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Journal ArticleDOI

Numerical investigation of rarefied vortex loop formation due to shock wave diffraction with the use of rorticity

16 Jun 2021-Physics of Fluids (AIP Publishing LLC AIP Publishing)-Vol. 33, Iss: 6, pp 067112
TL;DR: In this paper, the authors provide numerical results of the vortex loop formation caused by shock wave diffraction around a 90° corner using the direct simulation Monte Carlo method and the compressible Navier-Stokes equations with the appropriate Maxwell velocity slip and the von Smoluchowski temperature jump boundary conditions.
Abstract: When compressed gas is ejected from a nozzle into a low-pressure environment, the shock wave diffracts around the nozzle lip and a vortex loop will form. The phenomenon has been widely investigated in the continuum flow regime, but how the shock diffraction and vortex behave under rarefied flow conditions has not received as much attention. It is necessary to understand this transient flow in rarefied environments to improve thrust vector control and avoid potential contamination and erosion of spacecraft surfaces. This work provides numerical results of the vortex loop formation caused by shock wave diffraction around a 90° corner using the direct simulation Monte Carlo method and the compressible Navier–Stokes equations with the appropriate Maxwell velocity slip and the von Smoluchowski temperature jump boundary conditions. The Mach number and rarefaction effects on the formation and evolution of the vortex loop are discussed. A study of the transient structures of vortex loops has been performed using the rorticity concept. A relationship of mutual transformation between the rorticity and shear vectors has been discovered, demonstrating that the application of this concept is useful to understand vortex flow phenomena.

Summary (3 min read)

A. Background

  • In the past two decades, many micro-satellites, e.g., CubeSats, have been sent into space for purposes such as earth observation, telecommunication, and navigation 33 .
  • Non-electric propulsion systems (e.g. cold gas micro-thrusters) have been deployed extensively for orbit transfer and manoeuvring due to their high reliability 33 .
  • In the continuum flow regime, shock wave diffraction and compressible vortex loop formation and development have attracted attention for decades.
  • There are also some novel investigations of shock wave diffraction, such as diffraction from a curved exit from the shock tube 29 , the interaction of two perpendicular diffracting shock waves 44 and vortex merging caused by shock wave diffraction in three dimensions 6 .

B. Usage of rorticity

  • This definition is ambiguous, and in some circumstances, the vorticity concentration can not be explained to be the existence of a vortex, e.g. in turbulent flow.
  • It is unable to distinguish between rotational and irrotational structures, but both rotation and local shear movements can create vorticity.
  • In Ref.47, the building of the vortex vector or rorticity is not derived from the continuum assumption of fluid flow so that rorticity is a purely mathematical concept and can be used in both continuum and rarefied flow.
  • Since the new concepts are defined purely mathematically, and the calculation is based on linear algebra, there are no assumptions related to the fluid.
  • Hence, the concepts are appropriate for all flow conditions.

1. Mesh and time-step independence

  • The time derivative is discretised using a first-order implicit Euler scheme and a second-order central-upwind differencing scheme is applied to discretise the gradient terms, divergence terms, and diffusive terms, and also to interpolate the cell centre values to the cell faces.
  • Since the fine mesh is not computationally expensive, and the spatial resolution is better, the fine mesh with a resolution of 0.25 mm is used in the remainder of this work.

III. SIMULATION DETAILS

  • The working gas in all simulations of this work is nitrogen, and the non-dimensional parameters, including the Knudsen number and the Reynolds number, are based on the half-height of the shocktube, which is 0.01 m.
  • The mean free path of the Knudsen number is calculated according to the conditions downstream of the primary shock wave inside the shock tube.
  • The Rankine-Hugoniot relations are used to calculate the macroscopic parameters defined in each case of this work; the case setup is shown in Table II and III.
  • In the dsmcFoamPlus solver, the NTC method is used for collision partner selection, and the variable hard sphere model with Larsen-Borgnakke energy redistribution (with an inverse rotational energy collision number of 5) is used to perform the collisions.
  • The size of the domain outside the shock tube exit is varied for different cases because the computational costs are different.

IV. RESULTS AND DISCUSSION

  • There is no visible shock structure within the vortex until the shock Mach number reaches 1.5.
  • It it well-known that shock wave thickness increases with flow rarefaction and that the Navier-Stokes equations are unable to capture this physics.
  • As illustrated in experiments 19 24 and simulations 53 31 , the vortex-induced shock pair occurs due to the high-velocity magnitude within the shear layer of the primary vortex loop in the continuum flow regime.
  • As the Knudsen number increases, the embedded shock and the shear layer degenerate further.
  • It can be concluded that the high rarefaction level will simplify the inner structures of the compressible vortex loop.

1. Velocity and pressure field comparisons

  • An increase in local Knudsen number can be found in the primary shock, embedded shock, and the expansion at the exit.
  • Through the comparison between Figures 8 and 9 , the difference of the results in Figure 8 between the two solvers can be explained.
  • The axial pressure distribution profile differs between the slip flow and the higher Knudsen number regimes.

2. Rorticity and shear vector field

  • The authors call the region formed by closed-loop streamlines the vortex atmosphere.
  • Figures 11 and 12 present contours of the rorticity field and streamlines before and after the formation of the isolated rorticity region, or rorticity loop cross-section, with different shock Mach numbers, in the near continuum flow regime, as calculated using hy2Foam and dsmcFoamPlus.
  • The high rorticity magnitude at the corner in the expansion fan indicates that a fluid element experiences a significant increase of rotational kinetic energy because the rorticity is defined as twice the fluid-rotational angular velocity, according to Equation (2.23) in Ref. 47.
  • Such a phenomenon can also be confirmed from the algorithm validation in the DNS simulation result of a 2D Blasius-profile mixing layer flow in Ref. 47.
  • The vortex centre, defined from the streamlines, is constrained within the circle of influence of the shear vector field, suggesting that the core of a vortex loop built on streamlines has both rotational and shear movements but the strength of the rotational movement is much stronger than that of the shear movement.

C. Evaluation of the rotational strength of the vortices

  • With the introduction of rorticity, substituting Equation (1) into Equation ( 6), the following is obtained: EQUATION and thereby EQUATION ) Still, there are no strict rules on the limitation of this circuit size, resulting in the introduction of errors.
  • The rorticity region within a vortex atmosphere is connected to the rorticity sheet at the tube exit (.
  • The left-hand side of Equation ( 13) is the difference between the total shear vector flux and the remaining shear vector flux within a rorticity loop.

D. Geometrical characteristics of rarefied vortex loop

  • Calculating the geometrical parameters of rarefied vortex loops is helpful in the understanding of the vortex loop coverage in the flow field.
  • The authors can define the region with rorticity in the vortex described by streamlines as the rotational core of a vortex loop.
  • This equivalent diameter intuitively describes the size of a circular ring's cross-section with the same area.
  • When the shock Mach number is 1.4 and 1.3, a highly linear relationship between the equivalent radius of the rorticity ring and non-dimensional time is found.
  • The expansion caused by low pressure outside the tube leads to growth of the vortex loop in the radial direction.

E. Failure of vortex loop formation

  • Figure 23 displays the variation of the velocity and the streamlines due to the increase of rarefaction level with a shock Mach number of 1.6 in the slip flow regime.
  • The vortex loop will be diluted during its propagation and should eventually disappear when the Knudsen number limit for formation is reached.
  • Outside the tube, Figure 26 (b) presents the dimensionless tangential velocity profile calculated from DSMC at x =12 mm, which is 2 mm away from the tube exit, and t =0.12 ms.
  • The thickness of the vortex sheet increases, and the tangential velocity difference decreases significantly with an increase of the Knudsen number.
  • Typically, the Prandtl-Meyer expansion fan will not intrude into the shock tube, or the expansion will stay outside the tube exit.

V. CONCLUSIONS

  • Attention has been primarily paid to the vortex loop formation caused by shock wave diffraction over a rectangular corner in dilute gas flows.
  • Transient DSMC and compressible CFD simulations has been performed, and comparisons have been made between the results from the dsmcFoamPlus and hy2Foam where the Knudsen number allowed.
  • An increase in flow rarefaction results in the innerstructure of vortex loops becoming simpler and the flow patterns in all the simulations of this work are laminar.
  • The increase of rorticity flux in the isolated rorticity loop with time is attributed to the transformation from the shear vector to rorticity.
  • Further investigation can be conducted to explore the relationship between the vortex formation and the Knudsen number.

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Cao, Z., White, C. and Kontis, K. (2021) Numerical investigation of
rarefied vortex loop formation due to shock wave diffraction with the use of
rorticity. Physics of Fluids, 33(6), 067112. (doi: 10.1063/5.0054289).
This is the author’s final accepted version.
There may be differences between this version and the published version.
You are advised to consult the publisher’s version if you wish to cite from
it.
http://eprints.gla.ac.uk/241432/
Deposited on: 12 May 2021
Enlighten Research publications by members of the University of Glasgow
http://eprints.gla.ac.uk

Numerical investigation of rareed vortex loop formation due to shock wave
diraction with the use of rorticity
Ziqu Cao (曹子曲),
1, a)
Craig White,
1, b)
and Konstantinos Kontis
1, c)
James Watt School of Engineering, University of Glasgow, Glasgow G12 8QQ,
UK
(Dated: May 9, 2021)
When compressed gas is ejected from a nozzle into a low-pressure environment, the shock wave diracts
around the nozzle lip and a vortex loop will form. The phenomenon has been widely investigated in the
continuum ow regime, but how the shock diraction and vortex behave under rareed ow conditions has
not received as much attention. It is necessary to understand this transient ow in rareed environments
to improve thrust vector control and avoid potential contamination and erosion of spacecraft surfaces. This
work provides numerical results of the vortex loop formation caused by shock wave diraction around a 90
corner using the direct simulation Monte Carlo method and the compressible Navier-Stokes equations with
the appropriate Maxwell velocity slip and the von Smoluchowski temperature jump boundary conditions. The
Mach number and rarefaction eects on the formation and evolution of the vortex loop are discussed. A study
of the transient structures of vortex loops has been performed using the rorticity concept. A relationship of
mutual transformation between the rorticity and shear vectors has been discovered, demonstrating that the
application of this concept is useful to understand vortex ow phenomena.
Keywords: rareed; DSMC; vortex loop; shock diraction; rorticity
I. INTRODUCTION
A. Background
In the past two decades, many micro-satellites, e.g.,
CubeSats, have been sent into space for purposes such as
earth observation, telecommunication, and navigation
33
.
The continuous growth of applications for cost-eective
micro-satellites in low Earth orbit (LEO) is leading to
a requirement for specialized thruster systems that can
provide thrusts in the micro- and mili-Newton range,
in order to control their motions and orbits
8
. Micro-
propulsion systems can be classed as electric and non-
electric types
48
. The hardware of electric propulsion sys-
tems, such as pulsed plasma thrusters, is more compli-
cated than that of the non-electric type, which includes
cold gas, liquid, and solid rocket propulsion systems
48
.
The technologies used in electric propulsion systems must
be validated to be reliable before extensive practical us-
age. Non-electric propulsion systems (e.g. cold gas
micro-thrusters) have been deployed extensively for orbit
transfer and manoeuvring due to their high reliability
33
.
A common point shared by both of these propulsion tech-
nologies is that they operate by ejecting a mass of gas
from a nozzle at high velocity to produce thrust.
During the transient period as a thruster begins to re,
the sudden ejection of relatively high-pressure supersonic
gas from a nozzle into a low-pressure environment gener-
ates a shock diraction around the lip of the nozzle, re-
sulting in lateral vortex formation. This phenomenon has
a)
Electronic mail: z.cao.2@research.gla.ac.uk
b)
Electronic mail: Craig.White.2@glasgow.ac.uk
c)
Electronic mail: kostas.kontis@glasgow.ac.uk
not been studied in any great detail, particularly under
rareed ow conditions such as those found in LEO. Gen-
erally, there are three issues are related to the diracted
shock and the lateral vortex loop formation and propaga-
tion: sound generation, transport and mixing, and vortex
interactions
38
. The low Reynolds number, as a result of
the low density, in a rareed gas will inuence the sound
eld generated from the vortices. The sound pressure
level and sound frequency spectrum will be signicantly
dierent to that generated in a continuum gas and may
cause unexpected vibrations. Secondly, the ejected gas
from a nozzle contains some solid particles and liquid
droplets. There is a possibility that these fast-moving
particles and droplets may impinge on the surfaces of
satellites and spacecraft downstream of the thrusters or
against surfaces parallel to the thruster axis with the help
of the shock diraction and the vortex loop. It has been
reported that these droplets and particulates can signif-
icantly inuence operations on the International Space
Station
25
and satellites
28
, but the transient ow phenom-
ena that cause the contamination or erosion and the re-
sulting reduction of service life are not fully understood.
It is also necessary to understand the propagation ability
and coverage of a vortex loop with rarefaction eects to
improve the thrust vector control. Even if the propulsion
system used is a cold gas thruster, the lack of knowledge
of rareed transient ows could lead to an improper es-
timation of disturbing forces and heat loads. If multiple
thrusters are deployed, shock interactions and vortex in-
teractions will occur, and these complicated phenomena
cannot be well understood without fundamental knowl-
edge of rareed vortex loops.
Typical examples can be found not only in the noz-
zle of micro-thrusters but also in the pulsed discharge
of thrusters of spacecraft for attitude and orbit control
and adjustment
22
. The ow from a convergent-divergent

2
micro-nozzle will experience ow-regime variation from
the continuum regime (Kn < 0.001) within the combus-
tion or stagnation chamber, to the slip and transition
ow regime (0.01 < Kn < 10) as the ow expands in the
diverging section, to the free-molecular regime (Kn > 10)
far downstream of the nozzle exit
20
. The Knudsen num-
ber Kn here is dened as
Kn =
λ
L
where
λ
is the gas mean free path and L is a characteristic
length scale. In the limit of continuum ow, a shock wave
will diract around the corner of the nozzle exit lip as the
ow establishes, and the vortex sheet will roll up to form
a vortex loop. However, if the mean free path at the
nozzle exit is large enough, or the physical size of the
nozzle is small enough, the high Knudsen number will
inuence the ow development.
In the continuum ow regime, shock wave diraction
and compressible vortex loop formation and development
have attracted attention for decades. Brouillette and
Hébert
4
examined the propagation and interaction of
compressible vortex loops with Mach numbers ranging
from 1.0 to 2.0 using shadowgraph and schlieren pho-
tography. They classied the compressible vortex loop
structure according to the shock Mach number. If the
shock Mach number M
s
is lower than 1.43, then there is
no shock wave in the vortex loop, if 1.43 M
s
1.6 there
is an embedded shock wave, and if M
s
< 1.6, a secondary
vortex loops form ahead of the primary vortex loop.
Sun and Takayama
46
numerically studied the circula-
tion production in shock wave diraction around convex
corners. They suggested that the vorticity increases with
the wall angle and that there is a surge of vorticity at
the corner with angles from 15
to 45
, with the vorticity
tending to a constant value when the wall angle exceeds
90
. The vorticity produced by the slipstream is consid-
ered a vital portion of the total vorticity.
Shear layer development in shock wave diraction was
reported by Skews et al.
43
. The authors enlarged the
uid domain outside the shock tube exit to allow for a
longer time scale and avoid reections of expansion waves
and incident shock waves. They reported the existence
of a lambda shock and suggested that the angle between
the shear layer and the wall changes during the shear
layer development when the wall angle is larger than 20
,
which indicates that it is not self-similar.
Gnani et al.
11
employed splitters with a spike-shaped
structure to produce shock diraction and visualized the
shock and turbulence phenomena using schlieren photog-
raphy. They observed that the reected shock waves were
distorted during the process of passing through the vor-
tex, but remained continuous and were not cut o by the
vortex. The interactions between shock wave diraction
and a jet with two incident Mach numbers were investi-
gated for noise control applications
12
at a Reynolds num-
ber greater than 1 million. It was found that the co-ow
stretched the shock wave in the ow direction, and the
rounded splitter generated small periodic vortices.
There are also some novel investigations of shock wave
diraction, such as diraction from a curved exit from
the shock tube
29
, the interaction of two perpendicular
diracting shock waves
44
and vortex merging caused by
shock wave diraction in three dimensions
6
.
The studies outlined above are limited to the contin-
uum regime, with Knudsen number 0< Kn < 0.001, in
which the Reynolds number is high and the ow is often
idealised as inviscid, and therefore the Euler equation can
give a reasonably accurate model of the ow. However,
under rareed ow conditions, viscous eects remain im-
portant
27
.
In rareed conditions, shock waves have been investi-
gated numerically using various techniques, such as di-
rect simulation Monte Carlo (DSMC), kinetic solvers,
and by hybrid numerical methods in shock tubes
36,49,52
,
hypersonic ows
7,13,14,39–41
, and in plasma ow
26
. In-
vestigations of nozzle exit jet ow in the rareed condi-
tion are mainly restricted to steady-state ow, such as
hybrid numerical simulation of rareed supersonic ow
from micro-nozzles by Torre et al.
20
, rareed nozzle ow
by Deschenes and Grot
8
, and transition regime ow in a
diuser investigated by Groll
16
. To the best of the au-
thors’ knowledge, there are no reports of transient eects
during shock diraction in the rareed ow regime.
Therefore, this work will present an investigation of
shock wave diraction around a 90
corner under rar-
eed conditions with lateral vortex loop formation. The
rarefaction eect on the shock wave diraction at a Mach
number of 1.6 will be discussed. The results from a con-
tinuum solver utilising slip and jump boundary condi-
tions and a rareed gas ow solver will be compared. To
resolve the inner movements of uid elements in a vortex,
the rortex or vortex vector
47
is used in this work.
B. Usage of rorticity
Conventionally, a vortex is dened as a relatively high
vorticity region and the strength of a vortex is quantied
by the circulation Γ
47
, which is the sum of the vorticity
within a closed loop in the ow domain. However, this
denition is ambiguous, and in some circumstances, the
vorticity concentration can not be explained to be the
existence of a vortex, e.g. in turbulent ow. It is unable
to distinguish between rotational and irrotational struc-
tures, but both rotation and local shear movements can
create vorticity. For instance, in the boundary layer of
laminar ow, there is a velocity gradient along the wall’s
perpendicular direction so that the vorticity in this area
is non-zero, but no vortex exists. In Ref.47, the building
of the vortex vector or rorticity is not derived from the
continuum assumption of uid ow so that rorticity is
a purely mathematical concept and can be used in both
continuum and rareed ow.
The introduction of the rortex vector can eectively
separate the vorticity
ω
into a rotational part, rorticity

3
R, and an irrotational part, the shear vector
S, such that
ω
=
R +
S. (1)
Compared with the former eigenvalue-based vortex iden-
tication criteria, including Q-criterion and
λ
ci
-criterion,
the rorticity is a vector that will not be contaminated by
the shear movements and ow visualisation benets from
the rorticity eld and rorticity lines
9
. Since the new con-
cepts are dened purely mathematically, and the calcula-
tion is based on linear algebra, there are no assumptions
related to the uid. Hence, the concepts are appropriate
for all ow conditions.
II. NUMERICAL METHODS
A. Direct simulation Monte Carlo
The direct simulation Monte Carlo (DSMC) method
3
is a standard tool for investigating rareed ows with
moderate to high Knudsen number. It has been used
to simulate a wide range of rareed ow problems,
such as hypersonic vehicles
30
, rareed jets
23
, multiphase
plumes
18
, and even astrophysical ows
50
.
DSMC is a particle based stochastic method that em-
ulates the physics of real inter-particle processes and can
provide a solution to the Boltzmann equation. A large
number of real atoms/molecules are represented by each
statistically representative simulator particle, which re-
duces the computational cost of a simulation. Addition-
ally, the movements and collisions between representa-
tive particles are decoupled over a small time step, which
is a valid assumption so long as the timestep remains
much smaller than the local mean collision time of the
gas. Gas-surface interactions, e.g. a diuse reection,
are handled during this movement phase.
Once all of the particles have been moved, a stochastic
collision process takes place. The collisions must take
place between particles that are near-neighbours in order
to obtain a realistic transfer of mass, momentum, and
energy. In order to enforce this, a computational mesh
is used, in which the cells must be smaller than the local
mean free path of the gas. The computational cells are
also used to obtain volumes that are necessary to report
on ow properties such as density and temperature.
An overview of the basic algorithm that all DSMC
solvers follow can be given as:
1. Update the position of all particles in the uid do-
main using the particle tracking algorithm, which
also deals with the motion of particles across faces
of the mesh, and applies boundary conditions. Let-
ting
r represent the particle’s position,
v its veloc-
ity, and t the timestep, the mathematical form of
the movement for the i-th particle is:
r
i
(t + t) =
r
i
(t) +
v
i
(t)t =
r
i
(t) +
r
2. Prepare for the collision routine in each cell of the
domain by updating the list of particles in each cell.
3. Perform the collisions based on the collision partner
selection and binary collision models.
4. Sample the particle properties.
5. Go back to step 1 with the addition of t in time
until the end time is reached.
6. Calculate the macroscopic properties of the ow
eld
51
.
In this work, the dsmcFoamPlus solver, developed by
White et al.
51
and implemented in OpenFOAM, is used.
Transient ow simulations are performed, in which the
algorithm above is performed for each individual case
in the ensemble. Some additions to the dsmcFoamPlus
solver have been implemented to make it possible to per-
form multiple ensembles of the same simulation. These
are described in Appendix VI A.
As previously mentioned, in order to ensure near-
neighbour collisions, the cell size should be smaller than
the local mean free path. The virtual sub-cell technique
is used in dsmcFoamPlus, in which the numerical cells
are split into 8 individual collision cells. The number of
particles in each cell must be sucient to reduce the sta-
tistical error in the computed collision rates; typically at
least 20 particles per cell are required when using the no
time counter (NTC) method to calculate the number of
possible collision pairs
42
. The number of DSMC particles
in each cell can vary due to dierences in local number
density, and the cell size throughout the domain. Al-
ternative collision schemes comprise simplied and gen-
eralized Bernoulli Trial
34,45
collision schemes (SBT and
GBT), but for simplicity and as a preliminary investiga-
tion, we only consider the NTC method in this work.
The number of ensembles M to obtain a desired frac-
tional error in the local velocity E
u
for each transient case
in the DSMC method is determined according to the local
ow Mach number Ma
17
and is given by
M =
1
γ
Ac
2
N
ppc
Ma
2
E
2
u
, (2)
where
γ
is the ratio of specic heats, Ac is the acous-
tic number, and N
ppc
is the average number of DSMC
particles in the cell. In the current work, Ac can be ap-
proximated to 1. For instance, to obtain a 10% uncer-
tainty in the velocity for nitrogen gas at a Mach number
of 0.1, 286 ensembles with 25 particles in each cell are
required. If the Mach number is increased to 0.3, the
fractional error reduces to 3.33%. The fractional error of
the other volume-averaged quantities of interest; density
E
ρ
, temperature E
T
, and pressure E
P
, can be evaluated
by
E
ρ
=
1
p
MN
ppc
1
Ac
, (3)

4
E
T
=
1
p
MN
ppc
s
k
c
v
/N
A
, (4)
and
E
P
=
Ac
γ
p
MN
ppc
, (5)
respectively.
B. Navier-Stokes-Fourier Solver
The hy2Foam
5
solver is a density-based Navier-Stokes-
Fourier code, designed to solve hypersonic ow problems,
which can be characterised by high Mach number and
the presence of chemical reactions. A two-temperature
model assumes the translational and rotational temper-
ature is equal to a trans-rotational temperature, and
that the electron, electronic energy, and vibrational en-
ergy temperatures are equal to a vibrational-electron-
electronic temperature. It is specially designed to simu-
late high-speed ow in the near continuum regime, which
is computationally expensive with a DSMC solver. It has
been veried and validated with and without chemical
reactions in hypersonic ow conditions
5
. It is derived
from the rhoCentralFoam solver and is therefore based
on the central-upwind dierencing schemes of Kurganov
and Tadmor. A detailed description of hy2Foam can be
found in Ref. 5.
Behind a shockwave with high Mach number, the gas
molecules may, in general, have enough energy for the
vibrational mode to become excited and for chemical re-
actions, such as dissociation, to take place. The Mach
numbers are relatively low in the current work; hence,
the two-temperature model will be degraded to a conven-
tional single temperature. The temperature behind the
shock wave will not be high enough to promote chemi-
cal reactions. Due to the Knudsen numbers considered,
the no-slip boundary condition is not appropriate and
so the Maxwell velocity slip
5
and the Von Smoluchowski
temperature jump boundary conditions
5
will be used to
model the velocity slip and temperature jump phenom-
ena at the solid wall boundaries. Due to the small length
scales and relatively low density, the Reynolds number is
small, so the ow can be considered laminar.
1. Mesh and time-step independence
As the hy2Foam solver is based on the conventional
computational uid mechanics (CFD) method, a mesh
and time-step independence study of vortex loop forma-
tion caused by a shock wave diraction from a shock
tube at Kn = 0.005 and Ma
s
= 1.6 with dierent mesh
level and time-step conditions is carried out to ensure
that the discretisation errors have been minimized. The
computational geometry is shown in Figure 1. The time
Table I: Mesh and time-step independence study.
Case label Mesh name Grid resolution (mm) Time-step (s)
A Coarse 1.0 5 ×10
8
B Medium 0.33 5 ×10
8
C Fine 0.25 1 ×10
7
D Fine 0.25 5 ×10
8
E Fine 0.25 2.5 ×10
8
derivative is discretised using a rst-order implicit Euler
scheme and a second-order central-upwind dierencing
scheme is applied to discretise the gradient terms, diver-
gence terms, and diusive terms, and also to interpolate
the cell centre values to the cell faces.
Three levels of mesh density and time-steps were sim-
ulated, as shown in Table I. The axial pressure distribu-
tion at a time of 0.1 ms has been plotted in Figure 2. No
signicant dierence between the results of mesh density
in cases B and C is found. Since the ne mesh is not
computationally expensive, and the spatial resolution is
better, the ne mesh with a resolution of 0.25 mm is used
in the remainder of this work.
As the results with all time-steps in cases C-E are in
good agreement, as shown in Table I, a time step of 5 ×
10
8
s is chosen for all the hy2Foam cases to guarantee
that the CourantFriedrichsLewy number in all CFD
cases is around 0.1.
2. Validation of rorticity calculator code
The calculation of the rorticity elds generated in this
work is validated by studying a Burgers vortex super-
posed on a shearing motion
9
. The Burgers vortex is an
exact solution of the Navier-Stokes equation, and its ve-
locity eld is described by
u =
ξ
x
Γ
2
π
r
2
1 e
r
2
ξ
2
υ
y C
Re
ξ
˜r
2
0
y,
v =
ξ
y
Γ
2
π
r
2
1 e
r
2
ξ
2
υ
x,
w = 2
ξ
z
where
ξ
is the strain rate ,
υ
is the kinematic viscos-
ity, and the last term on the right hand side of the x-
component of velocity is a shearing motion superposed
on to the vortex eld. C is a user-dened constant, Re is
the Reynolds number which is dened as Re = Γ/(2
πυ
),
˜r
0
is a non-dimensional vortex size, equal to 1.5852. In
the validation, C = 1, Re = 10,
ξ
= 1, and the circulation
is specied as 63 m
2
/s. The spatial resolution in the XY
plane is 250 ×250.
Contours of constant vorticity are shown in Figure
3(a) and agree well with the results in Figure 3(b) from

Citations
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Proceedings ArticleDOI
03 Jan 2022
TL;DR: In this article , the authors revisited the derivation of the continuum models from the Boltzmann-Curtiss kinetic equations for diatomic and polyatomic gases, and investigated the effects of bulk viscosity, vibrational energy, and rarefaction on the hypersonic flow around a two-dimensional cylinder and supersonic flow past a cavity.
Abstract: The discrepancy between Navier Stokes Fourier-based CFD and aero thermodynamic flight test data becomes noticeable from altitudes above 50 km. It is argued that this gap is due to the shortcomings of the conventional Navier-Stokes-Fourier model. To investigate the cause of the shortcomings, we revisited the derivation of the continuum models from the Boltzmann-Curtiss kinetic equations for diatomic and polyatomic gases. We investigated the effects of bulk viscosity, vibrational energy, and rarefaction on the hypersonic flow around a two-dimensional cylinder and supersonic flow past a cavity. In addition, we analyzed the flow field in detail using a new complete compressible vorticity transport equation including the bulk viscosity.

1 citations

Book
26 Dec 2016
TL;DR: In this work, the Monte Carlo simulator OpenFOAM and Sparta have been studied and benchmarked against numerical and theoretical/experimental data for inert and chemically reactive flows and it is shown how a simulation with a mean value of one particle per cell gives sufficiently good results with very low computational resources.
Abstract: Hypersonic re-entry vehicles aerothermodynamic investigations provide fundamental information to other important disciplines like materials and structures, assisting the development of thermal protection systems (TPS) efficient and with a low weight. In the transitional flow regime, where thermal and chemical equilibrium is almost absent, a new numerical method for such studies has been introduced, the direct simulation Monte Carlo (DSMC) numerical technique. Verification and validation efforts are needed to lead to its acceptance. In this work, the Monte Carlo simulator OpenFOAM and Sparta have been studied and benchmarked against numerical and theoretical/experimental data for inert and chemically reactive flows. The results show the validity of the data found with the DSMC. It is shown how a simulation with a mean value of one particle per cell gives sufficiently good results with very low computational resources. This achievement aims to reconsider the correct investigation method in the transitional regime where both the direct simulation Monte Carlo (DSMC) and the computational fluid-dynamics (CFD) can work, can work, but with a different computational effort.

1 citations

Journal ArticleDOI
TL;DR: In this article , an opposing jet is applied to the space shuttle arc leading edge and the lifting body cone leading edge in the hypersonic condition of the rarefied flow field.
Abstract: In this paper, an opposing jet is applied to the space shuttle arc leading edge and the lifting body cone leading edge in the hypersonic condition of the rarefied flow field. The DSMC numerical algorithm is used to simulate and analyze the underlying working physics of the opposing jet. The results provide a reference for designing hypersonic vehicles in near space that reduce drag and protect against heat.

1 citations

Journal ArticleDOI
TL;DR: In this paper , the authors investigated the aerodynamic amplification effect of the initial compression wave (ICW) in circumferential cracks induced by high-speed trains and found that the ICW may be the main cause of crack propagation and concrete block formation.
Abstract: Spalling of concrete blocks from tunnel linings is a severe defect in high-speed railway tunnels (HSRTs). The amplified initial compression wave (ICW) in circumferential cracks induced by high-speed trains may be the main cause of crack propagation and concrete block formation. To investigate the aerodynamic amplification effect of the ICW in circumferential cracks, tunnel-crack models are established and solved based on the unsteady viscous k–ε turbulence method. A scaled indoor experiment is carried out to verify the reliability of the calculation method. The characteristics of amplified pressure and corresponding mechanisms are analyzed and revealed. Three influential parameters, including the crack width, crack depth, and train velocity, are analyzed and discussed in detail. The main conclusions are as follows: (1) the maximum amplified pressure in a typical circumferential crack is 5.68 times that of the ICW. (2) The maximum power spectrum density (PSD) of the aerodynamic pressure at the crack tip is 91.04 times that at the crack mouth. The crack tip suffers most from the aerodynamic impact of the fluctuating component of pressure waves, whereas the crack mouth is most susceptible to the average component. (3) The train velocity is the most influential parameter on the maximum pressure at the crack tip, followed by the crack depth. The power function with an exponent of 2.3087 is applicable for evaluating the relationship between the maximum pressure and train velocities. (4) The train velocity and crack depth are most influential parameters to the maximum PSD. The relationship between the maximum PSD and the crack widths, crack depths, and train velocities can be reasonably described by the power function. (5) The mechanism of pressure amplification is as follows: first, the superposition of the internal energy possessed by air molecules near crack surfaces. Second, the increase in the internal energy of air near the crack tip because of the gradually narrowing space. The results of our research may be applicable in analyzing the cracking behavior of tunnel lining cracks and preventing the spalling of concrete blocks in HSRTs.
References
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Journal ArticleDOI
TL;DR: In this article, the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms is analyzed and the errors depend on Mach number, Knudsen number, number of particles, etc.

276 citations

Journal ArticleDOI
TL;DR: In this article, the existence of the rotational axis is proved through real Schur decomposition, and a fast algorithm for calculating Rortex is also presented based on the real-Schur-decomposition.
Abstract: A vortex is intuitively recognized as the rotational/swirling motion of the fluids. However, an unambiguous and universally-accepted definition for vortex is yet to be achieved in the field of fluid mechanics, which is probably one of the major obstacles causing considerable confusions and misunderstandings in turbulence research. In our previous work, a new vector quantity which is called vortex vector was proposed to accurately describe the local fluid rotation and clearly display vortical structures. In this paper, the definition of the vortex vector, named Rortex here, is revisited from the mathematical perspective. The existence of the rotational axis is proved through real Schur decomposition. Based on real Schur decomposition, a fast algorithm for calculating Rortex is also presented. In addition, new vorticity tensor and vector decompositions are introduced: the vorticity tensor is decomposed to a rigidly rotational part and an anti-symmetric deformation part, and the vorticity vector is decomposed to a rigidly rotational vector and a non-rotational vector. Several cases, including 2D Couette flow, 2D rigid rotational flow and 3D boundary layer transition on a flat plate, are studied to demonstrate the justification of the definition of Rortex. It can be observed that Rortex identifies both the precise swirling strength and the rotational axis, and thus it can reasonably represent the local fluid rotation and provide a new powerful tool for vortex dynamics and turbulence research.

273 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of the possible rotational axis is proved through real Schur decomposition, and a fast algorithm for calculating Rortex is also presented, which can reasonably represent the local fluid rotation and provide a new powerful tool for vortex dynamics and turbulence research.
Abstract: A vortex is intuitively recognized as the rotational/swirling motion of the fluids. However, an unambiguous and universally accepted definition for vortex is yet to be achieved in the field of fluid mechanics, which is probably one of the major obstacles causing considerable confusions and misunderstandings in turbulence research. In our previous work, a new vector quantity that is called vortex vector was proposed to accurately describe the local fluid rotation and clearly display vortical structures. In this paper, the definition of the vortex vector, named Rortex here, is revisited from the mathematical perspective. The existence of the possible rotational axis is proved through real Schur decomposition. Based on real Schur decomposition, a fast algorithm for calculating Rortex is also presented. In addition, new vorticity tensor and vector decompositions are introduced: the vorticity tensor is decomposed to a rigidly rotational part and a non-rotationally anti-symmetric part, and the vorticity vector is decomposed to a rigidly rotational vector which is called the Rortex vector and a non-rotational vector which is called the shear vector. Several cases, including the 2D Couette flow, 2D rigid rotational flow, and 3D boundary layer transition on a flat plate, are studied to demonstrate the justification of the definition of Rortex. It can be observed that Rortex identifies both the precise swirling strength and the rotational axis, and thus it can reasonably represent the local fluid rotation and provide a new powerful tool for vortex dynamics and turbulence research.

262 citations

Journal ArticleDOI
TL;DR: In this article, an alternative eigenvector-based definition of Rortex is introduced, in which the direction of the possible axis of the local rotation is determined by the real eigen vector of the velocity gradient tensor.
Abstract: Most of the current Eulerian vortex identification criteria, including the Q criterion and the λci criterion, are exclusively determined by the eigenvalues of the velocity gradient tensor or the related invariants and thereby can be regarded as eigenvalue-based criteria. However, these criteria will be plagued with two shortcomings: (1) these criteria fail to identify the swirl axis or orientation; (2) these criteria are prone to severe contamination by shearing. To address these issues, a new vector named Rortex which represents the local fluid rotation was proposed in our previous work. In this paper, an alternative eigenvector-based definition of Rortex is introduced. The direction of Rortex, which represents the possible axis of the local rotation, is determined by the real eigenvector of the velocity gradient tensor. And then the rotational strength obtained in the plane perpendicular to the possible axis is used to define the magnitude of Rortex. This new equivalent definition allows a much more efficient implementation. Furthermore, a systematic interpretation of scalar, vector, and tensor versions of Rortex is presented. By relying on the tensor interpretation, the velocity gradient tensor is decomposed to a rigid rotation part and a non-rotational part including shearing, stretching, and compression, different from the traditional symmetric and anti-symmetric tensor decomposition. It can be observed that shearing always manifests its effect on the imaginary part of the complex eigenvalues and consequently contaminates eigenvalue-based criteria, while Rortex can exclude the shearing contamination and accurately quantify the local rotational strength. In addition, in contrast to eigenvalue-based criteria, not only the iso-surface of Rortex but also the Rortex vectors and the Rortex lines can be applied to investigate vortical structures. Several comparative studies on simple examples and realistic flows are studied to confirm the superiority of Rortex.

210 citations

Journal ArticleDOI
TL;DR: A hybrid numerical scheme designed for hypersonic non-equilibrium flows is presented which solves the Navier-Stokes equations in regions of near-equ equilibrium and uses the direct simulation Monte Carlo method where the flow is in non-Equilibrium.

173 citations

Frequently Asked Questions (12)
Q1. What have the authors contributed in "Numerical investigation of rarefied vortex loop formation due to shock wave diffraction with the use of rorticity" ?

In this paper, the authors investigated the relationship between the vortex formation and the Knudsen number, and compared the results from the dsmcFoamPlus and hy2FoAM simulations where the knudsen numbers allowed. 

During the transient period as a thruster begins to fire, the sudden ejection of relatively high-pressure supersonic gas from a nozzle into a low-pressure environment generates a shock diffraction around the lip of the nozzle, resulting in lateral vortex formation. 

As the rorticity field of a vortex loop core is an irregular shape, an equivalent diameter of a vortex is defined asdeq = √ AR π18H , where t∗0 is the time that the rorticity loop formed and it is 0.25ms, 0.18ms, 0.15ms,0.13ms, 0.1ms for Ms = 1.3, Ms = 1.4, Ms = 1.5, Ms = 1.6, Ms = 2.0, respectively. 

The number of particles in each cell must be sufficient to reduce the statistical error in the computed collision rates; typically at least 20 particles per cell are required when using the no time counter (NTC) method to calculate the number of possible collision pairs42. 

Increasing the shock Mach number causes the shape of the rorticity loop cross-section to change comma-like shape to a mushroom shape. 

The direct simulation Monte Carlo (DSMC) method3 is a standard tool for investigating rarefied flows with moderate to high Knudsen number. 

The circulation can be decomposed into a rorticity flux that describes the fluid-rotational strength of a vortex and a shear vector flux that represents the shear strength of a vortex. 

The continuous growth of applications for cost-effective micro-satellites in low Earth orbit (LEO) is leading to a requirement for specialized thruster systems that can provide thrusts in the micro- and mili-Newton range, in order to control their motions and orbits8. 

For instance, to obtain a 10% uncertainty in the velocity for nitrogen gas at a Mach number of 0.1, 286 ensembles with 25 particles in each cell are required. 

The authors enlarged the fluid domain outside the shock tube exit to allow for a longer time scale and avoid reflections of expansion waves and incident shock waves. 

The rorticity field always coincides with the closed-loop streamlines if the vortex is stationary in the plane perpendicular to the rotational axis, such as in a Taylor-Green vortex sheet or a Burgers vortex, and it does not coincide with the closed-loop streamlines if the vortex translates. 

An overview of the basic algorithm that all DSMC solvers follow can be given as:1. Update the position of all particles in the fluid domain using the particle tracking algorithm, which also deals with the motion of particles across faces of the mesh, and applies boundary conditions.