V.A. Titarev

Efficient deterministic modelling of rarefied gas flows

V.A. Titarev

Dorodnicyn Computing Centre of Russian Academy of Sciences

64 IUVSTA Workshop on Practical Applications and Methods of GasDynamics for Vacuum Science and Technology, May 16-19, Germany

V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 1 / 32

Outline of the presentation

1 Motivation:

why kinetic equations and what methods we need for industrial applications

2 Model application: gas flow in long micro channels of finite length

planar casecircular pipelinearised as well as non-linear problems, including flow into vacuum

3 Review of existing methods & results

steady-state iterationcomposite methods

4 Framework of new unstructured mesh solvers

mixed-element unstructured spatial solverimplicit time evolutionefficient HPC version

5 Calculations of flows in long finite-length channels

comparison with published data of Loyalka, Graur & Sharipovdata for long finite-length planar channeldata for long finite-length circular pipe


Why to use the Boltzmann kinetic equation?

At present, the Monte-Carlo statistical simulation method (DSMC) is the computationalmethod of choice. However,

Due to statistical fluctuations and 1st order not very suitable for unsteady flows,transitional and near-continuum flows, slow flows

computational efficiency is not optimal due to explicit time evolution

The Boltzmann equation is free of any limitations of the DSMC:

The equation is applicable across all flow regimes, i.e. from free molecular tonear-continuum flows

Unsteady flows can be treated in a straightforward manner.

The deterministic nature of the equation allows the development of efficienthigh-order accurate methods, including methods with implicit time evolution

It is possible to use special properties of the flow problem (e.g. asymptoticsolution) in construction of numerical methods


Requirements for industrial applications

Ability to compute flows with Knudsen layers in arbitrary geometries

multi-block structured meshesmixed-element unstructured meshes, pure tetras not sufficient

High-order of accuracy (at least 2nd order)

linear schemes not suitable (e.g. MacCormack)non-linear solution-adaptive methods are required

Conservation with respect to collision integral

for model equations various approaches available: Rykov et al 1994,Mieussens 2000, Titarev 2006

Rapid convergence to steady state

implicit time evolution

Good scalability on modern HPC machines

MPI modelhybrid MPI-OpenMP or MPI-GPU models


S-model kinetic equation

In the non-dimensional variables the kinetic equation takes the form:

∂f

∂t+ ξx

∂f

∂x+ ξy

∂f

∂y+ ξz

∂f

∂z= ν(f + − f ),

ν =8

5√π

nT

µ

1

Kn, f + = fM

(1 +

4

5(1− Pr)Sc(c2 − 5

2)

),

fM =n

(πT )3/2exp (−c2), c =

v√T, v = ξ − u, S =

2q

nT 3/2.

Macroscopic quantities defined as(n, nu, n(

3

2T + u2), q

)=

∫ (1, ξ, ξ2,

1

2vv 2

)fdξ.

Boundary condition on the surface:

f (x, ξ) = fw =nw

(πTw )3/2exp

(− ξ2

Tw

), ξn = (ξ, n) > 0,

nw = Ni/Nr , Ni = −∫

ξn<0

ξnfdξ, Nr = +

∫ξn>0

ξn1

(πTw )3/2exp

(− ξ2

Tw

)dξ.


Linearised kinetic equation

Linearise around the Maxwellian distribution f0 corresponding to average values ofdensity and temperature n0, T0:

f = f0(1 + h), h = h(x, ξ), |h| � 1,

f0 =n0

(πT0)3/2exp (−ξ2/T0) ≡ 1

π3/2exp (−ξ2).

Linearised macroscopic quantities:

n̂ =n − n0

n0=

∫f0hdξ, u =

∫ξf0hdξ,

T̂ =T − T0

T0=

2

3

∫ξ2f0hdξ − n̂, q = −5

4u +

1

2

∫ξξ2f0hdξ.

The evolution equation for the perturbation function h has the following form:

ξx∂h

∂x+ ξy

∂h

∂y+ ξz

∂h

∂z= ν0(h(S) − h), ν0 =

8

5√π

1

Kn,

h(S) = n̂ + 2uξ + (ξ2 − 3

2)T̂ +

8

5(1− Pr)(ξ2 − 5

2)qξ.


Existing methods: steady-state iteration

Typically used in two space dimensions. Let n be the iteration counter. Then:

ξx

(∂f

∂x

)n+1

+ ξy

(∂f

∂y

)n+1

= νn(f nm − f n+1)

Advantages:

Low storage requirement

Extends to axisymmetric (Shakhov, 1974) and 3D (Arkhipov & Bishaev, 2007)flows directly

Allows ”shock-fitting” type difference schemes

Disadvantages:

Difficult to implement with high order of accuracy

Non-conservative with respect to collision integral

Poor convergence for small Kn, each iteration adds only a Kn2 change (Bishaev &Rykov, 1975)


Composite methods

Basic idea: for Kn� 1 explicitly use the continuum Navier-Stokes solution in thenumerical method

1 Bishaev & Rykov, 1975: one-dimensional non-linear heat transfer problem

2 Sharipov & Subbotin, 1992: multi-dimensional linearised problem, typical ofmicrochannel flow

Consider the linearised equation:

ξ∇f = L̂f + g(x, ξ)

Now assumef = f0 + f̃ , f0 = lim

Kn→0f

Thenξ∇f̃ = L̂f̃ + g(x, ξ) + L̂f0 − ξ∇f0

The use of Chapman-Enskog method gives

f0 = 2ξ · u0

where u0 is the bulk velocity, found from Stoke’s equations:

µ4u0 = ∇p, divu0 = 0.


Existing 3D methods & codes for kinetic equations

These are time marching methods, in which steady solution is computed as a limit intime:

multi-block structured (Z.-H. Li & H.-X. Zhang.J. Comp. Phys.,v.193, 2004)

Unified Flow Solver (UFS) on Cartesian semi-structured meshes (V.I. Kolobovet. al. J. Comp. Phys., 223:589–608, 2007)

1st order tetrahedral (Yu.Yu. Kloss et. al. Atomic Physics, 105(4), 2008)

However, none of them satisfies all the requriements:

spatial meshes not very suitable for certain problems (e.g. long micro channels,Knudsen layer resolution)

slow convergence to steady state due to explicit time evolution

unclear scalability on HPC

More efficient & universal methods of numerical modelling are needed.


2D benchmark problem: flow in long planar micro channel

A planar microchannel of length 2l & width a connects large reservoirs filled withthe same monatomic gas at pressures p1,p2 and temperatures T1,T2, respectively.

∆P = p1 − p2 & ∆T = T1 − T2 cause the gas movement through channel.

The main quantity of interest: mass flow rate M:

M =

√2RT0

p02a

∫−a

aρ(x , y , z)u(x , y , z)dy , p0 =1

2(p1 + p2), T0 =

1

2(T1 + T2)


3D benchmark problem: flow through a long circular pipe

A microchannel (pipe) of length 2l & radius a connects large reservoirs filled withthe same monatomic gas at pressures p1,p2 and temperatures T1,T2, respectively.

Due to the spatial symmetry of the problem only quarter of the pipe is considered.

∆P = p1 − p2 & ∆T = T1 − T2 cause the gas movement through channel.

Non-dimensional mass flow rate:

M =

√2RT0

p0|A|

∫A

ρ(x , y , 0)w(x , y , 0)dxdy


Hierarchy of solutions

The solution to the problem may have the following levels of approximation:

1 Fully non-linear approximation for finite-length channel (arbitrary pressure &temperature jumps)

2 Linearised approximation for finite-length channel (small pressure & temperaturejumps)

3 Asymptotic approximation corresponding to l/a→∞gas motion is caused by constant pressure & temperature gradients actingalong the channelspatial dimension of the problem is reduced

Choice of flow model for the Boltzmann equation:

1 exact collision integral - in theory most accurate, in practice very difficult toachieve good accuracy numerically

2 BGK (or Krook) model - the simplest model collision integral, not accurate fornon-isothermal flows

3 S-model collision integral of Shakhov - presently the most accurate model equationfor microchannel flows


Existing results for planar microchannel flows

Infinitely long channel - a lot of studies, see e.g. F. Sharipov and V. Seleznev 1998.

F. Sharipov & G. Bertoldo - comparison of model & exact collision integrals

Finite-length channel, linearised flow - only short & moderate tubes, see e.g.

C. Cercignani & I. Neudachin 1979V.D. Akin’shin, A.M. Makarov, V.D. Seleznev and F.M. Sharipov, 1988, 1989E.M. Shakhov 1999,2000

Finite-length channel, nonlinear flow - Larina & Rykov 1996

Sharipov & Seleznev 1994: approximate method to calculate mass flow ratewithout taking into account end effects.

Calculations for really long finite-length channels are missing even in the linearisedregime.


Existing results for 3D microchannel flows

Infinitely long channel -most studies for the circular/rectangular cross section.

More recent results

I. Graur & F. Sharipov, 2009 - elliptic cross sectionL. Szalmas & D. Valougeorgis 2010 - triangular and trapezoidal cross sectionV. Titarev & E.M. Shakhov 2010 - arbitrary polygonal cross section

Linearised flow through a finite-length circular pipe: Shakhov 2000

Nonlinear flow:

Orifice flow Shakhov 1974Axisymmetric rarefied flow in the pipe caused by evaporation/condensationShakhov 1996, Larina & Rykov 1998aFlow through short tubes for arbitrary pressure ratios, including into vacuum:Varoutis et al 2008, 2009 (using Monte-Carlo)Anikin, Kloss, Martynov and Tcheremissine 2010: Numerical modelling ofKnudsen experiment

Accurate calculations for really long finite-length pipes are missing even in thelinearised regime.


Nesvetay 2D/3D

Framework of 2D/3D one-step implicit methods for deterministicmodelling of rarefied gas flows developed by the author from 2007onwards

Flow model - Boltzmann kinetic equation with various model collisionintegrals (Krook, Shakhov, Rykov etc)

The framework consists of the following blocks:

high-order TVD method on hybrid unstructured meshes

fully conservative procedure to calculate macroscopic quantities

one-step implicit time evolution method

both OpenMP and MPI parallelisation

Due to time restrictions, only 3D version will be presented


Conservative discrete velocity framework

March in time to steady state:

∂

∂tg = −ξ∇g + J(g), J = ν(g (S) − g),

where g is the distribution function f for the nonlinear case andperturbation h in the linearised case.

Replace the infinite domain of integration in the molecular velocityspace ξ by a finite computational domain.

The kinetic equation is replaced by a system of Nξ advectionequations for each of gα = g(x, ξα):

∂

∂tgα = −ξα∇gα + J(gα),

which are connected by the macroscopic parameters in the functiong (S) from the model collision integral J.


Approximation of model collision integral on ξ mesh

Let ωα be the weights of the second order composite quadrature rule used forintegration in ξ space. To compute the vector of primitive variables

U = (n, u1, u2, u3,T , q1, q2, q3)T

for each spatial cell we have the following system of equations

∑α

1ξξ2

vv 2

α

(f +α − fα)ωα +

000

2 Pr q

= 0.

Here subscripts i are n are omitted for simplicity.These eight equations are solved using the Newton iterations the initial guess for whichis provided by the direct (non-conservative) approximation

nnu

32nT + nu2

q

=∑α

1ξξ2

12vv 2

α

fαωα

Usually, one or two Newton iterations are sufficient for convergence.


Advection scheme

Introduce in the physical variables x = (x1, x2, x3) = (x , y , z) a computationalmesh consisting of elements (spatial cells) Vi .

Denote by |Vi | the cell volume, |A|il area of face l .

Omit subscript α for simplicity. Let ∆t = tn+1 − tn, gn = g(tn, x, ξ), ,

One-step explicit method can be written as:

gn+1 − gn

∆t= −ξ∇gn + Jn

The implicit one-step method has the following form:

(1 + ∆tνn + ∆tξ∇)gn+1 − gn

∆t= −ξ∇gn + Jn


Fully discrete method

Basic idea:

integrate over Vi

them discretize left-hand side with first-order upwind spatial differences

the right-hand side Lnα is approximated using a TVD method.

The result is a system of linear equations for increments of the solution δn = gn+1 − gn:

(1 + νni ∆t)δni +∆t

|Vi |∑l

ξnlFl(δni , δ

nσl (i)

)|Ail | = {−(ξ∇gn)i + Jni }∆t

where σl(i) the cell index of the cell adjacent to the face l of cell Vi .

Using divergence theorem → sum of face fluxes:

(ξ∇gn)i =1

|Vi |∑l

Φnil , Φn

il =

∫Ail

ξnlg(tn, x, ξ)dS .

The values at the next time level are given by gn+1i = gn

i + δni .


Stencils for piece-wise linear reconstruction method


Time evolution

An approximate factorization of the system is carried out using the approachsuggested in Men’shov & Nakamura 1995, 2000.

As a result, the computational cost of one time step of the implicit method is only25% larger than the computational cost of an explicit method.

In calculations, the value of the time step ∆t is evaluated according to theexpression

∆t = C mini

di/ξ0,

where C is the prescribed CFL number, di the characteristic linear size of thecell Vi .

C ≤ 13

corresponds to the conventional explicit method


Results

1 Calculations are run for both 2D and 3D channels

2 Both linearised & non-linear formulations are considered

3 Efficiency of implicit time evolution is assessed

4 Parallel scalability tests are carried out

5 All calculations are run on the HPC Facility ’Astral’ of the Cranfielduniversity, which is a Hewlett Packard HPC, comprising 856 IntelWoodcrest cores (3.0GHz).


Results: pressure driven & creep flow

Calculations have been carried out for

l/a = 10, 100 and 1000 in 2Dl/a = 1, 10 and 100 in 3D

Main calculated characteristic of the flow - mass flow rate as function of ν0, l/aand ratios of temperature and pressure in reservoirs

Pressure driven flow: movement is caused by pressure difference only

Linearised problem: solution is independent of pressure differenceNon-linear problem: pressure ratio p1/p2 defines the flow

Creep flow movement is caused by temperature gradient; pressure difference is

zero.

Only linearised problem is considered

To compare results for different l/a and p1/p2 consider mass flow rate scaled withthe average pressure gradient:

Mp = − 2l

∆PM|∆T=0, MT =

2l

∆TM|∆P=0


Mass flow rate in linearised 2D problem


Efficiency of implicit time evolution in 3D

The solution of the nonlinear problem with pressure ratio p1/p2 = 2 is computed forν0 = 1 and hexa mesh with 55000 cells


Scalability studies using 163 velocity mesh in 3D

Spatial mesh of 55000 hexa cells and 163 velocity mesh are used, performancenormalised by the result on 16 cores

Weak scaling (fixed size per core) Strong scaling (fixed problem size)


Asymptotic case l/a =∞ & circular pipe

Flow is described by spatially two-dimensional kinetic equation with a source termSolid line - Lo & Loyalka, 1982, semi-analytic methodCrosses – Graur & Sharipov, 2009, 106 spatial cells, composite schemeRed circles - Titarev & Shakhov, 2010, 103 − 7× 103 spatial cells, direct solution


Mass flow rate in linearised 3D problem


Flow in a 3D composite tube

Channel: circular and pentagonal sections smoothly connected

B.C.: z = 0 - evaporation, z = 5 – complete condensation.

Mixed element mesh with Knudsen layer: 10051 spatial cells intotal, of which 6401tetras, 880 hexa, 2344 prisms and 426 pyramids.

Velocity mesh: 163 cells.

Calculations run on a PC


Mass flow ρw for Kn = 0.1

Cuts along the axis (left) and symmetry plane (right)


Conclusions

1 A brief summary of existing numerical methods for kinetic equationshas been presented

2 Review of existing results for 2D and 3D microchannels shows thatthere are virtually no numerical results for long finite-length channels

3 A new framework for analyzing rarefied flows in complex 2D/3Dgeometries has been presented, which is fully unstructured, efficientand scalable up to 512 cores

4 Using the proposed approach, for the first time an accuratecalculation of the mass flow rate through long microchannels hasbeen performed for various flow regimes


Publications

1 V.A. Titarev. Conservative numerical methods for model kinetic equations //Computers and Fluids, 36(9):1446 – 1459, 2007.

2 V.A. Titarev. Numerical method for computing two-dimensional unsteady rarefiedgas flows in arbitrarily shaped domains // Computational Mathematics andMathematical Physics, 49(7):1197–1211, 2009.

3 V.A. Titarev. Implicit numerical method for computing three-dimensional rarefiedgas flows using unstructured meshes // Computational Mathematics andMathematical Physics, 50(10):1719–1733, 2010.

4 V.A. Titarev. Implicit unstructured-mesh method for calculating Poiseuille flows ofrarefied gas // Communications in Computational Physics, 8(2):427–444, 2010.

5 V.A. Titarev and E.M. Shakhov. Nonisothermal gas flow in a long channelanalyzed on the basis of the kinetic S-model // Computational Mathematics andMathematical Physics, 50(12):2131–2144, 2010.

6 V.A. Titarev. Efficient deterministic modelling of three-dimensional rarefied gasflows, submitted to Communications in Computational Physics.

7 V.A. Titarev. Linearised problem of rarefied gas flow through a long circular pipeof finite length, in preparation.


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V.A. Titarev