 1 − u 1. In planar jets the front thickens with distance achieving thickness of order unity at axial distances x "−(1+)Rea forcing the near-axis fluid to change slowly its velocity and temperature, being necessary distances x "−2Rea to reach values of T and 1−u of order unity. In round jets the front remains at r = a up to distances x "−(1−)(log "−1)2 where the front is forced to move radially towards the axis of the jet, reaching r = 0 at x "−1 log "−1Rea. The arrival of the front forces the change of the velocity and temperature in the near-axis region, reaching values of order unity in a far field region of characteristic length x "−1 log "−1Rea, distance comparable to that needed by the front to achieve the axis. In both geometries, the distance necessary for the fully development of the cold jet is considerably longer than that required by the isothermal jet x Rea

Liñán Martínez, Amable

Sánchez Pérez, Antonio Luis

Sánchez Sanz, Mario

New Trends in Fluid Mechanics Research is the proceedings of the Fifth International Conference on Fluid Mechanics (ICFM-V); it is the primary forum for the presentation of technological advances and research results in the fields of theoretical, experimental, and computational Fluid Mechanics. Following the previous conferences in Beijing (1987, 1993 and 1998) and Dalian (2004) organized by the Chinese Society of Theoretical and Applied Mechanics, the Scientific Committee for ICFM presents ICFM-V to provide a forum for researchers to exchange original ideas and recent advances in Fluid Mechanics and relevant interdisciplinary subjects. Topics include: flow instability and turbulence, aerodynamics and gas dynamics, hydrodynamics, industrial and environmental fluid mechanics, biofluid mechanics, geophysical fluid mechanics, plasma and magneto-hydrodynamics, multiphase flows, non-Newtonian flows and flows in porous media, flow of reacting fluid, microscale flow and others

Zhuang, F. G

Li, J. C

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Proceedings of the Fifth International Conference on Fluid Mechanics (Shanghai, 2007)

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NEW TRENDS IN FLUID MECHANICS RESEARCH
Proceeding of the Fifth International Conference on Fluid Mechanics, Aug.15-19, 2007, Shanghai, China
c©2007 Tsinghua University Press & Springer
Laminar Gas Jets in High-Temperature Atmospheres
M. Sa´nchez-Sanz1∗, A. L. Sa´nchez1, A. Lin˜a´n2
1 Area de Meca´nica de Fluidos, Universidad Carlos III de Madrid, 28911 Legane´s, Spain
2 E. T. S. I. Aerona´uticos, Pl. Cardenal Cisneros 3, 28040 Madrid, Spain
Email: mssanz@ing.uc3m.es
Abstract Numerical and asymptotic methods are used to describe the structure of low-temperature
laminar gas jets discharging into a hot atmosphere of the same gas in the limit of small jet-to-ambient
temperature ratios ε = Tj/To  1. In the limit ε → 0, heat conduction cannot modify significantly
the temperature in the cold gas, leading to a two-region flow structure consisting of a neatly defined
unperturbed cold jet for r < rf(x), where T = T
′/To = ε and u = U
′/Uj = 1, surrounded by a hot gas.
These two region are separated by a transition layer where T − ε ∼ ε and 1− u  1. In planar jets
the front thickens with distance achieving thickness of order unity at axial distances x ∼ ε−(1+σ)Rea
forcing the near-axis fluid to change slowly its velocity and temperature, being necessary distances
x ∼ ε−2Rea to reach values of T and 1−u of order unity. In round jets the front remains at r = a up
to distances x ∼ ε−(1−σ)(log ε−1)2 where the front is forced to move radially towards the axis of the
jet, reaching r = 0 at x ∼ ε−1 log ε−1Rea. The arrival of the front forces the change of the velocity and
temperature in the near-axis region, reaching values of order unity in a far field region of characteristic
length x ∼ ε−1 log ε−1Rea, distance comparable to that needed by the front to achieve the axis. In
both geometries, the distance necessary for the fully development of the cold jet is considerably longer
than that required by the isothermal jet x ∼ Rea.
Keywords: laminar, cold gas jet, front, compressible, asymptotic
1 Introduction and formulation
This paper describes the structure of plane and round laminar gas jets discharging into a hot atmo-
sphere of the same gas in the limit of small jet-to-ambient temperature ratios ε = Tj/To  1. The
boundary-layer approximation is used to describe the slender steady solution that emerges for mod-
erately large values of Reynolds number Re = ρoUja/µo, where Uj represents the exit jet velocity, a is
its characteristic transverse dimension (the initial radius for the round jet and the initial half-width
for the planar jet) and ρo and µo are the ambient values of the density and viscosity.
The convection-diffusion balance indicates that at downstream distances from the jet exit of order
Rea the acceleration due to the shear stresses and the cooling due to heat conduction have reached
transverse distances of order a in the ambient fluid, generating a coflowing stream surrounding the
cold jet where the velocity is of order Uj. In formulating the problem in dimensionless form, we choose
to use the scales of this initial development region. Thus, the density and viscosity ρ and µ are scaled
with their ambient values, while the streamwise and transverse coordinates x and r are scaled with
Re a and a, respectively, and the axial and radial velocity components u and v are scaled with Uj and
µo/(aρo), leading to a formulation independent of the Reynolds number.
In terms of the dimensionless variables selected, the problem reduces to that of integrating
∂
∂x
(ρu) +
1
ri
∂
∂r
(ρriv) = 0 (1)
ρu
∂u
∂x
+ ρv
∂u
∂r
=
1
ri
∂
∂r
(
riµ
∂u
∂r
)
(2)
ρu
∂T
∂x
+ ρv
∂T
∂r
=
1
Pr
1
ri
∂
∂r
(
riµ
∂T
∂r
)
(3)
with initial conditions at x = 0{
0 ≤ r ≤ 1 : u− 1 = T − ε = 0
r > 1 : u− uc = T − 1 = 0 (4)
and with boundary conditions for x > 0
{
r = 0 : ∂u/∂r = v = ∂T/∂r = 0
r →∞ : u− uc = T − 1 = 0. (5)
In previous equation i = 0 for the planar jet and i = 1 in the round jet. As can be seen, the
formulation considers the presence of a coflowing outer stream, with uc representing the coflow-to-jet
velocity ratio. The Prandtl number, Pr, is assumed to be constant in the analysis. The continuity,
momentum and energy conservation equations need to be supplemented with
ρT = 1 and µ = T σ (6)
corresponding, respectively, to the equation of state written in the near isobaric approximation and
the presumed power-law dependence for the temperature variation of the viscosity and heat conduc-
tivity, where the exponent will be taken equal to σ = 0.70. Note that the solution of the previous
problem must satisfy 2ε
∫
∞
0
riu(u− uc)/Tdr = 1− uc and 2ε
∫
∞
0
riu(1− T )/Tdr = (1− ε), obtained
from the radial integration of (2) and (3) once written in conservative form.
The solution to (1)–(5) can be simplified in the case uc = 1, when u = 1 everywhere in the flowfield.
Integration of (3) with the boundary condition at r = 0 then provides v = Pr−1T σ∂T/∂r, which can
be substituted into (1) to give
∂T
∂t
− T
2
ri
∂
∂r
(
riT (σ−1)
∂T
∂r
)
= 0. (7)
This equation, to be integrated with the initial and boundary conditions indicated in (4) and (5),
describes the time evolution of a symmetric pocket of cold gas in a hot atmosphere, with t = x/Pr
representing the dimensionless time. Note that with i = 2 the equation describes the evolution of a
spherical gas pocket, a problem previously investigated by Tarifa, Crespo & Fraga [2]
They noticed that, in the limit ε  1, heat conduction cannot modify significantly the temperature
in the cold gas, leading for t  1 to a two-region flow structure consisting of a neatly defined un-
perturbed cold pocket for r < rf (t), where T = ε and v = 0, surrounded by a nearly steady region
where the temperature is given in the first approximation by T σ = 1 − rf/r, as corresponds to a
uniform mass flux Prρvr2 = rf/σ with negligible mass accumulation for r > rf . These two regions,
of comparable size, are separated by a thin transition region where T − ε ∼ ε, whose inner structure
is also quasisteady when described in terms of the local coordinate (r − rf )/εσ.
Fig. 1: Velocity and temperature evolution at the axis in plane (left plots) and round jets (right plots)
in the far field. For plane jets we compare the evolution of velocity and temperature in the
limit ε = 0 (solid line) with calculations made with ε = 0.1 (dotted lines) and ε = 0.05 (dashed
lines). For round jets (right plots) we represent the velocity deficit and temperature at the
axis for different coflow velocities uc and ε = 0.01 (solid lines), ε = 0.03 (dot-dashed lines)
and ε = 0.05 (dashed lines). The dot placed at the horizontal axis marks the point, predicted
from the asymptotic analysis carried out in the limit ε→ 0, where the front achieves the axis
xfε/ log ε
−1 = Prσ/4.
The solution for cold jets downstream from the jet exit shows also an unperturbed cold core where
T = ε and u = 1 surrounded by an outer region where the temperature and the velocity are seen to
evolve to reach its boundary values T = 1 and u = uc as r → ∞. Significant differences with the
evolution of a cold spherical pocket are however encountered.
2 The plane jet
In contrast to what happens in the spherical case, the surrounding outer gas never reaches a quasi-
steady balance. Instead, axial convection and radial transport are seen to balance everywhere in the
outer region, whose transverse extent increases continuously with downstream distance according to
(r − 1) ∼ √x. The same balance is found in the thickening transition layer separating the outer
region from the unperturbed core, which remains centered at r = 1, with a thickness that increases
continuously with distance according to (r−1) ∼ ε(1+σ)/2√x. Both regions admit self-similar solutions,
that remain valid until the unperturbed core and the transition layer merge at distances x ∼ ε−(1+σ),
where we find T − ε ∼ 1 − u ∼ ε at the jet axis. In this far field, the outer region extends over
large radial distances of order r ∼ ε−(1+σ)/2, resulting in small transverse gradients of temperature
and velocity and, consequently, reduced rates of heating and deceleration of the near-axis fluid. As a
result, the final development of the jet to attain axial values of T and 1−u of order unity is postponed
to large downstream distances of order x ∼ ε−2, giving the evolution plotted in figure .
3 The round jet
Initially, the heat coming from the hot ambient does not suffice to warm up the cold fluid emerging
from the jet. As a result, a thickening layer of very low density gas surrounds initially the jet front,
which remains at r = 1 for x ∼ O(1), much as in the case of the planar jet. However, a quasi-steady
balance with negligible axial convection eventually settles outside the jet for x  1 as the mass flux
emerging from the jet decreases. At distances x ∼ ε−(1−σ)(ln ε−1)2, the front starts moving radially
towards the axis as the cold gas is ablated from the cold jet surface, much as in the case of a spherical
gas pocket [2]. The front, with thickness r− rf ∼ εσ log(
√
x), moves slowly reaching the center of the
jet at distances that can be seen to be xf = σPr/4ε
−1 ln ε−1. Unlike the spherical case, the arrival
of the front does not mark the end of the jet evolution. The temperature and the velocity near the
axis, of order T ∼ 1−u ∼ ε when the front arrives, evolve slowly downstream to reach values of order
unity in a far field region of characteristic length ε−1 ln ε comparable to the length needed for front
propagation to the axis.
Acknowledgments
This collaborative research was supported by the Spanish MEC under Projects# ENE2005-08580-
C02-01 and ENE2005-09190-C04-01.
REFERENCES
1. Abramowitz M, Stegun A. Handbook of Mathematical functions. Dover, New York. 1965
2. Tarifa CS, Crespo A, Fraga E. A Theoretical Model for the Combustion of Droplets in Super-critical
Conditions and Gas Pockets. Astronautica Acta, 1971;17: 685-692.
3. Pai SI. Axially symmetrical jet mixing of a compressible fluid. Quar. Appl. Math., 1971;10:141-
148.
4. Pai SI. Fluid Dynamics of Jets. D. Van Nostrand Company, Toronto. 1954


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