Grid Free Lagrangian Blobs Vortex Method With Brinkman Layer Domain Embedding Approach for Heterogeneous Unsteady Thermo Fluid Dynamics Problems

C. Golia, B. Buonomo & A. Viviani
International Journal of Engineering, (IJE) Volume (3): Issue (3) 313
Grid Free Lagrangian Blobs Vortex Method with Brinkman Layer
Domain Embedding Approach for Heterogeneous Unsteady
Thermo Fluid Dynamics Problems
Carmine GOLIA carmine.golia@unina2.it
Faculty of Engineering, Department of
Aerospace and Mechanical Engineering
Second University of Naples
Via Roma 29, Aversa, 81031 Italy
Bernardo BUONOMO bernardo.buonomo@unina2.it
Antonio VIVIANI antonio.viviani@unina2.it
Abstract
Modeling unsteady thermal–viscous flows inside/around complicated geometries
containing multiphase sub-systems (fluid–porous–solid) and multi-physics
phenomena (diffusion; forced/free/mixed convection; time variations of velocity,
temperatures and heat fluxes sources; still and moving bodies) is an ambitious
challenge in many applications of interest in science and engineering. Scope of
this exploratory work is to investigate if the combination of a grid free Lagrangian
Blob method with a Brinkman layer domain embedding approach can be useful
for the preliminary analysis of heterogeneous unsteady thermal buoyant
problems, where easiness, readiness, short computational times, good
qualitative and sufficient quantitative accuracy are the most important aspects. In
this work we couple a grid free unsteady Lagrangian Thermal-Vortex Blob
method with a double penalization method that considers solid bodies contoured
by a fictitious buffer thin boundary layer, described by a porous Brinkman model.
The model problem in this study is the interaction of a thermal buoyant plume
with a solid body, still or in motion. Both solid body and Brinkman boundary layer
are described by volume penalization applied by an unsteady mask method.
After a description of this novel approach, preliminary analyses for validations are
presented for various thermal buoyant steady/unsteady problems relative to
thermal and thermal-vortical patches, fixed, free and in presence of a still or
moving body. Comments on pros, contras and further work, conclude the paper.
Keywords: Buoyant Plume/Body Interaction, Lagrangian Methods, Thermal/Vortex Blobs, Grid Free,
Volume Penalization, Brinkman Domain Embedding

1. INTRODUCTION
The Brinkman domain embedding approach introduces a penalty term [treated as a continuous]
directly in the set of global equations over the whole flow domain, this term takes different order
of magnitudes according to the particular sub domain. This approach makes the viscous body
analysis almost independent on the grid domain, coordinate system, since it allows the analysis
of heterogeneous problems with quite complicated geometries by using simple mask functions,
and it can be used almost independently on the solution method used (FEM, FDM, Spectral
method, etc.).
The advantages of this formulation are:
• Single (Cartesian or other) mesh approach to the global field, avoiding body-fitted
unstructured mesh,
• Physical description of the sub domain (shape, position, size, dynamic and diffusive
parameters) by simple mask functions [that can be still or can move according to the
dynamics of the body].
• The warranted continuity of the diffusive fluxes across the interfaces implicitly embedded in
the global equation. This automatically takes account of conjugated heat transfer problems
and of coupling problems among heterogeneous media.
FIGURE 1: Embedding Domains.
This approach, proposed by Arquis & Caltagirone [1] and validated in Ref.s [2], [3], [4], [5], was
widely used as penalization techniques for the simulation of flow around solid bodies in the
contest of grid dependant computation methods, i.e. adaptive wavelet method by Schneider et al.
([6], [7], [8], [9], [10]); hybrid wavelet method by Vasilyev and Kevlahan [11], Kevlahan et al. [12];
spectral methods by Kevlahan and Ghidaglia [13].
The method has also been successfully used to simulate transitional and turbulent flows by an
array of cylinders using Cartesian grids by Bruneau and Mortazavi [14] and extended to
compressible flows by Chiavassa and Donat [15].
2. PRELIMINARIES
The physical idea of the penalization technique is to model the obstacle as a porous medium with
porosity tending to zero. This corresponds to a Brinkman-type model with a permeability that
varies according to the space domain. We can consider the whole field, treated as a global
Brinkman porous medium, with properties locally varying according to the particular body treated
as a specific porous media. The Navier-Stokes system is modified by adding a supplementary

term with the idea of forcing the velocity to satisfy the no-slip condition on the boundary of the
obstacle. The new system is then solved in an obstacle–free computational domain.
Momentum (Navier-Stokes):
( ) ( ) ( )
 µ −∂    + • ∇ = −∇ + ∇ • µ ∇ + ∇ − + ∆ρ +
    ∂ ρ
  
bodyt
T
rif
V VV 1 1
V V p V V g f
t Re ReDa K
(1)
The value of the local specific permeability, K, models the specific embedded medium, i.e.:
( )
f
p
s
K if r fluid region
K t,r K if r porousregion
K 0 if r solid region+
 → +∞ ∈
= ∈

→ ∈
(2)
• In case of fluid, the “Darcy drag“is negligible with respect to the other terms and (1) reduces
to the classical Navier Stokes equation.
• In case of solid, we consider a medium with porosity almost unitary and permeability nearly
zero.
• In case of porous region, the “Darcy drag” term takes the role of a penalty term that imposes
low velocity, so that the convective terms become negligible and (1) reduces to the classical
Brinkman equation used in porous media theory:
( ) ( )
( )
 µ ∇ − ∇ • µ ∇ + ∇ + = ∆ρ + 
   ρ  
t
T
p o
V1 1
p V V g f
Re ReDa K
(3)
Note that the prolongation of the flow parameters [pressure, velocity and temperature inside the
solid media] makes the continuity conditions on the velocity and temperature field and diffusive
fluxes satisfied on and across the boundaries.
Problems may arise from the fact that in the regions where penalization constraints are imposed,
the formulation is first order [as in all penalizations] and the equations become very stiff to
integrate in time.
2.1 Brinkman double penalization method
Angot et al. [3] first proposed a “double penalization method” where a solid body is bounded by a
porous Brinkman layer. This approach was used by Bruneau and Mortazavi [16, 17] to compute
the viscous flow around a ground vehicle surrounded by a thin layer of porous material. Carbou
[18], [19] explained the good performances of this method characterizing the boundary layer that
appears over the obstacle, and in [20] by using a BKW method performed an asymptotic
expansion of the solution when a little parameter, measuring the thickness of the thin layer and
the inverse of the penalization coefficient, tending to zero, proved that the method is equivalent to
the computations using greater thickness model by a standard Brinkman Model
2.2 Lagrangian Blob Method
The blob concept [21] is based on an attempt to make discrete the free space Dirac
representation of a generic function “ f ” at the location (x) and time (t):
( ) ( ) ( )= δ −∫f x,t f x ',t x x ' dx ' (4)
This representation can be made discrete, in an Eulerian formulation, as follows:
( ) ( )
− 
=  
 
 
∫
p q
p q
x x
f x ,t f x ,t W ,h d
h
V
V (5)

where the kernel function W( r , h) satisfies given properties in order that, in the limit h→0, the
two representations must coincide ( h can be regarded as the grid size ).
The Lagrangian Blob formulation is aimed to represent the value of the given field function “ f ” for
the p-th particle that at time (t) is located in the position (xp). This requires further modifications of
eq. (5) by considering, for a finite number of particle within a cluster around the pth-particle, the
following definition:
( ) ( )
∈
− 
= ∆ 
 
 
∑ p q
qp q
q Cluster
aroundp
x x
f x ,t f x ,t W ,h Vol
h
(6)
where ∆Volq is the finite elementary volume associated with the q-th particle.
FIGURE 2: Kernel Function for the Blob Representation.
It is easy to observe that the approximations deriving from the Lagrangian Blob formulation
depends on three facts:
1. The choice of the kernel function W(r, h) that must be rapidly decay with r (either Gaussian
like or a polynomials compact over x/r usually equal to 1-2 ),
2. The computation of the integral in eq. (5) (approximated by a simple summation),
3. The finite number of particles considered in the cluster (depending on the cluster’s radius).
It must be pointed out that:
• in order to allow a suitable overlap of the volumes associated to adjoining blobs a further
“mollification” is performed as:
( ) ( ) p q
p p q q q
q Cluster
around p
r - r
f r = f r W , h ∆Vol
σ∈
 
 
 
 
∑,t ,t (7)
where " σ = h γ “ is defined as the blob’s radius and "γ " is an overlap parameter (usually of
the order of 1 – 1.5).
• In order to be dimensionally consistent the kernel function W(r, h) must have the dimension
of the inverse of a Volume (according to the dimension “d” of the space considered). Since it
is usual to consider as kernels non-dimensional functions rapidly decaying, the "mollification"
takes the form:

( ) ( ){ } ( )p q p q
p p q q q q
q Cluster q Cluster
around p around p
r - r r - r
f r = W ,h f r ∆Vol W , h r
σ σσ σ∈ ∈
   
   =
   
   
∑ ∑ qd d
1 1
,t ,t F ,t (8)
where Fq(rq,t)= fq ∆Volp is considered as the “field intensity” of the variable “fq” over the finite
volume ∆Vp.
Note that the blob representation is defined as a space average (i.e. it is an implicit sub-scale
model), that is: the Lagrangian blob method for the Helmholtz formulation of the complete Navier-
Stokes equations, under the limit of validity of the Boussinesq approximation, represents the
space averaged equivalent equations:
( ) ( )
( )
2
2
ω
+ • V ω = • ω V + ν ω -β g T
t
T
+ • V T = α T
t
∂  ∇ ∇ ∇ • ∇   ∂

∂
∇ ∇ ∂
(9)
So that, if the “h” parameter and the time integration step are small enough with respect to
turbulent scale, the Lagrangian Blob method does not need any turbulent stress term, and it can
be considered to represent a DNS method.
The Blob particles representation is linked to important Lagrangian conservation properties. For a
general advection problem Lu=g , the blob representation satisfy the “self-adjoint-ness” condition:
( ) ( ) ( )
T T
0 0
Lu (•,t) , W •, t dt = u (•,0) ,L* W •,0 + g(t) ,L*W •, t dt∫ ∫ (10)
where W,Lu is the (space averaged) blob representation of the operator Lu, with L* its adjoint.
Obviously the Lagrangian Blob formulation of a differential problem needs the representation of
all the differential operators in the problems, namely divergence, gradient and Laplacian. This
may be a problem that is solved in various manners, as it will be reported later.
In conclusion Blob methods represent exact weak solution for any admissible test function (local
averaged equation), i.e blob particle method achieve some (implicit) sub-grid scale model. Blobs
are "Dirac particles" that directly translate with flow and transport extensive properties. They
move according to velocity field, and exchange momentum and energy with neighborhood blob
particles according to diffusive processes.
Blob methods then differ from classical grid techniques since they do not involve projection of the
equation in a finite dimensional space. They are suitable for any advection and diffusion problem
such as: Navier Stokes equations ; Multiphase media; Gas dynamics; Vlasov-Maxwell equation.
3. THE MODEL PROBLEM
The objective is to consider an obstacle–free computational domain and in the frame work of a
low order unsteady scheme we are aimed to use a vorticity– velocity– temperature particle blob
approach with a Lagrangian formulation that is typically second order.
The present approach considers the analysis of the Helmholtz equations that are obtained as the
curl of the momentum term of the Navier Stokes. In this formulation the pressure disappear and
the velocity field is implicitly solenoidal. The problem of solenoidality that deals with the vorticity
field does not arise in 2D problems that are the scope of this explorative analysis.
This approach avoid the imposition of the divergence free velocity field and of the asymptotic
pressure closure condition needed for fixing mass flow in buoyant problems that is very difficult to
impose and represent one of the weak points of many industrial codes.

The formulation, Golia et al. [22], [23], [24], [25], furthermore makes use of a splitting technique of
Chorin [26], [27], that separates explicitly convective and diffuses steps, recasting the problem in
a hyperbolic one for the particle and in parabolic problems for the diffusive and source
phenomena that occur along each particle path.
We do not consider flows around solid body by Brinkman double penalization, Fig. (3). In this
regard we shall consider 3 zones (fluid, Brinkman layer, solid body) that are defined by a time
varying mask function.Then for the generic p-th particle located at rp at time t, we must solve the
following initial value problem (in case of constant diffusive properties):
Convective steps (trajectory of the p-th particle):
=
p
p
d r
U(r ,t)
dt
(11)
Diffusive Step (variations of the field properties of the p-th particle along its trajectory):
( )
2
pp
p
p body2
p
p body
1
g T fluid
Re
V V1t
Brink. Layer
Re
V Solid Body
  ∇ ω − β ∧ ∇  
∂ω 
 =  ∇ ∧ −∂   ∇ ω −
  η
   
ω = ∇ ∧
(12)
p 2
p
T 1
T
t Re Pr
∂
= ∇
∂
(13)
FIGURE 3: The Brinkman Layer.
Note that:
• in the buoyant term ‘β’ denotes the thermal coefficient of expansion
• η <<1 is a penalization parameter
• we have not considered the dissipative term in the energy equation since it is very small
and it is usually discarded in case of buoyancy.
• we do not consider buoyancy in the Brinkman Layer

• the error introduced, see [3], is expected to be of the order η
-1/4
The Blob representation of the eqns. (11), (12), (13) - detailed elsewhere [22], [23], [24], [25] -
considers as field unknowns the p-th particle local “Vortex intensity” Γp=ωp ∆Vp (i.e. local velocity
circulation) and the p-th particle local “Thermal intensity” Θp=Τp ∆Vp .
The velocity field needed in (12) is computed using an Helmholtz decomposition, i.e. as sum of a
potential velocity field and a vortical velocity field computed with a generalized Biot-Savart law.
This is a weak point of the Blob Vorticity Methods since the velocity of a given blob, induced by all
the N vortex blobs present in the field, represents a classical N-Body problem that require O(N
2
)
calculations. The deriving computational burden is obviously not acceptable in problems where N
increase very rapidly in time. To overcome this fact, we use an "in house" Adaptive Fast Multipole
Method (FMM) [28] that results to be a O(N) algorithm. The FMM was devised to be capable of
self organize in order to reach optimal computation conditions while N is varying and able
warrant an imposed error level on the calculation of the velocity field induced by the vortex blobs.
The blob representations of the Laplacian diffusive terms and of the Gradient terms present in
(12) and (13) are performed according to the Particle Strength Exchange (PSE) method proposed
by Degond & Mas-Gallic [29]:
( ) ( )
=
∇ Γ = Γ − Γ −∑
N
2
p p q p q
q 1
Lapl r r ,h (14)
( )( ) ( )
=
∇ Θ = − Θ + Θ −∑
N
p p qp q p q
q 1
r r Grd r r ,h (15)
The kernels Lapl(•,•) and Grd(•,•) used here are the high order ones according to Eldredge et alia
[30]. The PSE discretization of the differential operators are quite accurate for blobs regularly
distributed in the field and away from boundaries/discontinuities. Usually, after a number of
integration steps, to avoid inaccuracy due to the typical Lagrangian distortion of the particle field,
a re-gridding process is performed to project the field on a regular mesh. New particles are
allocated on the new mesh points and their values are interpolated from the cluster of the
neighborhood ones, on the old disordered lattice, with a suitable kernel function:
( )
=
 = −
  ∑
Neighborhood
Cluster
new oldnew old
p q regrid p q
q 1
F F W r r , h (16)
The kernel Wregrid(•,•) used here is compact over r/h=2.5 and is third order accurate [31].
After the discretization of each term of (11), (12), (13), it results in an initial value problem (I.V.P.)
problem that requires the time integration of N-equations. This is performed, in general with a 2nd
order algorithm.
It must be said that for particles located within the Brinkman Layer the problem is extremely stiff.
The time integration can produce oscillations or inaccuracies if the time step “dt” is not enough
smaller then the value of the penalty parameter “η”. This is a burden since in order to have
sufficient accuracy the penalty parameter “η” must be at least of order of 10
-4
.
In this work we use different strategies for the time integration for particles located in particular
volume phase. For fluid particle we use a 3rd order Adams-Bashforth method that computes the
"n+1 time step" values using the old values at the "n time step" and at the "n-1 time step".
For particles located in the Brinkman Layer we use a first order implicit method. Future work will
test various methods to improve accuracy in order to use larger values of the ratios of time steps
over penalty parameters.

4. THE VALIDATION
Validations are needed to ascertain the preliminary accuracy of the original Thermal-Blob method
before the use of the Double Brinkman Penalization. In the following we shall present various
tests that consecutively validate each particular segments of the method and of the relative code.
4.1 Inviscid elliptical vortex blob patches
Scope of this test is to validate the accuracy of the routines for computation of the velocity field
induced by the vortex blobs and for the time integration of the blob’s particle paths.
We do consider a classical inviscid Elliptical Vortex Blob Patches (Lambs’ vortex) that [32] will
rotate steadily and as rigid body, according to the value of the (constant) patch vorticity and to the
values of the semi axes a, b, with the angular velocity:
( )
rot patch 2
ab
a b
Ω = ω
+
(17)
The time frames depicted in Fig. 4 show the almost perfect rigid body rotation (no regrinds), thus
validating the computation of the induced velocity and of the time integration routines.
FIGURE 4: Inviscid Elliptic Patch at Various Times.
Nx=Ny=81; dx=dy=0.00625; dt=0.00125, ωpatch = π , Vel. Acc.= 10
-4
4.2 Free buoyant thermal blob patches
Scope of this test is the validation of the routines that compute Laplacian (diffusion processes)
and gradients, the correctness of the buoyant term and the effectiveness of the regrid processes.
We consider an initially Thermal Free Patch in a gravity field undergoing buoyant and diffusion
processes. The patch rising will split in two anti rotating cores due to the vorticity created by the
buoyancy term in eqn.(12). Geometrical symmetry is expected and global vorticity must be
conserved to zero.
Fig. 5 represents the real computational field and the evolution of the blobs particles, colored
according to their temperature. It is interesting to note as the Lagrangian formulation allows the
code to be completely adaptively and grid free. The blobs naturally go where needed and the
code automatically creates new particles according to the vorticity generations.
The run corresponds to the following parameters: Gr=0.65107; Pr=0.748; h=0.01; dt=0.0125. At
t=0, the initial, number of blobs is 2511 and grows spontaneously up to 8919 , at t=5.

FIGURE 5: Free Plume Convection of Thermal Free Patch Initially Circular.
4.3 Free buoyant thermal-vortex blob patches
Scope of this test is the validation of the coupling of the thermal buoyant process with the
dynamics effects of a non zero initial vortex patch. Here we consider an initially circular Free
Patch of blob particles, having both Thermal and Vortex content, placed in a gravity field, that
undergoes buoyant and diffusion processes.
Due to the vorticity created by the buoyancy term in (12), the rising patch will split in two anti
rotating cores. But in this case geometrical symmetry is not maintained due to the fact that global
vorticity must be conserved to the non zero initial value.
FIGURE 6 : Free Plume Convection of Vortex-Thermal Free Patch Initially Circular.

FIGURE 7: Free Buoyant Rising Plume from a Thermal Blob Patch at Different Grashof Numbers.
As shown in [33], integral balance laws impose according to the sign of the initial vorticity, that
one of the two cores rotates more (i.e. will have larger vorticity) than the other.
Fig. 6 reports the evolution of the iso-vorticity contours. The computational parameters are:
Pr=0.748; h=0.1, dt = 0.00125, Vref=0.125 10-3; Lref = 0.1188, Rey=1, regrid process is performed
after 10 time steps.
It is interesting to note that, as predicted, the initial circular blob path rising is split in two counter-
rotating ones, the left one having a large vorticity content of the other, then having higher angular
rotation velocity. Since, the global of vorticity content must be constant as the initial one, the
rotation difference of the two cores will vary in time. Therefore the iso-vorticity contours undergo
higher relative changes with respect to the thermal ones.
4.4 Fixed buoyant thermal blob patch
Scope of the test is to verify the free buoyant rising plume that stems from a Thermal Blob Patch
which is maintained fixed and at a constant temperature. The same Patch will be hereafter
considered to generate a thermal plume that shall interact with solid bodies (still or moving).
Fig.7 reports the thermal contours of the rising plume at various times, for different Grashof
numbers: the top row refers to Gr=0.25 10
8
, the bottom row refers to Gr=0.625 10
8
. Temperature
level refers to temperature rise with respect to the asymptotic one. Both case are relative to an
elliptical fixed thermal patch having a=0.0198 and b=3a. The run corresponds to the following
parameters: Gr = 0.65 10
7
; ReyΩD = 0.243 10
4
; Pr = 0.748 ; h = 0.01 ; ∆t = 0.0125. The initial (at
t=0) number of blobs is 41311, and grows spontaneously up to 67220, at t=3,.
Even if the value of the reference blob distance "h" is not small enough to reach turbulent scale, it
is evident, from the figure, the capacity of the method and of the code to catch the instability of
the rising plume that is verified at a certain rising distance from the standing thermal patch.
5. PLUME-BODY INTERACTIONS
In this paragraph we consider the interaction of the thermal buoyant plume (described in the
previous paragraph) with a body using the Brinkman Layer as per Fig. 3. In the following, we shall
present the interactions of the rising plume with a still blunt body, with a still slender body and
with a slender moving body (linear motion and rotation).
5.1 Plume interaction with a still blunt body
The data of the thermal plume are as the ones of the top row of Fig. 7.

The body is an ellipse b/a=3 staying still above a thermal patch that originate a buoyant plume.
The body is at zero temperature rise and the Brinkman layer has a thickness twice the blob
nominal distance "h".
Fig. 8 shows three frames of a video clip referring to the touching of the plume mushroom with
the Brinkman Layer of the body and two other significant later states, namely the breaking of the
mushroom and the lateral migration of the two counter-rotating vortices generated by the
interaction.
FIGURE 8: Plume Interaction with a Still Blunt Body Simulated with Brinkman Layer.
5.2 Plume interaction with a still slender body
The same comments as the ones of paragraph 5.1 apply, but here the elliptical body, as depicted
in Fig. 9, is rotated by 90° normally to the gravity vector to represent a slender obstacle for the
rising plume. In this case a quite lighter interaction is expected and the migration of the two
counter-rotating vortices generated by the interaction is quite faster. The technique used to
highlight the temperature contours emphasizes the automatic dynamic development of the
computation field that, at the initial stages, does not contain the body.
FIGURE 9: Plume Interaction with a Still Slender Body Simulated with Brinkman Layer.
5.3 Plume interaction with a transverse moving body
The data of the thermal plume are as the ones of the top row of Fig. 7. The body is an ellipse with
b/a=3, oscillating horizontally (normally to the gravity vector) with an amplitude of 0.2 and a
period of 0.82. The body is at zero temperature rise and the Brinkman layer has a thickness twice
the blob nominal distance "h".
The images in Fig. 10 report the blobs position and the velocity vector. In the top row the blobs
are colored according to the local vorticity, in the bottom row the blobs are colored according the
local temperature rise.

As a check, the invariance of the zero total vorticity over a full period of oscillation is verified. The
oscillatory motion of the solid body is synchronized such that at time t=1 the plume top mushroom
hits the bottom of the body.
FIGURE 10: Thermal Plume Interacting with a Slender Body Oscillating Laterally (contours).
Figure 10 reports some shots that reveal the strike and the relative hysteresis phenomena of the
flow pattern. The t = 2.64 shot refers to the ellipse moving west ward, the t = 2.96 shot refers to
the ellipse moving eastwards. Obviously the running of the video clip can better reveal the
dynamics of the interaction and the realistic behaviors of such a complicate unsteady interaction.
5.4 Plume interaction with a rotating body
In this case we consider a thermal plume generated by a thermal fixed particle cluster posed
below a circular body that rotates with a given angular speed Ωb anti-clockwise.
Fig.11 refers to the case Ωb= 10 [rad/s] with hot blob cluster aligned below the rotating body.
Fig.12 refers to the case Ωb= 20 [rad/s] with hot blob cluster aligned below the rotating body.
Fig.13 refers to the case Ωb= 30 [rad/s] with hot blob cluster misaligned below the rotating body.
The six shots of each Figure refer to different times and display blobs colored according the
specific temperature value. We can notice:

FIGURE 11: Thermal Plume Interacting with a Slender Body Rotating Anti-Clockwise at 10 rad/s.
Plume Generated by a Thermal Fixed Cluster Aligned with the Rotating Body.
Blobs Colored with the Specific Thermal Level.
• the capacity of the approach to generate automatically blobs, as needed, that moves in the
region where thermal and vorticity effects are relevant.
• the different dynamic behavior of the plume:
o In the initial stage the steam of the plume is linked with the mushroom cap that contains
anti-rotating vortical structures: the sting furnish energy to the cup enough to overcame
the loss of thermal and kinetic energy due to heat conduction and to viscous effects: it
derives a certain stability for the sting;
o In the latter stage a pinch-off phenomena is verified: the sting separates for the cup that
is no longer feden;
o It derive that the cup tends to decrease the rotation speed and to fragmentize whereas
the sting loses it stability and oscillates laterally (as a candle) due, from one side, to the
rotation of the top body and from the other side, to the need to obey to conservation of its
global vorticity;
o This is evident in the last shots of all figures, but in particular in Fig. 13 where the sting is
first moved away by the anti-clockwise rotation of the body and after is attracted.

Plume Generated by a Thermal Fixed Cluster Aligned with the Rotating Body.
6. CONCLUSIONS & FUTURE WORK
The use of the Brinkman penalization method to simulate the layer over a body considered as a
porous media appears compatible with the grid free Lagrangian Vortex and Thermal Method
developed by the authors. This method seems quite sound for simulating the viscous interaction
of plumes and bodies since is based on fluid dynamics considerations that confine diffusive
phenomena in segregated regions near the body.
The first experience is positive but further works are required mainly in relation to the time
integration of the diffusion processes for particles within the Brinkman Layer. The use of implicit
scheme was experienced to be necessary, but further analysis is recommended in order to
optimize peculiar schemes to improve the ratio of the time step interval over the penalization
parameter. It must be underlined that dealing with unsteady motion of the body was quite simple
and the relative effort limited mainly to the development of an efficient masking procedure, and to
the of the time integration step that must be small enough with respect to the oscillation period.
In this preliminary analysis we did not consider the thermal interaction between the plume and the
thermal characteristics of the body (considered at zero temperature rise). This is a quite large
field of analysis that comprise, among other, all the various thermal conditions on the body, and
the insertion of heat flux within the body itself and others.
It is obvious that further work should be reserved also to forced and mixed thermal convection
and to real porous bodies. The working agenda is certainly quite full.
We do want to conclude this paper by stating that the first experience is definitely positive since
the mix of the adaptive Lagrangian Blobs method with Brinkman penalization generates a

computational environment quite handy and efficient for the analysis of quite complicated
problems of real interest in engineering science.
Plume Generated by a Thermal Fixed Cluster Misaligned with the Rotating Body.
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