In this paper we propose a simple, unstructured mesh generation technique that is capable of handling the large and complex blade motion that is encountered in certain rotating machines, such as vertical axis wind turbines or cycloidal propellers. The technique is characterised by localised re-meshing and interpolation, so as to keep the mesh generation cost and interpolation error as low as possible

Bonfiglioli, A.

Paciorri, Renato

English

PACIORRI, Renato

Archivio della ricerca- Università di Roma La Sapienza

1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of ATI 2015doi: 10.1016/j.egypro.2015.12.022  Energy Procedia  82 ( 2015 )  209 – 214 ScienceDirectATI 2015 - 70th Conference of the ATI Engineering AssociationA local, un-structured, re-meshing technique capable ofhandling large body-motion in rotating machinery.A. Bonfigliolia,1, R. PaciorribaScuola di Ingegneria, Universita` degli Studi della Basilicata, V.le dell’Ateneo Lucano 10, Potenza, 85100, ItalybDip.to Ingegneria meccanica e aerospaziale, Universita` di Roma ”La Sapienza” , Via Eudossiana 18, Rome, 00184, ItalyAbstractIn this paper we propose a simple, unstructured mesh generation technique that is capable of handling the large andcomplex blade motion that is encountered in certain rotating machines, such as vertical axis wind turbines or cycloidalpropellers. The technique is characterised by localised re-meshing and interpolation, so as to keep the mesh generationcost and interpolation error as low as possible.c© 2011 Published by Elsevier Ltd.Keywords: Rotating machines, Unstructured mesh generation, Arbitrary Lagrangian Eulerian1. IntroductionIn certain rotating machinery, such as the Voith Cycloidal Rudder (VCR), the blades undergo a fairlycomplex relative motion. When the hydrodynamic behaviour of these devices is numerically simulated bymeans of CFD, the shape and volume of the domain occupied by the fluid change with time and the largedisplacement of the wetted surface might even lead to topological changes in the computational domain.Under such circumstances, mesh deformation algorithms [1] might be inapplicable and overlapping grids [2]or global re-meshing techniques should be used instead. The time-dependent re-meshing of the fluid domaincan however amount to a significant portion of the total computational cost. Moreover, the transfer ofinformation between the grids generated at subsequent time levels or between overlapping meshes is likelyto lead to conservation violation and loss of resolution.We propose a mesh generation technique that is characterised by localised re-meshing and interpolation,so as to keep the mesh generation cost and interpolation error as low as possible.In the proposed technique, a body-fitted, unstructured grid is attached to each of the moving bladesand each blade, along with the grid that is attached to it, is allowed to move over a fixed, backgroundtriangulation that covers the entire computational domain. At each time-step, the cells of the backgroundmesh which are overlap by the body-fitted grids are temporarily removed and local re-meshing is appliedonly in the neighbourhood of the outer boundary of the body-fitted meshes.1Corresponding author: Tel. +39-0971-205203. Email Address: aldo.bonfiglioli@unibas.it.Available online at www.sciencedirect.com© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of ATI 2015210   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 Nomenclatureα angle of attackΔt time-step lengthw nodal grid velocityx array of the Cartesian coordinates of thegrid-pointsω angular frequencyΦ Inviscid flux balanceΦei signals sent to vertex i of cell eρ densityc airfoil’s chordCi median dual celld dimension of the spaceH total enthalpyI identity matrix of order d + 2k reduced frequencyKei inflow parameterM Mach numberp static pressureT static temperatureT e triangle eU conserved variablesu horizontal velocity componentU∞ freestream velocity magnitudev vertical velocity componentZ parameter vector = √ρ (1,H, u, v)TAs far as the discretization of the governing PDEs is concerned, an Arbitrary Lagrangian Eulerian (ALE)scheme is required in order to account for the non-zero grid velocities that arise within the body-fittedmeshes and along their interface with the background, stationary triangulation.2. The numerical methodThe numerical method we propose consists in the loose coupling between an unstructured grid generatorthat is used generate computational meshes around the moving bodies and within the fluid domain and anunstructured CFD solver that is used to discretise the governing PDEs using the boundary-conforming andnon-overlapping meshes created by the mesh generator.The two codes are loosely coupled in the sense that the grid generator invokes the CFD solver as a blackbox. This has obvious consequences in terms of algorithmic simplicity, since it allows to re-use any existinggas-dynamic code, as long as its discretization is vertex centred.In the next two sections we will describe the unstructured solver and the mesh generator algorithm.2.1. The eulfs unstructured-grid solverThe eulfs code is an in-house, unstructured CFD solver that has been developed over the last fifteenyears: see [3] for a detailed description of its basic features and [4] for more recent developments. It relieson Fluctuation Splitting (FS), or Residual Distribution [5, 6, 7] schemes for the spatial discretisation. In theFS approach the dependent variables are stored at the vertices of the computational mesh which is madeup of triangles in the 2D space, and tetrahedra in 3D and are assumed to vary linearly and continuously inspace. The inviscid flux balance Φe (also referred to as the cell residual or cell fluctuation) is evaluated overeach triangular/tetrahedral element e by means of a conservative linearisation [8] based on the parametervector Z, and scattered to the element vertices using signals Φei . Within a cell e, the signals have to sum upto the net flux for conservation:∑i∈eΦei = Φe. The nodal residual is then assembled by collecting fractionsΦei of the net fluxes Φe associated with all the elements by which the node i is surrounded. The variousFS schemes proposed in the literature differ by the way cell residuals are split into signals. In this paper, A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 211XY(a) The domain and the moving body.XY(b) The d-mesh.XY(c) The b-mesh.Fig. 1. Domain, moving body and meshes.we use the FS version [9, 10] of the popular Lax-Wendroff (LW) scheme, because this is the simplest two-time-levels, explicit scheme that is second-order-accurate both in space and time. Using the LW scheme,the signals sent to vertex i of cell e are:Φei =[1d + 1 I +12Δt|T e|Kei]Φe (1)where |T e| denotes the area of the triangle and Kei , the so-called inflow parameter, is a matrix that dependsupon the cell-averaged Jacobian matrix of the inviscid fluxes and the normal to the edge opposite vertex i,see [3] for details. When the grid moves and/or deforms, matrix Kei also depends upon the cell-averagedgrid velocity.The following explicit update formula is obtained for the Arbitrary Lagrangian Eulerian (ALE) versionof the LW scheme:|Ci|n+1Un+1i = |Ci|nUni + Δt∑eiΦei (2)whereCi the median dual cell centred about grid-point i and U is the vector of the conserved variables whichcan be computed from parameter vector using the following identity:U =12(∂U∂Z)Z.The Geometric Conservation Law is satisfied as long as the signals in Eq. (1) are evaluated on the mesh attime level n+ 12 and the telescoping property of the fluxes is guaranteed using the approach described in [11].2.2. The mesh generator algorithmIn order to illustrate the algorithmic features of the proposed mesh generation algorithm, we consider theflow produced by a body, such as the circular cylinder shown in Fig. 1(a), that moves inside a fluid. At timet, the body is located in an arbitrary position, shown using the solid line in Fig. 1(a); over the time intervalΔt, the body moves to a new location, shown by the dashed line in Fig. 1(a). As shown in Fig. 1(b), the flowdomain is discretised using a triangular mesh that does not take into account the presence of the body. Thismesh, which is fixed in space, will be hereafter denoted as the “d-mesh”. Another mesh, that will be calledthe “b-mesh”, is generated around the body, as shown in Fig. 1(c). This body-fitted mesh, which is entirelyindependent of the d-mesh, moves rigidly with the body. We assume that at time t = nΔt the solution vectorZn is known at all points of both the d-mesh (except for those nodes that are hidden by the body) and theb-mesh. The process that leads from the known solution Zn at time t to the updated solution Zn+1 at timet + Δt can be split into the three steps that will be described in detail in the following sub-sections.212   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 XY(a) d-mesh with the boundary of b-meshat time t.XY(b) d-mesh after the cell removal process.XY(c) Insertion of the b-mesh.Fig. 2. Cell removal, followed by the insertion of the b-mesh.2.2.1. Generation of the computational meshAt time t, the b-mesh covers a certain region of the flow domain; this is shown in Fig. 2(a), where theouter boundary of the b-mesh has been highlighted in red. All cells of the d-mesh that have one of theirvertices either: i) falling within the outer boundary of the b-mesh or ii) outside of it, but closer to it than apreset distance, are removed. The reason for removing the grid-points that meet the criterion ii) is that thesenodes, if not removed, might be overcome by the b-mesh as it moves from its location at time t to the one attime t + Δt. All the nodes of the d-mesh that have been removed will be hereafter called: “phantom” nodes.Figure 2(b) shows the appearance of the “d-mesh” following the removal of the phantom nodes.At this stage, the b-mesh can be inserted inside the hole dug into the d-mesh by the cell removal step; thetwo meshes are however still disjoint, as shown in Fig. 2(c). The two meshes are then mutually connectedby means of a local constrained Delaunay triangulation (CDT) which does not introduce any additionalgrid-point. This CDT is constrained in the sense that the edges that belong to the boundary of the hole andthose that make up the outer boundary of the b-mesh are forced to be part of the final triangulation, seeFig. 3(a). The new “computational” mesh that has been generated by the local CDT will be used to advancethe solution from time t to time t + Δt. The solution vector Zn that is stored in the grid-points of both the d-and b- meshes can be easily transferred to the vertices of the computational mesh because every node of thecomputational mesh belongs to either the d-mesh or the b-mesh.2.2.2. Evolution of the solution on the computational meshThe solution Zn and the computational mesh at time t, shown in Fig. 3(a), are provided as input to theeulfs code in order to advance the solution (and the computational mesh) from time t to time t + Δt. Sincesome of the nodes of the computational mesh, more precisely those that also belong to the b-mesh, movewith the body, a nodal grid velocity must also be supplied to the CFD code. More precisely, for each node iof the computational mesh, the eulfs code needs the nodal grid velocity wi computed at time t+Δt/2 usingthe finite difference formula:wn+1/2i =(xn+1i − xni )Δt.The motion of the grid-points of the computational mesh causes a rigid motion of the triangles that alsobelong to the b-mesh and a deformation of the cells that have been generated by the local CDT, see Fig. 3(b).The output of the CFD code is the updated solution Zn+1 within all grid-points of the computational mesh attime t + Δt, shown in Fig. 3(b).2.2.3. Projection of the solution available on the computational mesh on the d- and b-meshesThe computed solution at time t+Δt has to be transferred from the computational mesh back to the d- andb-meshes. Concerning the b-mesh, the projection is trivial, because all grid-points of the b-mesh also belongto the computational mesh. By contrast, the projection of the solution Zn+1 on the d-mesh requires caution. A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 213XY(a) Computational mesh at time t.XY(b) Computational mesh at time t + Δt.XY(c) Interpolation of phantom points.Fig. 3.This is because not all the grid-points belonging to the d-mesh also belong to the computational mesh: thisis the case of the phantom nodes that have been identified and removed in step 2.2.1. Nonetheless the nodalstate within the phantom nodes must also be updated to time t+Δt, since the set of the phantom nodes couldchange between subsequent time intervals. This is because, due to the motion of the body, some of the pointsof the d-mesh that were close the boundary of the b-mesh and therefore declared phantom when integratingfrom t to t + Δt, might re-appear during the subsequent time interval [t + Δt, t + 2Δt]. The interpolationof the phantom nodes of the d-mesh, which is made using the nodal values available on the computationalmesh, allows to compute the state of all phantom nodes, except for those that are hidden by the body, seeFig. 3(c).At this stage, the numerical solution has correctly been updated at time level t + Δt on both the b- andd-meshes, so that the next time level can be computed re-starting from the first step 2.2.1 of the algorithm.3. Numerical exampleThe verification of the algorithm described in Sect. 2 has been carried out by looking at the temporalevolution of the aerodynamic coefficients of a NACA 0012 airfoil that is pitching about its leading edge withthe following law:α (t) = 0.016◦ + 2.51◦ sin (ω t) . (3)The angular frequency ω that appears in Eq. (3) can be expressed as a function of the dimension-lessreduced frequency:k = ωc2U∞= 0.0814.Free-stream conditions are: T∞ = 273 K, p∞ = 101325 Pa, M∞ = 0.5.The choice of a law of body motion, such as Eq. (3), that gives rise to a moderate displacement of thebody is motivated by the fact that, by doing so, it is also possible to perform the same calculation using agrid with fixed connectivity in which only the subset of nodes which is closest to the body undergoes a rigidmotion (according to Eq. (3)), while the remaining grid-points remain fixed in space. This same approachwould easily lead to a folding grid, if larger displacements were used. It is this calculation, which has beenperformed using the same eulfs code, that as been used as the reference calculation.Figure 4 compares the temporal evolution of the lift coefficient computed using the newly developedmesh generation approach (MUMs) and the reference calculation; it is evident that the two approaches giveidentical results.4. ConclusionsIn this paper we a propose a simple mesh generation algorithm that can easily handle the large blademotion encountered in certain rotating machines. The proposed technique is characterised by localised re-214   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 Fig. 4. Pitching airfoil: temporal evolution of the lift coefficientmeshing and interpolation, so as to keep the mesh generation cost and interpolation error as low as possible.The technique is also versatile, in the sense that it can be coupled with virtually any vertex-centred CFDcode that is capable of solving the governing PDEs, written using an ALE formulation.References1. Jasak, H., Tukovic´, v.. Automatic mesh motion for the unstructured finite volume method. Transactions of Famena2006;30(2):1–20.2. Hadzˇic´, I., Hennig, J., Peric´, M., Xing-Kaeding, Y.. Computation of flow-induced motion of floating bodies. AppliedMathematical Modelling 2005;29(12):1196 – 1210. doi:http://dx.doi.org/10.1016/j.apm.2005.02.014.3. Bonfiglioli, A.. Fluctuation splitting schemes for the compressible and incompressible Euler and Navier-Stokes equations.International Journal of Computational Fluid Dynamics 2000;14:21–39.4. Bonfiglioli, A., Paciorri, R.. A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parame-ter vector. International Journal of Computational Fluid Dynamics 2013;27(4-5):210–227. doi:10.1080/10618562.2013.813491.5. Deconinck, H., Paille`re, H., Struijs, R., Roe, P.. Multidimensional upwind schemes based on fluctuation-splitting for systemsof conservation laws. Computational Mechanics 1993;11(5/6):323–340.6. van der Weide, E., Deconinck, H., Issman, E., Degrez, G.. A parallel, implicit, multi-dimensional upwind, residual distributionmethod for the Navier-Stokes equations on unstructured grids. Computational Mechanics 1999;23:199–208.7. Abgrall, R.. Residual distribution schemes: Current status and future trends. Computers and Fluids 2006;35(7):641 – 669.doi:10.1016/j.compfluid.2005.01.007.8. Deconinck, H., Roe, P., Struijs, R.. A Multi-dimensional Generalization of Roe’s Flux Difference Splitter for the EulerEquations. Computers and Fluids 1993;22(2/3):215–222.9. Struijs, R.. A multi-dimensional upwind discretization method for the Euler equations on unstructured grids. Ph.D. thesis;Technische Universiteit Delft, The Nederlands; 1994.10. Rossiello, G., De Palma, P., Pascazio, G., Napolitano, M.. Second-order-accurate explicit fluctuation splitting schemes forunsteady problems. Computers & Fluids 2009;38(7):1384 – 1393. doi:DOI: 10.1016/j.compfluid.2008.01.021.11. Michler, C., De Sterck, H., Deconinck, H.. An arbitrary lagrangian eulerian formulation for residual distribution schemes onmoving grids. Computers & Fluids 2003;32(1):59 – 71. doi:10.1016/S0045-7930(01)00095-0.Aldo Bonfiglioli graduated in 1992 summa cum laude in Mechanical Engineer-ing from the TUBari, obtained the Diploma Course from the VKI and the PhDfrom the TUBari. Since 2007 he is Associate Professor of Fluid Machines at theUniversity of Basilicata. His current research interests include the development ofdiscretization schemes on unstructured meshes.

A local, un-structured, re-meshing technique capable of handling large body-motion in rotating machinery

AbstractIn this paper we propose a simple, unstructured mesh generation technique that is capable of handling the large and complex blade motion that is encountered in certain rotating machines, such as vertical axis wind turbines or cycloidal propellers. The technique is characterised by localised re-meshing and interpolation, so as to keep the mesh generation cost and interpolation error as low as possible

Paciorri, R.

Elsevier - Publisher Connector 

A Local, Un-structured, Re-meshing Technique Capable of Handling Large Body-motion in Rotating Machinery 

1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Scientific Committee of ATI 2015
doi: 10.1016/j.egypro.2015.12.022 
 Energy Procedia  82 ( 2015 )  209 – 214 
ScienceDirect
ATI 2015 - 70th Conference of the ATI Engineering Association
A local, un-structured, re-meshing technique capable of
handling large body-motion in rotating machinery.
A. Bonfigliolia,1, R. Paciorrib
aScuola di Ingegneria, Universita` degli Studi della Basilicata, V.le dell’Ateneo Lucano 10, Potenza, 85100, Italy
bDip.to Ingegneria meccanica e aerospaziale, Universita` di Roma ”La Sapienza” , Via Eudossiana 18, Rome, 00184, Italy
Abstract
In this paper we propose a simple, unstructured mesh generation technique that is capable of handling the large and
complex blade motion that is encountered in certain rotating machines, such as vertical axis wind turbines or cycloidal
propellers. The technique is characterised by localised re-meshing and interpolation, so as to keep the mesh generation
cost and interpolation error as low as possible.
c© 2011 Published by Elsevier Ltd.
Keywords: Rotating machines, Unstructured mesh generation, Arbitrary Lagrangian Eulerian
1. Introduction
In certain rotating machinery, such as the Voith Cycloidal Rudder (VCR), the blades undergo a fairly
complex relative motion. When the hydrodynamic behaviour of these devices is numerically simulated by
means of CFD, the shape and volume of the domain occupied by the fluid change with time and the large
displacement of the wetted surface might even lead to topological changes in the computational domain.
Under such circumstances, mesh deformation algorithms [1] might be inapplicable and overlapping grids [2]
or global re-meshing techniques should be used instead. The time-dependent re-meshing of the fluid domain
can however amount to a significant portion of the total computational cost. Moreover, the transfer of
information between the grids generated at subsequent time levels or between overlapping meshes is likely
to lead to conservation violation and loss of resolution.
We propose a mesh generation technique that is characterised by localised re-meshing and interpolation,
so as to keep the mesh generation cost and interpolation error as low as possible.
In the proposed technique, a body-fitted, unstructured grid is attached to each of the moving blades
and each blade, along with the grid that is attached to it, is allowed to move over a fixed, background
triangulation that covers the entire computational domain. At each time-step, the cells of the background
mesh which are overlap by the body-fitted grids are temporarily removed and local re-meshing is applied
only in the neighbourhood of the outer boundary of the body-fitted meshes.
1Corresponding author: Tel. +39-0971-205203. Email Address: aldo.bonfiglioli@unibas.it.
Available online at www.sciencedirect.com
© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Scientific Committee of ATI 2015
210   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 
Nomenclature
α angle of attack
Δt time-step length
w nodal grid velocity
x array of the Cartesian coordinates of the
grid-points
ω angular frequency
Φ Inviscid flux balance
Φei signals sent to vertex i of cell e
ρ density
c airfoil’s chord
Ci median dual cell
d dimension of the space
H total enthalpy
I identity matrix of order d + 2
k reduced frequency
Kei inflow parameter
M Mach number
p static pressure
T static temperature
T e triangle e
U conserved variables
u horizontal velocity component
U∞ freestream velocity magnitude
v vertical velocity component
Z parameter vector = √ρ (1,H, u, v)T
As far as the discretization of the governing PDEs is concerned, an Arbitrary Lagrangian Eulerian (ALE)
scheme is required in order to account for the non-zero grid velocities that arise within the body-fitted
meshes and along their interface with the background, stationary triangulation.
2. The numerical method
The numerical method we propose consists in the loose coupling between an unstructured grid generator
that is used generate computational meshes around the moving bodies and within the fluid domain and an
unstructured CFD solver that is used to discretise the governing PDEs using the boundary-conforming and
non-overlapping meshes created by the mesh generator.
The two codes are loosely coupled in the sense that the grid generator invokes the CFD solver as a black
box. This has obvious consequences in terms of algorithmic simplicity, since it allows to re-use any existing
gas-dynamic code, as long as its discretization is vertex centred.
In the next two sections we will describe the unstructured solver and the mesh generator algorithm.
2.1. The eulfs unstructured-grid solver
The eulfs code is an in-house, unstructured CFD solver that has been developed over the last fifteen
years: see [3] for a detailed description of its basic features and [4] for more recent developments. It relies
on Fluctuation Splitting (FS), or Residual Distribution [5, 6, 7] schemes for the spatial discretisation. In the
FS approach the dependent variables are stored at the vertices of the computational mesh which is made
up of triangles in the 2D space, and tetrahedra in 3D and are assumed to vary linearly and continuously in
space. The inviscid flux balance Φe (also referred to as the cell residual or cell fluctuation) is evaluated over
each triangular/tetrahedral element e by means of a conservative linearisation [8] based on the parameter
vector Z, and scattered to the element vertices using signals Φei . Within a cell e, the signals have to sum up
to the net flux for conservation:
∑
i∈eΦ
e
i = Φ
e. The nodal residual is then assembled by collecting fractions
Φei of the net fluxes Φ
e associated with all the elements by which the node i is surrounded. The various
FS schemes proposed in the literature diﬀer by the way cell residuals are split into signals. In this paper,
 A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 211
X
Y
(a) The domain and the moving body.
X
Y
(b) The d-mesh.
X
Y
(c) The b-mesh.
Fig. 1. Domain, moving body and meshes.
we use the FS version [9, 10] of the popular Lax-Wendroﬀ (LW) scheme, because this is the simplest two-
time-levels, explicit scheme that is second-order-accurate both in space and time. Using the LW scheme,
the signals sent to vertex i of cell e are:
Φei =
[
1
d + 1 I +
1
2
Δt
|T e|K
e
i
]
Φe (1)
where |T e| denotes the area of the triangle and Kei , the so-called inflow parameter, is a matrix that depends
upon the cell-averaged Jacobian matrix of the inviscid fluxes and the normal to the edge opposite vertex i,
see [3] for details. When the grid moves and/or deforms, matrix Kei also depends upon the cell-averaged
grid velocity.
The following explicit update formula is obtained for the Arbitrary Lagrangian Eulerian (ALE) version
of the LW scheme:
|Ci|n+1Un+1i = |Ci|nUni + Δt
∑
ei
Φei (2)
whereCi the median dual cell centred about grid-point i and U is the vector of the conserved variables which
can be computed from parameter vector using the following identity:
U =
1
2
(
∂U
∂Z
)
Z.
The Geometric Conservation Law is satisfied as long as the signals in Eq. (1) are evaluated on the mesh at
time level n+ 12 and the telescoping property of the fluxes is guaranteed using the approach described in [11].
2.2. The mesh generator algorithm
In order to illustrate the algorithmic features of the proposed mesh generation algorithm, we consider the
flow produced by a body, such as the circular cylinder shown in Fig. 1(a), that moves inside a fluid. At time
t, the body is located in an arbitrary position, shown using the solid line in Fig. 1(a); over the time interval
Δt, the body moves to a new location, shown by the dashed line in Fig. 1(a). As shown in Fig. 1(b), the flow
domain is discretised using a triangular mesh that does not take into account the presence of the body. This
mesh, which is fixed in space, will be hereafter denoted as the “d-mesh”. Another mesh, that will be called
the “b-mesh”, is generated around the body, as shown in Fig. 1(c). This body-fitted mesh, which is entirely
independent of the d-mesh, moves rigidly with the body. We assume that at time t = nΔt the solution vector
Zn is known at all points of both the d-mesh (except for those nodes that are hidden by the body) and the
b-mesh. The process that leads from the known solution Zn at time t to the updated solution Zn+1 at time
t + Δt can be split into the three steps that will be described in detail in the following sub-sections.
212   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 
X
Y
(a) d-mesh with the boundary of b-mesh
at time t.
X
Y
(b) d-mesh after the cell removal process.
X
Y
(c) Insertion of the b-mesh.
Fig. 2. Cell removal, followed by the insertion of the b-mesh.
2.2.1. Generation of the computational mesh
At time t, the b-mesh covers a certain region of the flow domain; this is shown in Fig. 2(a), where the
outer boundary of the b-mesh has been highlighted in red. All cells of the d-mesh that have one of their
vertices either: i) falling within the outer boundary of the b-mesh or ii) outside of it, but closer to it than a
preset distance, are removed. The reason for removing the grid-points that meet the criterion ii) is that these
nodes, if not removed, might be overcome by the b-mesh as it moves from its location at time t to the one at
time t + Δt. All the nodes of the d-mesh that have been removed will be hereafter called: “phantom” nodes.
Figure 2(b) shows the appearance of the “d-mesh” following the removal of the phantom nodes.
At this stage, the b-mesh can be inserted inside the hole dug into the d-mesh by the cell removal step; the
two meshes are however still disjoint, as shown in Fig. 2(c). The two meshes are then mutually connected
by means of a local constrained Delaunay triangulation (CDT) which does not introduce any additional
grid-point. This CDT is constrained in the sense that the edges that belong to the boundary of the hole and
those that make up the outer boundary of the b-mesh are forced to be part of the final triangulation, see
Fig. 3(a). The new “computational” mesh that has been generated by the local CDT will be used to advance
the solution from time t to time t + Δt. The solution vector Zn that is stored in the grid-points of both the d-
and b- meshes can be easily transferred to the vertices of the computational mesh because every node of the
computational mesh belongs to either the d-mesh or the b-mesh.
2.2.2. Evolution of the solution on the computational mesh
The solution Zn and the computational mesh at time t, shown in Fig. 3(a), are provided as input to the
eulfs code in order to advance the solution (and the computational mesh) from time t to time t + Δt. Since
some of the nodes of the computational mesh, more precisely those that also belong to the b-mesh, move
with the body, a nodal grid velocity must also be supplied to the CFD code. More precisely, for each node i
of the computational mesh, the eulfs code needs the nodal grid velocity wi computed at time t+Δt/2 using
the finite diﬀerence formula:
wn+1/2i =
(xn+1i − xni )
Δt
.
The motion of the grid-points of the computational mesh causes a rigid motion of the triangles that also
belong to the b-mesh and a deformation of the cells that have been generated by the local CDT, see Fig. 3(b).
The output of the CFD code is the updated solution Zn+1 within all grid-points of the computational mesh at
time t + Δt, shown in Fig. 3(b).
2.2.3. Projection of the solution available on the computational mesh on the d- and b-meshes
The computed solution at time t+Δt has to be transferred from the computational mesh back to the d- and
b-meshes. Concerning the b-mesh, the projection is trivial, because all grid-points of the b-mesh also belong
to the computational mesh. By contrast, the projection of the solution Zn+1 on the d-mesh requires caution.
 A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 213
X
Y
(a) Computational mesh at time t.
X
Y
(b) Computational mesh at time t + Δt.
X
Y
(c) Interpolation of phantom points.
Fig. 3.
This is because not all the grid-points belonging to the d-mesh also belong to the computational mesh: this
is the case of the phantom nodes that have been identified and removed in step 2.2.1. Nonetheless the nodal
state within the phantom nodes must also be updated to time t+Δt, since the set of the phantom nodes could
change between subsequent time intervals. This is because, due to the motion of the body, some of the points
of the d-mesh that were close the boundary of the b-mesh and therefore declared phantom when integrating
from t to t + Δt, might re-appear during the subsequent time interval [t + Δt, t + 2Δt]. The interpolation
of the phantom nodes of the d-mesh, which is made using the nodal values available on the computational
mesh, allows to compute the state of all phantom nodes, except for those that are hidden by the body, see
Fig. 3(c).
At this stage, the numerical solution has correctly been updated at time level t + Δt on both the b- and
d-meshes, so that the next time level can be computed re-starting from the first step 2.2.1 of the algorithm.
3. Numerical example
The verification of the algorithm described in Sect. 2 has been carried out by looking at the temporal
evolution of the aerodynamic coeﬃcients of a NACA 0012 airfoil that is pitching about its leading edge with
the following law:
α (t) = 0.016◦ + 2.51◦ sin (ω t) . (3)
The angular frequency ω that appears in Eq. (3) can be expressed as a function of the dimension-less
reduced frequency:
k = ωc
2U∞
= 0.0814.
Free-stream conditions are: T∞ = 273 K, p∞ = 101325 Pa, M∞ = 0.5.
The choice of a law of body motion, such as Eq. (3), that gives rise to a moderate displacement of the
body is motivated by the fact that, by doing so, it is also possible to perform the same calculation using a
grid with fixed connectivity in which only the subset of nodes which is closest to the body undergoes a rigid
motion (according to Eq. (3)), while the remaining grid-points remain fixed in space. This same approach
would easily lead to a folding grid, if larger displacements were used. It is this calculation, which has been
performed using the same eulfs code, that as been used as the reference calculation.
Figure 4 compares the temporal evolution of the lift coeﬃcient computed using the newly developed
mesh generation approach (MUMs) and the reference calculation; it is evident that the two approaches give
identical results.
4. Conclusions
In this paper we a propose a simple mesh generation algorithm that can easily handle the large blade
motion encountered in certain rotating machines. The proposed technique is characterised by localised re-
214   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 
Fig. 4. Pitching airfoil: temporal evolution of the lift coeﬃcient
meshing and interpolation, so as to keep the mesh generation cost and interpolation error as low as possible.
The technique is also versatile, in the sense that it can be coupled with virtually any vertex-centred CFD
code that is capable of solving the governing PDEs, written using an ALE formulation.
References
1. Jasak, H., Tukovic´, v.. Automatic mesh motion for the unstructured finite volume method. Transactions of Famena
2006;30(2):1–20.
2. Hadzˇic´, I., Hennig, J., Peric´, M., Xing-Kaeding, Y.. Computation of flow-induced motion of floating bodies. Applied
Mathematical Modelling 2005;29(12):1196 – 1210. doi:http://dx.doi.org/10.1016/j.apm.2005.02.014.
3. Bonfiglioli, A.. Fluctuation splitting schemes for the compressible and incompressible Euler and Navier-Stokes equations.
International Journal of Computational Fluid Dynamics 2000;14:21–39.
4. Bonfiglioli, A., Paciorri, R.. A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parame-
ter vector. International Journal of Computational Fluid Dynamics 2013;27(4-5):210–227. doi:10.1080/10618562.2013.813491.
5. Deconinck, H., Paille`re, H., Struijs, R., Roe, P.. Multidimensional upwind schemes based on fluctuation-splitting for systems
of conservation laws. Computational Mechanics 1993;11(5/6):323–340.
6. van der Weide, E., Deconinck, H., Issman, E., Degrez, G.. A parallel, implicit, multi-dimensional upwind, residual distribution
method for the Navier-Stokes equations on unstructured grids. Computational Mechanics 1999;23:199–208.
7. Abgrall, R.. Residual distribution schemes: Current status and future trends. Computers and Fluids 2006;35(7):641 – 669.
doi:10.1016/j.compfluid.2005.01.007.
8. Deconinck, H., Roe, P., Struijs, R.. A Multi-dimensional Generalization of Roe’s Flux Diﬀerence Splitter for the Euler
Equations. Computers and Fluids 1993;22(2/3):215–222.
9. Struijs, R.. A multi-dimensional upwind discretization method for the Euler equations on unstructured grids. Ph.D. thesis;
Technische Universiteit Delft, The Nederlands; 1994.
10. Rossiello, G., De Palma, P., Pascazio, G., Napolitano, M.. Second-order-accurate explicit fluctuation splitting schemes for
unsteady problems. Computers & Fluids 2009;38(7):1384 – 1393. doi:DOI: 10.1016/j.compfluid.2008.01.021.
11. Michler, C., De Sterck, H., Deconinck, H.. An arbitrary lagrangian eulerian formulation for residual distribution schemes on
moving grids. Computers & Fluids 2003;32(1):59 – 71. doi:10.1016/S0045-7930(01)00095-0.
Aldo Bonfiglioli graduated in 1992 summa cum laude in Mechanical Engineer-
ing from the TUBari, obtained the Diploma Course from the VKI and the PhD
from the TUBari. Since 2007 he is Associate Professor of Fluid Machines at the
University of Basilicata. His current research interests include the development of
discretization schemes on unstructured meshes.


BONFIGLIOLI, Aldo

Archivio della Ricerca - Università della Basilicata

1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of ATI 2015doi: 10.1016/j.egypro.2015.12.022  Energy Procedia  82 ( 2015 )  209 – 214 ScienceDirectATI 2015 - 70th Conference of the ATI Engineering AssociationA local, un-structured, re-meshing technique capable ofhandling large body-motion in rotating machinery.A. Bonfigliolia,1, R. PaciorribaScuola di Ingegneria, Università degli Studi della Basilicata, V.le dell’Ateneo Lucano 10, Potenza, 85100, ItalybDip.to Ingegneria meccanica e aerospaziale, Università di Roma ”La Sapienza” , Via Eudossiana 18, Rome, 00184, ItalyAbstractIn this paper we propose a simple, unstructured mesh generation technique that is capable of handling the large andcomplex blade motion that is encountered in certain rotating machines, such as vertical axis wind turbines or cycloidalpropellers. The technique is characterised by localised re-meshing and interpolation, so as to keep the mesh generationcost and interpolation error as low as possible.c© 2011 Published by Elsevier Ltd.Keywords: Rotating machines, Unstructured mesh generation, Arbitrary Lagrangian Eulerian1. IntroductionIn certain rotating machinery, such as the Voith Cycloidal Rudder (VCR), the blades undergo a fairlycomplex relative motion. When the hydrodynamic behaviour of these devices is numerically simulated bymeans of CFD, the shape and volume of the domain occupied by the fluid change with time and the largedisplacement of the wetted surface might even lead to topological changes in the computational domain.Under such circumstances, mesh deformation algorithms [1] might be inapplicable and overlapping grids [2]or global re-meshing techniques should be used instead. The time-dependent re-meshing of the fluid domaincan however amount to a significant portion of the total computational cost. Moreover, the transfer ofinformation between the grids generated at subsequent time levels or between overlapping meshes is likelyto lead to conservation violation and loss of resolution.We propose a mesh generation technique that is characterised by localised re-meshing and interpolation,so as to keep the mesh generation cost and interpolation error as low as possible.In the proposed technique, a body-fitted, unstructured grid is attached to each of the moving bladesand each blade, along with the grid that is attached to it, is allowed to move over a fixed, backgroundtriangulation that covers the entire computational domain. At each time-step, the cells of the backgroundmesh which are overlap by the body-fitted grids are temporarily removed and local re-meshing is appliedonly in the neighbourhood of the outer boundary of the body-fitted meshes.1Corresponding author: Tel. +39-0971-205203. Email Address: aldo.bonfiglioli@unibas.it.Available online at www.sciencedirect.com© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of ATI 2015210   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 Nomenclatureα angle of attackΔt time-step lengthw nodal grid velocityx array of the Cartesian coordinates of thegrid-pointsω angular frequencyΦ Inviscid flux balanceΦei signals sent to vertex i of cell eρ densityc airfoil’s chordCi median dual celld dimension of the spaceH total enthalpyI identity matrix of order d + 2k reduced frequencyKei inflow parameterM Mach numberp static pressureT static temperatureT e triangle eU conserved variablesu horizontal velocity componentU∞ freestream velocity magnitudev vertical velocity componentZ parameter vector =√ρ (1,H, u, v)TAs far as the discretization of the governing PDEs is concerned, an Arbitrary Lagrangian Eulerian (ALE)scheme is required in order to account for the non-zero grid velocities that arise within the body-fittedmeshes and along their interface with the background, stationary triangulation.2. The numerical methodThe numerical method we propose consists in the loose coupling between an unstructured grid generatorthat is used generate computational meshes around the moving bodies and within the fluid domain and anunstructured CFD solver that is used to discretise the governing PDEs using the boundary-conforming andnon-overlapping meshes created by the mesh generator.The two codes are loosely coupled in the sense that the grid generator invokes the CFD solver as a blackbox. This has obvious consequences in terms of algorithmic simplicity, since it allows to re-use any existinggas-dynamic code, as long as its discretization is vertex centred.In the next two sections we will describe the unstructured solver and the mesh generator algorithm.2.1. The eulfs unstructured-grid solverThe eulfs code is an in-house, unstructured CFD solver that has been developed over the last fifteenyears: see [3] for a detailed description of its basic features and [4] for more recent developments. It relieson Fluctuation Splitting (FS), or Residual Distribution [5, 6, 7] schemes for the spatial discretisation. In theFS approach the dependent variables are stored at the vertices of the computational mesh which is madeup of triangles in the 2D space, and tetrahedra in 3D and are assumed to vary linearly and continuously inspace. The inviscid flux balance Φe (also referred to as the cell residual or cell fluctuation) is evaluated overeach triangular/tetrahedral element e by means of a conservative linearisation [8] based on the parametervector Z, and scattered to the element vertices using signals Φei . Within a cell e, the signals have to sum upto the net flux for conservation:∑i∈eΦei = Φe. The nodal residual is then assembled by collecting fractionsΦei of the net fluxes Φe associated with all the elements by which the node i is surrounded. The variousFS schemes proposed in the literature differ by the way cell residuals are split into signals. In this paper, A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 211XY(a) The domain and the moving body.XY(b) The d-mesh.XY(c) The b-mesh.Fig. 1. Domain, moving body and meshes.we use the FS version [9, 10] of the popular Lax-Wendroff (LW) scheme, because this is the simplest two-time-levels, explicit scheme that is second-order-accurate both in space and time. Using the LW scheme,the signals sent to vertex i of cell e are:Φei =[1d + 1I +12Δt|T e|Kei]Φe (1)where |T e| denotes the area of the triangle and Kei , the so-called inflow parameter, is a matrix that dependsupon the cell-averaged Jacobian matrix of the inviscid fluxes and the normal to the edge opposite vertex i,see [3] for details. When the grid moves and/or deforms, matrix Kei also depends upon the cell-averagedgrid velocity.The following explicit update formula is obtained for the Arbitrary Lagrangian Eulerian (ALE) versionof the LW scheme:|Ci|n+1Un+1i = |Ci|nUni + Δt∑eiΦei (2)where Ci the median dual cell centred about grid-point i and U is the vector of the conserved variables whichcan be computed from parameter vector using the following identity:U =12(∂U∂Z)Z.The Geometric Conservation Law is satisfied as long as the signals in Eq. (1) are evaluated on the mesh attime level n+ 12 and the telescoping property of the fluxes is guaranteed using the approach described in [11].2.2. The mesh generator algorithmIn order to illustrate the algorithmic features of the proposed mesh generation algorithm, we consider theflow produced by a body, such as the circular cylinder shown in Fig. 1(a), that moves inside a fluid. At timet, the body is located in an arbitrary position, shown using the solid line in Fig. 1(a); over the time intervalΔt, the body moves to a new location, shown by the dashed line in Fig. 1(a). As shown in Fig. 1(b), the flowdomain is discretised using a triangular mesh that does not take into account the presence of the body. Thismesh, which is fixed in space, will be hereafter denoted as the “d-mesh”. Another mesh, that will be calledthe “b-mesh”, is generated around the body, as shown in Fig. 1(c). This body-fitted mesh, which is entirelyindependent of the d-mesh, moves rigidly with the body. We assume that at time t = nΔt the solution vectorZn is known at all points of both the d-mesh (except for those nodes that are hidden by the body) and theb-mesh. The process that leads from the known solution Zn at time t to the updated solution Zn+1 at timet + Δt can be split into the three steps that will be described in detail in the following sub-sections.212   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 XY(a) d-mesh with the boundary of b-meshat time t.XY(b) d-mesh after the cell removal process.XY(c) Insertion of the b-mesh.Fig. 2. Cell removal, followed by the insertion of the b-mesh.2.2.1. Generation of the computational meshAt time t, the b-mesh covers a certain region of the flow domain; this is shown in Fig. 2(a), where theouter boundary of the b-mesh has been highlighted in red. All cells of the d-mesh that have one of theirvertices either: i) falling within the outer boundary of the b-mesh or ii) outside of it, but closer to it than apreset distance, are removed. The reason for removing the grid-points that meet the criterion ii) is that thesenodes, if not removed, might be overcome by the b-mesh as it moves from its location at time t to the one attime t + Δt. All the nodes of the d-mesh that have been removed will be hereafter called: “phantom” nodes.Figure 2(b) shows the appearance of the “d-mesh” following the removal of the phantom nodes.At this stage, the b-mesh can be inserted inside the hole dug into the d-mesh by the cell removal step; thetwo meshes are however still disjoint, as shown in Fig. 2(c). The two meshes are then mutually connectedby means of a local constrained Delaunay triangulation (CDT) which does not introduce any additionalgrid-point. This CDT is constrained in the sense that the edges that belong to the boundary of the hole andthose that make up the outer boundary of the b-mesh are forced to be part of the final triangulation, seeFig. 3(a). The new “computational” mesh that has been generated by the local CDT will be used to advancethe solution from time t to time t + Δt. The solution vector Zn that is stored in the grid-points of both the d-and b- meshes can be easily transferred to the vertices of the computational mesh because every node of thecomputational mesh belongs to either the d-mesh or the b-mesh.2.2.2. Evolution of the solution on the computational meshThe solution Zn and the computational mesh at time t, shown in Fig. 3(a), are provided as input to theeulfs code in order to advance the solution (and the computational mesh) from time t to time t + Δt. Sincesome of the nodes of the computational mesh, more precisely those that also belong to the b-mesh, movewith the body, a nodal grid velocity must also be supplied to the CFD code. More precisely, for each node iof the computational mesh, the eulfs code needs the nodal grid velocity wi computed at time t+Δt/2 usingthe finite difference formula:wn+1/2i =(xn+1i − xni )Δt.The motion of the grid-points of the computational mesh causes a rigid motion of the triangles that alsobelong to the b-mesh and a deformation of the cells that have been generated by the local CDT, see Fig. 3(b).The output of the CFD code is the updated solution Zn+1 within all grid-points of the computational mesh attime t + Δt, shown in Fig. 3(b).2.2.3. Projection of the solution available on the computational mesh on the d- and b-meshesThe computed solution at time t+Δt has to be transferred from the computational mesh back to the d- andb-meshes. Concerning the b-mesh, the projection is trivial, because all grid-points of the b-mesh also belongto the computational mesh. By contrast, the projection of the solution Zn+1 on the d-mesh requires caution. A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 213XY(a) Computational mesh at time t.XY(b) Computational mesh at time t + Δt.XY(c) Interpolation of phantom points.Fig. 3.This is because not all the grid-points belonging to the d-mesh also belong to the computational mesh: thisis the case of the phantom nodes that have been identified and removed in step 2.2.1. Nonetheless the nodalstate within the phantom nodes must also be updated to time t+Δt, since the set of the phantom nodes couldchange between subsequent time intervals. This is because, due to the motion of the body, some of the pointsof the d-mesh that were close the boundary of the b-mesh and therefore declared phantom when integratingfrom t to t + Δt, might re-appear during the subsequent time interval [t + Δt, t + 2Δt]. The interpolationof the phantom nodes of the d-mesh, which is made using the nodal values available on the computationalmesh, allows to compute the state of all phantom nodes, except for those that are hidden by the body, seeFig. 3(c).At this stage, the numerical solution has correctly been updated at time level t + Δt on both the b- andd-meshes, so that the next time level can be computed re-starting from the first step 2.2.1 of the algorithm.3. Numerical exampleThe verification of the algorithm described in Sect. 2 has been carried out by looking at the temporalevolution of the aerodynamic coefficients of a NACA 0012 airfoil that is pitching about its leading edge withthe following law:α (t) = 0.016◦ + 2.51◦ sin (ω t) . (3)The angular frequency ω that appears in Eq. (3) can be expressed as a function of the dimension-lessreduced frequency:k =ωc2 U∞= 0.0814.Free-stream conditions are: T∞ = 273 K, p∞ = 101325 Pa, M∞ = 0.5.The choice of a law of body motion, such as Eq. (3), that gives rise to a moderate displacement of thebody is motivated by the fact that, by doing so, it is also possible to perform the same calculation using agrid with fixed connectivity in which only the subset of nodes which is closest to the body undergoes a rigidmotion (according to Eq. (3)), while the remaining grid-points remain fixed in space. This same approachwould easily lead to a folding grid, if larger displacements were used. It is this calculation, which has beenperformed using the same eulfs code, that as been used as the reference calculation.Figure 4 compares the temporal evolution of the lift coefficient computed using the newly developedmesh generation approach (MUMs) and the reference calculation; it is evident that the two approaches giveidentical results.4. ConclusionsIn this paper we a propose a simple mesh generation algorithm that can easily handle the large blademotion encountered in certain rotating machines. The proposed technique is characterised by localised re-214   A. Bonfi glioli and R. Paciorri /  Energy Procedia  82 ( 2015 )  209 – 214 Fig. 4. Pitching airfoil: temporal evolution of the lift coefficientmeshing and interpolation, so as to keep the mesh generation cost and interpolation error as low as possible.The technique is also versatile, in the sense that it can be coupled with virtually any vertex-centred CFDcode that is capable of solving the governing PDEs, written using an ALE formulation.References1. Jasak, H., Tuković, v.. Automatic mesh motion for the unstructured finite volume method. Transactions of Famena2006;30(2):1–20.2. Hadžić, I., Hennig, J., Perić, M., Xing-Kaeding, Y.. Computation of flow-induced motion of floating bodies. AppliedMathematical Modelling 2005;29(12):1196 – 1210. doi:http://dx.doi.org/10.1016/j.apm.2005.02.014.3. Bonfiglioli, A.. Fluctuation splitting schemes for the compressible and incompressible Euler and Navier-Stokes equations.International Journal of Computational Fluid Dynamics 2000;14:21–39.4. Bonfiglioli, A., Paciorri, R.. A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parame-ter vector. International Journal of Computational Fluid Dynamics 2013;27(4-5):210–227. doi:10.1080/10618562.2013.813491.5. Deconinck, H., Paillère, H., Struijs, R., Roe, P.. Multidimensional upwind schemes based on fluctuation-splitting for systemsof conservation laws. Computational Mechanics 1993;11(5/6):323–340.6. van der Weide, E., Deconinck, H., Issman, E., Degrez, G.. A parallel, implicit, multi-dimensional upwind, residual distributionmethod for the Navier-Stokes equations on unstructured grids. Computational Mechanics 1999;23:199–208.7. Abgrall, R.. Residual distribution schemes: Current status and future trends. Computers and Fluids 2006;35(7):641 – 669.doi:10.1016/j.compfluid.2005.01.007.8. Deconinck, H., Roe, P., Struijs, R.. A Multi-dimensional Generalization of Roe’s Flux Difference Splitter for the EulerEquations. Computers and Fluids 1993;22(2/3):215–222.9. Struijs, R.. A multi-dimensional upwind discretization method for the Euler equations on unstructured grids. Ph.D. thesis;Technische Universiteit Delft, The Nederlands; 1994.10. Rossiello, G., De Palma, P., Pascazio, G., Napolitano, M.. Second-order-accurate explicit fluctuation splitting schemes forunsteady problems. Computers & Fluids 2009;38(7):1384 – 1393. doi:DOI: 10.1016/j.compfluid.2008.01.021.11. Michler, C., De Sterck, H., Deconinck, H.. An arbitrary lagrangian eulerian formulation for residual distribution schemes onmoving grids. Computers & Fluids 2003;32(1):59 – 71. doi:10.1016/S0045-7930(01)00095-0.Aldo Bonfiglioli graduated in 1992 summa cum laude in Mechanical Engineer-ing from the TUBari, obtained the Diploma Course from the VKI and the PhDfrom the TUBari. Since 2007 he is Associate Professor of Fluid Machines at theUniversity of Basilicata. His current research interests include the development ofdiscretization schemes on unstructured meshes.

A Local, Un-structured, Re-meshing Technique Capable of Handling Large Body-motion in Rotating Machinery

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A local, un-structured, re-meshing technique capable of handling large body-motion in rotating machinery

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