A Review of Vortex Methods and Their Applications: From Creation to Recent Advances
Abstract
:1. Introduction
2. Vortex Methods for Inviscid Incompressible Flows
2.1. General Description and Formulation
2.2. Convergence Issues
3. Viscous Vortex Methods
3.1. The Viscous Splitting and Random Walk Method
- (1)
- advection:
- (2)
- diffusion:
Random Walk Method (RW)
3.2. Deterministic Viscous Vortex Schemes
3.2.1. Particle Strength Exchange Method (PSE)
- Approximate the diffusion operator by an integral operator:
- The particle approximation of the diffusion equation can be obtained from the numerical integration of (35). The resulting vortex scheme is given by:
3.2.2. Core-Spreading Method (CSM)
3.2.3. Diffusion Velocity Method (DVM)
4. Spatial Adaptation and Hybrid Vortex Methods
4.1. The Meshless Rezoning Methods
4.2. The Remeshing Methods
4.2.1. Remeshing Kernels
The Ordinary Interpolation Kernels
The B-Splines-Based Kernels
Extrapolation of B-Splines-Based Kernels
High Order Kernels
- P1: W has support in ,
- P2: W is even and piecewise polynomial of degree M in intervals of the form ,
- P3: W is of class ,
- P5: W satisfies the interpolation property (55).
4.2.2. The Directional Splitting
- advection + remeshing in the X-direction on
- advection + remeshing in the Y-direction on
- advection + remeshing in the X-direction on
4.3. Hybrid Vortex Methods
4.3.1. The Eulerian-Lagrangian Domain Decomposition Method
4.3.2. From Vortex-In-Cell (VIC) to Semi-Lagrangian Vortex Methods
- Poisson equation:
- (a)
- Assign particle vorticity values to the grid using a P2M formula.
- (b)
- Compute the velocity field solving the following equation on the grid:
- Advection:
- (a)
- Interpolate the velocity field on the particles using a M2P formula.
- (b)
- Perform a Lagrangian advection of the particles and get their new positions and vorticity:
- (c)
- Assign vorticity values and element volumes to the grid using a P2M formula.
- Stretching (in three-dimensions only):
- (a)
- Solve the stretching equation on the grid by differentiation of the velocity field:Note: One can also write the stretching equation in its conservative form, namely
- Diffusion:
- (a)
- Solve the diffusion equation on the grid and get the final grid values for the vorticity field:
- (b)
- Interpolate the particle values from the grid using a M2P formula.
- Possibility to use Fast Poisson solvers for the velocity computation instead of direct summations using the Biot-Savart integral,
- Easy treatment of the viscous effects using finite-differences schemes on the grid (or finite-volumes schemes, spectral schemes) for the resolution of the diffusion equation,
- Easiness of implementation compared to domain decomposition methods,
- Basis for mixing VM with other methods, easy model coupling,
- Prevalence of the less restrictive Lagrangian CFL condition (Equation (25)) for the whole stability,
- Satisfaction of the particle overlapping condition all along the simulation through remeshing procedures, whose computational cost may be greatly decreased by the use of directional splitting schemes (see Section 4.2.2),
- Limitation of the diffusive effects in the advection step thanks to the Lagrangian framework (provided the use of regular and high order remeshing kernels like or more generally ).
5. Treatment of the Boundary Conditions
5.1. Boundary Treatment Based on Viscous Splitting and Vorticity Creation
- Computation of u on the wall () and creation of new sheets at the boundary with strengths in order to impose the no-slip boundary condition (these strength values directly come from the integration of Equation (65)). As explained by Chorin in [106], the velocity field is here extended across the wall by the anti-symmetry . As a consequence and a vortex sheet is generated at .
- Diffusion of the vortex sheets in the y direction using a Random Walk approach (see Section 3.1). If a sheet arises in the far-field region (), it is transformed into a vortex blob. If a sheet crosses the boundary, moving into the body, it is reflected on the other side.
5.2. Boundary Treatment Based on Vorticity Flux Conditions
5.2.1. The Continuous Problem
5.2.2. The Integral Formulation
- The diffusion is solved in the domain with a PSE scheme for instance (the diffusion scheme needs to be based on a change of vorticity strength),
- The PSE scheme in complemented by distributing this vorticity flux onto the existing vortex blobs so that vorticity enters the domain.
5.2.3. Improvements of Vorticity Flux Boundary Conditions
5.3. Boundary Treatment Based on Immersed Boundary Methods
5.3.1. The Brinkman Vortex Penalization Method
5.3.2. The Immersed Interface Vortex Method
6. Algorithmic Issues
6.1. Fast Vortex Methods
6.2. Performance-Computing Improvements Using GPUs
6.3. Adapted Grids, Multiresolution Aspects and LES Models
6.3.1. Adaptive Mesh Refinement (AMR)
6.3.2. Subgrid Scale Vortex Methods
7. Applications and Issues
7.1. Hydrodynamics
7.2. Aerodynamics
7.3. Scalars Transport
7.4. Porous Flows
Application | Flow Regime | Governing Equations | Numerical Method | Particle Remeshing | Poisson Solver | Boundary Treatment | Grid Type | Simulation | Processor Type | |
---|---|---|---|---|---|---|---|---|---|---|
[9] | Swimming of articulated bodies | , | Navier-Stokes equations | VPM with multi-body solver | Tensorial scheme | FFT solver | Brinkman penalization | Cartesian uniform | 2-D DNS | CPU |
[7,86] | Optimization of fish schooling | Navier-Stokes equations | VPM with CMA-ES algorithm | Tensorial scheme | FFT solver | Brinkman penalization | Cartesian uniform | 3-D DNS | CPU | |
[152] | Reinforcement learning for swimmers | Navier-Stokes equations | VPM with RL algorithm | Tensorial scheme | FMM solver | Brinkman penalization | Wavelets-based AMR | 2-D DNS | CPU | |
[14] | Dynamics of aircraft wings wake | Navier-Stokes equations | VPM based on SGS model | Tensorial scheme | FMM solver | - | Cartesian uniform | 3-D LES | CPU | |
[12] | Dynamics of rotorcraft wakes | Navier-Stokes equations | VPM based on SGS and ILL model | Tensorial scheme | FFT solver | ILL method | Cartesian uniform | 3-D LES | CPU | |
[13] | Dynamics of rotorcraft and aircraft wakes | , | Navier-Stokes equations | Hybrid method based on SGS model | - | FMM-GPU solver | BEM method | Body-fitted grid (near-body) | 3-D LES | CPU-GPU |
[153] | Passive scalar turbulent transport | Navier-Stokes equations | Semi Lagrangian VM | Directional scheme | FFT solver | - | Cartesian uniform | 3-D DNS | CPU-GPU | |
[141] | Sediment flow | , 7, 14, 28 | Navier-Stokes equations | Semi Lagrangian VM | Directional scheme | FFT solver | - | Cartesian uniform | 3-D DNS | CPU-GPU |
[20,155] | Flow control using porous media | , | Navier-Stokes equations | Semi Lagrangian VM | Directional scheme | FFT solver | Brinkman penalization | Cartesian uniform | 3-D DNS | CPU |
[156] | Rock dissolution at pore-scale | Dissolution regime () | Darcy Brinkman-Stokes equations | Semi Lagrangian method (for reaction equations) | Directional and schemes | FFT solver | penalization | Cartesian uniform | 3-D DNS | CPU-GPU |
8. Conclusions and Perspectives
8.1. Conclusions
8.2. Future Research Perspectives
- Turbulent and high Reynolds number flow simulations. The recent researches highlight the road map for this target with two possibilities. The first option consists in proposing modern turbulence models for the velocity-vorticity filtered Navier-Stokes equations (i.e., LES-like approaches), based for example on those already developed by Mansfield et al. [148] on one side or Cocle et al. [151] on the other side. The second view, that takes its idea from the inherent nature of vortex methods, is the multi-resolution approach (based on the pioneer works of Cottet and Wray [147,157]) which consists in using two different levels of resolution for vorticity (fine level) and the velocity (coarse level). This multi-resolution would have a deep interest in problems where important Schmidt numbers are involved like in heat transfer (e.g., thermal conduction and convection [158]), passive-scalar transport [153] or sedimentation flows [141].
- Fluid Structure Interactions (FSI). The handling of no-slip boundary conditions has long been one of the main difficulties of vortex methods. The implementation of the Brinkman penalization method has significantly improved this drawback in the context of Vortex-Particle-Mesh methods. However, this approach is limited because of its low order and accuracy. The very recent introduction of immersed interface method (IIM) [131,132] and immersed lifting line approach (ILL) [12,99] introduced a real improvement and allows us to start a real progress in the challenging field of FSI.
- Environmental and industrial particular flows. Vortex methods belong to the family of particle methods with direct treatment of particles. Therefore, their application to problems like sediment transport [141,159] or internal multi-phase flows in industry with particle mixing, dispersion, deposition, and particle-wall collision [160], seems natural and deserves a special place in the future perspectives.
- Multi-phase flows. Besides the industrial fluid/particles two-phase flows evoked above, vortex methods also have a role to play in the handling of multi-phase flows in general, with for instance the development of models able to capture interfaces between two phases and to handle surface tension, variable fluid properties and high mass density-ratio flows. To the authors knowledge, only few works based on vortex methods have been dedicated to this topic [161].
- Finally, future vortex computations need to be continually strengthened with novel high performance computing parallelism techniques (hybrid multi-CPU/multi-GPU) and powerful algorithms (like the very recent 2D-3D Poisson solver released in 2021 by Caprace et al. [162]) in order to achieve more complex flow approximations.
Author Contributions
Funding
Conflicts of Interest
Abbreviations
CFD | Computational fluids dynamics |
ODE | Ordinary differential equation |
VM | Vortex methods |
SPH | Smooth particle hydrodynamics |
CFL | Courant-Friedricks-Levy |
LCFL | Lagrangian Courant-Friedricks-Levy |
RW | Random walk |
PSE | Particle strength exchange |
CSM | Core spreading method |
DVM | Diffusion velocity method |
CCF | Convolution of cut-off function |
VRM | Vortex redistribution method |
RBF | Radial basis functions |
RPD | Regular point distributions |
DVH | Diffused vortex hydrodynamics |
VPM | Vortex-particle-mesh |
VIC | Vortex-in-cell |
P2M | Particle to Mesh |
M2P | Mesh to Particle |
FMM | Fast multiple method |
DNS | Direct numerical simulation |
LES | Large eddy simulation |
BEM | Boundary elements method |
IBM | Immersed boundary method |
FFT | Fast Fourier Transform |
GPU | Graphics processing unit |
CPU | Central processing unit |
AMR | Adaptive mesh refinement |
FWT | Fast wavelet transform |
LTS | Local time stepping |
SGS | Sub-grid scale |
SFS | Sub-filter scale |
CMA-ES | Covariance Matrix Adaptation - Evolutionary Strategy |
RL | Reinforcement learning |
ILL | Immersed lifting line |
HPC | High performance computing |
FSI | Fluid structure interaction |
Nomenclature
t | time |
x, y | spacial locations |
Q | solution vector in given continuous system |
F | source term in given continuous system |
Ω | computational domain |
u | velocity field |
δ | Dirac distribution |
p | particle |
volume of particle p (denoted in 2D) | |
location of particle p | |
denomination of the smoothing radial symmetric function in Lagrangian methods | |
ε | radius of the smoothing radial symmetric function |
Γ | circulation |
× | cross product |
ω | vorticity field (denoted ω in 2D) |
initial vorticity field (denoted in 2D) | |
quantity approximation | |
far-field velocity | |
local circulation around (denoted in 2D) | |
⋆ | convolution product |
G | Green’s function |
K | |
ζ | smooth cutoff function |
smoothing radial symmetric function in Vortex blobs methods (based on ζ and on blob radius ε) | |
quantity evaluated on mollified blob particles | |
h | particle spacing or grid spacing |
time step | |
ν | fluid viscosity |
Reynolds number. Also , , and for based on length, chord, diameter and width respectively | |
mollified diffusion kernel of given order, used in PSE schemes | |
diffusion velocity | |
⊗ | tensor product |
viscous vorticity | |
W | interpolation/remeshing kernel |
S | support of kernel W |
half of grid points constituting the support of kernel W () | |
ordinary interpolation/remeshing kernel of maximum order p | |
B-splines interpolation/remeshing kernel of maximum order p | |
Monaghan interpolation/remeshing kernel | |
interpolation/remeshing kernel of regularity r and maximum order p | |
basis functions in interpolation/remeshing process | |
velocity of the solid immersed body | |
any location on the solid surface | |
wall normal vorticity flux | |
surface of the solid body | |
vortex sheet element at the surface | |
strength of vortex sheet element | |
H | Heavyside function |
s | tangent vector to the surface |
surface curvature at point s | |
basis vectors in cylindrical coordinate system | |
χ | characteristic function of the solid body |
λ | non-dimensionalized Brinkman penalization factor |
threshold for refinement and compression, respectively, in wavelet-based AMR method | |
spatial filter of size Δ in LES | |
τ | subgrid scale stress tensor in LES |
eddy viscosity in Smagorinsky model | |
C | coefficient in Smagorinsky model |
rate-of-strain tensor in Smagorinsky model | |
low-pass filtered quantities in LES model | |
subgrid scale vorticity stress in LES | |
subfilter scale torque in LES | |
Schmidt number | |
κ | diffusivity of the advected scalar in transport problems. |
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Mimeau, C.; Mortazavi, I. A Review of Vortex Methods and Their Applications: From Creation to Recent Advances. Fluids 2021, 6, 68. https://doi.org/10.3390/fluids6020068
Mimeau C, Mortazavi I. A Review of Vortex Methods and Their Applications: From Creation to Recent Advances. Fluids. 2021; 6(2):68. https://doi.org/10.3390/fluids6020068
Chicago/Turabian StyleMimeau, Chloé, and Iraj Mortazavi. 2021. "A Review of Vortex Methods and Their Applications: From Creation to Recent Advances" Fluids 6, no. 2: 68. https://doi.org/10.3390/fluids6020068