Other CFD Methods Smooth Particle Hydrodynamicsgtryggva/CFD-Course/2013-Lecture-30.pdf · 2013. 4....
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Computational Fluid Dynamics Other CFD Methods Grétar Tryggvason Spring 2013 http://www.nd.edu/~gtryggva/CFD-Course/ Computational Fluid Dynamics Smooth Particle Hydrodynamics Computational Fluid Dynamics Smooth Particle Hydrodynamics: the fluid mass is lumped into smoothed blobs that are moved using Newton’s second law directly, without an underlying mesh Smooth Particle Hydrodynamics x 1 x 2 x 3 Computational Fluid Dynamics In SPH the fluid is modeled as a collection of smooth “blobs” or particles A(r) = A(r ')δ ∫ | r − r '| ( ) dr ' We start by the relationship And then approximate the delta function by a smoother “kernal” A(r) = A(r ') W (| r − r ' |, h) ∫ dr ' W (r, h) ∫ dr = 1 & W (r, h) h→0 ⎯ → ⎯⎯ δ r () Where: W (r, h) = 1 π 3/2 h exp r 2 h 2 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ We could select the kernal to be a Gaussian, although usually we desire a compact support Smooth Particle Hydrodynamics Computational Fluid Dynamics ρ i = ρ(r i ) = m k k ∑ ρ k ρ k W (| r i − r k |, h) = m k k ∑ W (r i − r k , h), A(r) = A(r ') W (| r − r ' |, h) ∫ dr ' = A(r ') ρ(r ') ρ(r ') W (| r − r ' |, h) ∫ dr ' = m k k ∑ A k ρ k W (| r − r k |, h) Smooth Particle Hydrodynamics m k ≈ ρ(r ') ∫ dr ' Approximate the integral by a sum over the smoothed particles: For density we have Computational Fluid Dynamics Smooth Particle Hydrodynamics The introduction of the smoothed kernal allows us to compute the gradient The introduction of the smoothed kernal allows us to compute the gradient ∇A(r) = m k k ∑ A k ρ k ∇W (| r − r k |, h)
Transcript of Other CFD Methods Smooth Particle Hydrodynamicsgtryggva/CFD-Course/2013-Lecture-30.pdf · 2013. 4....
Computational Fluid Dynamics
Other!CFD Methods!
Grétar Tryggvason!Spring 2013!
http://www.nd.edu/~gtryggva/CFD-Course/!Computational Fluid Dynamics
Smooth Particle Hydrodynamics!
Computational Fluid Dynamics
Smooth Particle Hydrodynamics: the fluid mass is lumped into smoothed blobs that are moved using Newton’s second law directly, without an underlying mesh!
Smooth Particle Hydrodynamics!
x1 x2 x3
Computational Fluid Dynamics
In SPH the fluid is modeled as a collection of smooth “blobs” or particles !
A(r) = A(r ')δ∫ | r − r ' |( )dr '
We start by the relationship!
And then approximate the delta function by a smoother “kernal”!
A(r) = A(r ')W (| r − r ' |,h)∫ dr '
W (r,h)∫ dr = 1 & W (r,h) h→0⎯ →⎯⎯ δ r( )
Where:!
W (r,h) =1
π 3/2hexp
r2
h2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
We could select the kernal to be a Gaussian, although usually we desire a compact support!
Smooth Particle Hydrodynamics!
Computational Fluid Dynamics
ρi = ρ(ri ) = mk
k∑ ρk
ρk
W (| ri − rk |,h) = mkk∑ W (ri − rk ,h),
A(r) = A(r ')W (| r − r ' |,h)∫ dr '
= A(r ')ρ(r ')ρ(r ')
W (| r − r ' |,h)∫ dr '
= mkk∑ Ak
ρk
W (| r − rk |,h)
Smooth Particle Hydrodynamics!
mk ≈ ρ(r ')∫ dr '
Approximate the integral by a sum over the smoothed particles:!
For density we have!
Computational Fluid Dynamics Smooth Particle Hydrodynamics!
The introduction of the smoothed kernal allows us to compute the gradient!
The introduction of the smoothed kernal allows us to compute the gradient!
∇A(r) = mk
k∑ Ak
ρk
∇W (| r − rk |,h)
Computational Fluid Dynamics
1ρ∇P = ∇
Pρ
⎛⎝⎜
⎞⎠⎟+
Pρ2 ∇ρ
∇ρ = mk
k∑ ∇W (| r − rk |,h)
∇
Pρ
⎛⎝⎜
⎞⎠⎟= mk
k∑ Pk
ρk2 ∇W (| r − rk |,h)
For the pressure we write:!
And using that!
The momentum equation is!
dui
dt= − mk
k∑ Pk
ρk2 +
Pi
ρi2
⎛
⎝⎜
⎞
⎠⎟ ∇W (| ri − rk |,h)
∇A(r) = mk
k∑ Ak
ρk
∇W (| r − rk |,h)
dudt
=−1ρ∇P
Start with the momentum equation!
Computational Fluid Dynamics
∇ρ = mk
k∑ ∇W (| r − rk |,h)
Similarly, for density we start with!
And using that!
The mass conservation equation is!
dρi
dt= − ρ∇ ⋅u( )i
ρ∇ ⋅u( ) = ∇ ⋅ ρu( ) − u ⋅∇ρ
∇A(r) = mk
k∑ Ak
ρk
∇W (| r − rk |,h)
dρi
dt= mk
k∑ uk − ui( ) ⋅∇W (| ri − rk |,h)
Sometimes an alternative form is used!
dρi
dt= ρi
mk
ρkk∑ uk − ui( ) ⋅∇W (| ri − rk |,h)
Computational Fluid Dynamics
∇ρ = mk
k∑ ∇W (| r − rk |,h)
The energy equation is !
Using that!
gives!
dEdt
= −Pρ∇ ⋅u
Pρ∇ ⋅u = ∇ ⋅
Pρ
u⎛⎝⎜
⎞⎠⎟− u ⋅∇
Pρ
⎛⎝⎜
⎞⎠⎟
∇A(r) = mk
k∑ Ak
ρk
∇W (| r − rk |,h)
dEi
dt= mk
Pk
ρk2
k∑ uk − ui( ) ⋅∇W (| ri − rk |,h)
Computational Fluid Dynamics
dui
dt= − mk
k∑ Pk
ρk2 +
Pi
ρi2
⎛
⎝⎜
⎞
⎠⎟ ∇W (| ri − rk |,h)
dri
dt= ui
dρi
dt= ρi
mk
ρkk∑ uk − ui( ) ⋅∇W (| ri − rk |,h)
dEi
dt= mk
Pk
ρk2
k∑ uk − ui( ) ⋅∇W (| ri − rk |,h)
Summary of governing equations!
Mass!
Momentum!
Energy!
Advect particles!
Smooth Particle Hydrodynamics!
P = P(ρ,e)Plus an equation of state!
Computational Fluid Dynamics
W (s) =10
7πh2
1−32
s2 +34
s3 0 ≤ s ≤ 1
14
(2 − s)2 1< s ≤ 2
0 2 < s < ∞
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
Smooth Particle Hydrodynamics!
Several different kernals have been used. The following is a popular one:!
For a more complete introduction, see: J. J. Monaghan. “An introduction to SPH.” Computer Physics Communications 48, 1988, Pages 89-96.!
SPH has been modified to include viscous effects by “local averaging” as well as incompressibility.!
Computational Fluid Dynamics
Other Particle Methods!
Computational Fluid Dynamics
Particle methods cover a large range of physical situations, some overlapping with more conventional methods. Particle methods include:!
!Smooth Particle Hydrodynamics!
!Point vortex/vortex blob methods!
!Particle-in-Cell codes!
!Direct Simulation Monte Carlo (rarified flow)!
!Dissipative particle methods !
!Molecular dynamics!
!and many more!
In many cases a grid is used also, either for efficiency or to account for some of the physics!
Not really for continuum flows!
Computational Fluid Dynamics
A method for rarified (low density) gases, where non-continuum effects are important!
!High altitude flight!!Nano-scales devices!
Such flows can be described by the Boltzmann equation which evolves the probability density function in six dimensional space and momentum space.!
For dense gases and liquids, the Boltzmann equation equation leads to the usual continuum equations !
Computational Fluid Dynamics
Since it is difficult to solve the six-dimensional equation, several approximate methods have been used, usually by incorporating randomness directly. In the Direct Simulation Monte Carlo method particles, representing clusters of molecules are advected through a domain resolved by a fixed grid and collision are modeled using a random number generator (MC). !
Not a molecular dynamic simulations since the particles are not as many as the molecules they represent.!
Computational Fluid Dynamics
Dissipative Particles Dynamics: particles represents molecules or fluid blobs but while the method has similarities with SPH, it is generally used to model mesoscale behavior of complex fluids and more liberties are taken in modeling the interactions of the particles!
Molecular Dynamics: Not really a CFD technique but can be used to simulate small (really small) fluid regions!
Computational Fluid Dynamics
There is no doubt that new solution strategies will continue to be developed. However, incremental advances of current approaches are likely to be the main vehicle for future advances in the computations of complex flows. The finite volume approach currently at the core of CFD has, in particular, proven to be exceedingly robust and versatile.!
Computational Fluid Dynamics
Advanced CFD!
http://users.wpi.edu/~gretar/me612.html!
Computational Fluid Dynamics
We can expect that the development of numerical methods for computing fluid flow will continue and we will see methods that are!
• More accurate!• More robust!• More versatile!
than current methods.!
We are also seeing new trends and a shift in emphasize to topics that in the past were on the sidelines.!
Computational Fluid Dynamics
New Directions for Computational Physics:!
Computational predictivity !
Multiscale/multiphysics!
Integrated simulations of complex systems!
Computational Fluid Dynamics
Computational predictivity: !This includes relatively well-defined tasks such as verification of the correctness of computer codes and uncertainty quantification as well as more nebulous ones like validations of the model being used. As codes become more complex their verification becomes more challenging. Methods such as the Method of Manufactured Solutions are one way. In its simplest form, uncertainty quantification is simply the propagation of uncertainties in parameters, properties and models to the final solution. Although conceptually simple, this is a formidable task both because we need to know all elementary uncertainties and because of the number of computations involved. Examination of statistical variations certainly has proved its value in many areas, such as quality control in manufacturing and uncertainty quantification is likely to become increasingly more important in the use of simulations in design. !
Computational Fluid Dynamics
Multiscale/multiphysics: !Multiscale is a broad term that usually means what the user intends it to. In most cases, however, it is used to mean phenomenon where some aspects of the physics that we wish to compute must be described by a different physical model. This can include contact lines in multiphase flow simulations represented by molecular or phase field models, reaction zones, shocks in rarified gases and so on. While we often think of multiscale representing different physical processes, such as continuum and non-continuum descriptions, it also applies to the same physics but modeled in different ways, such as when small drops are modeled as point particles. Numerical challenges include how to blend one description with another. !
Computational Fluid Dynamics
Integrated simulations of complex systems: !Engineers have long desired to have computational models that describe systems consisting of many coupled components. At the simplest level such simulators model the dynamics of connected rigid bodies, lumped models of chemical and power plants and so on. As computers become more powerful we are seeing growing efforts to attempt much more complex modeling, such as of rockets (the Illinois ASCI center) or a nuclear power plant (CASL), and other DOE funded efforts. Other examples include the Human Body Simulator Project in Japan (lead by S. Takagi) and possibly the recently announced Living Earth Simulator proposal by D. Helbing. Overall there is very limited theoretical basis for how to do the coupling (with some exceptions such as for solid/fluid problems) and that there is considerable room for significant progress. As the ASCI programs, CASL and other effort suggest, this is going to be a very significant area in the future.!
Computational Fluid Dynamics
A Few Advanced Scientific Computing sites!
http://science.energy.gov/ascr/facilities/incite/!
http://www.csar.illinois.edu/!
http://www.csafe.utah.edu/!
https://asc.llnl.gov/alliances/!
http://nnsa.energy.gov/!
http://www.sandia.gov/NNSA/ASC/univ/psaap.html!
Computational Fluid Dynamics
CASL: The Consortium for Advanced Simulation of Light Water Reactors!
http://www.casl.gov/!
http://www.ornl.gov/sci/nsed/docs/CASL_Overview.pdf!
http://www.ornl.gov/sci/nsed/docs/CASL_Project_Summary.pdf!
http://www.er.doe.gov/ascr/ascac/Meetings/Aug10/Kothe.pdf!
http://computing.ornl.gov/SC10/documents/Kothe_CASL_Overview_SC10.pdf!
Computational Fluid Dynamics
Computational Fluid Dynamics
From: http://www.ornl.gov/sci/nsed/docs/CASL_Project_Summary.pdf!
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