a novel numerical scheme for reactive flows at low mach numbers

Computer Physics Communications 129 (2000) 267–274www.elsevier.nl/locate/cpc

A novel numerical scheme for reactive flowsat low Mach numbers

Olga Filippova∗, Dieter HaenelInstitute of Combustion and Gasdynamic, University of Duisburg, D-47048 Duisburg, Germany

Abstract

For the simulation of low Mach number reactive flows with significant density changes a novel numerical scheme is proposedwhich contains modified lattice BGK model in combination with the conventional convective-diffusion solvers for equationsof temperature and species. Together with boundary-fitting formulation and local 2nd order grid refinement (embedded grids)the scheme enables the accurate consideration of low Mach number combustion in complex geometry in the wide range ofReynolds and Damköler numbers on the simplest Cartesian grids. 2000 Elsevier Science B.V. All rights reserved.

Keywords:Lattice BGK models; Low Mach number; Reactive flows

1. Introduction

Reactive flows at low Mach numbers are described by the whole system of Navier–Stokes equations withchemical source terms coupled with the transport equations for species. The feature that requires to define “lowMach number flows” as the special class for numerical simulations is connected with the fact that entropy, vorticityand acoustic modes of this system of equations in the case of low Mach numbers are of the greatly differentfrequencies. If only steady-state and low-frequency flows are of the main interest then a numerical solutionprocedure of the whole system of equations becomes non-efficient since the tracking of acoustic fluctuationsrequires very small time steps. To overcome this difficulty the low Mach number approximation (LMNA) ofNavier–Stokes equations was proposed in [1] and by others. In the reduced system of governing equations acousticwaves are filtered out, which allows to avoid the severe restriction on the time step.

In absence of temperature gradients the system of LMNA reduces to the system of Navier–Stokes equationsfor incompressible flow and transport equations for species. So the methods of solution are usually extensions ofmethods developed for the solution of incompressible flows. Pressure relaxation methods [2] widely used for thecomputations of slow combustion include the solution of a Poisson equation for the pressure at every time step,which needs time-consuming iteration procedure.

Recently a new class of incompressible solvers – the lattice Boltzmann methods, here used with the collision termof Bhatnagar–Gross–Krook (BGK) form [3,4], are created on the basis of gas-kinetic representation of fluid flow.These schemes don’t deal with the discretized system of Navier–Stokes equations, but describe the evolution of

∗ Corresponding author.

0010-4655/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0010-4655(00)00113-2

268 O. Filippova, D. Haenel / Computer Physics Communications 129 (2000) 267–274

discrete distribution functions in form of relaxation equations. Hydrodynamic variables are moments of the discretedistribution functions. It can be shown that LBGK schemes provide solutions for incompressible flows with second-order accuracy in Knudsen number for steady-state flows and flows in low-frequency limit. With improvements asboundary-fitting formulation and local 2nd order grid refinement (embedded grids) [5] the scheme enables to solveefficiently incompressible flows in complex geometry in wide range of Reynolds numbers on the simplest Cartesiangrids.

The idea to extend lattice-Boltzmann schemes to solution of combustion problems belongs to Succi et al. [6]. Inthe limiting case of infinite fast reactions and “cold” flames with little heat release the authors have described thereactive flow dynamics with 24 speeds LBE-FCHC scheme including in 2D two passive scalars as mixture fractionof the fuel and temperature. As it was mentioned in [6] the next class of problems to be addressed is low-Machreacting flows in which density is allowed to respond to temperature changes over a significant dynamical range ofvalues. The first scheme of this kind dealing with the system of LMNA equations in the case of simplified chemistrywas proposed by the present authors in [7]. It has contained the modified LBGK model for solution of continuityand momentum equations and the conventional convective-diffusion solver for transport equations for temperatureand species. With this scheme reactive flows in a wide region of Damköhler numbers were considered.

An improved variant of the original scheme is proposed in this paper which simplifies the introduction ofembedded grids in zones of chemical reactions and makes the whole algorithm more stable and efficient. Themodification introduces new variables and additional terms in the expression for equilibrium distribution function.An interesting aspect of the both schemes is that the value of dynamic part of the pressure is not obtained explicitlyduring the computations. It is included together with divergenges of velocity and flux of the mixture in the valueof zero-order moment of distribution functions. To obtain the value of dynamic part of the pressure explicitly onemust apply the usual finite-difference technique to this moment.

The locality in definition of pressure allows the use of different time-stepping in the zones of reactions and zonesof pure transport what results in the efficient resolution of reactive flows with thin zones of reactions.

2. Low Mach number approximation of Navier–Stokes equations (LMNA)

The governing equations for low Mach number reactive flows are derived from the Navier–Stokes equations ofa compressible gas together with transport equations for species by expanding the primitive variables(Eu,p,h) in aseries of square Mach numbersM2 and neglecting the higher order terms [1]. The low Mach number approximationof Navier–Stokes equations (LMNA) together with transport equations for species results in the following systemof equations:

∂tρmix + ∂α(ρmixuα)= 0, (1)

∂tρmixuα + ∂β(ρmixuβuα)+ ∂αp(1) − ∂βµ(∂βuα + ∂αuβ − 2

3δαβ∂γ uγ)= 0,

ρmixCp(∂tT + Eu∇T )= ∂γ Cpχρmix∂γ T + hiwi −∇ρmixT ·Cp,ini EVi + ∂p(0)

∂t,

(2)∂tni + Eu∇ni = 1

ρmix∂γDiρmix∂γ ni + wi

ρmix,

N∑i=1

ni = 1, p(0) = ρmixRT,

whereρmix andEu are the density and velocity of the mixture,µ is dynamic viscosity,χ is the coefficient of thermaldiffusivity, ni is the mass fraction andDi is the diffusivity of theith chemical species in a mixture,Cp,i is theconstant-pressure specific heat capacity of speciesi andCp is the constant-pressure specific heat capacity of themixture, EVi is the diffusion velocity of speciesi, wi andhi are the rate of production and heat of formation ofith

O. Filippova, D. Haenel / Computer Physics Communications 129 (2000) 267–274 269

species, respectively. Summation over repeated indices is assumed. By this way the pressure splits in two parts:in a thermodynamic partp(0)(t) which is constant in an open system and in a dynamic partp(1)(t, Er) associatedwith the gas motion. The fact that the thermodynamic part of pressure is constant in an open system results for themixture containingN species with different molecular weightsWi in the following algebraic relationship

ρmixT a = ρ0T0a0, a =N∑i=1

ni

Wi

.

Below the values of density of the mixtureρmix, temperatureT and molar density of the mixturea arenondimensional, divided by reference valuesρ0, T0 anda0 accordingly.

3. A modified LBGK scheme for solution of continuity and momentum equations of LMNA

The first combined LBGK-Finite-Difference scheme proposed in [7] has dealt with the simplified system ofLMNA (it was assumed that all species have the same molecular weights and constant-pressure specific heatcapacities). At the same time the dependence of dynamic viscosity of the mixture on the temperature wastaken into account. This dependence makes relaxation parameter in LBGK part of the scheme the field variabledepending on the temperature. By this way it was recognized that in the case of strong temperature gradientsthis dependence can stabilize the scheme. The problem under consideration was, whether it’s possible to makethe dependence of relaxation parameterω(T ) on the temperature stronglier conserving nevertheless physicallynatural dependenceµ(T ). It is achieved in the present scheme through the change of the expression for effectiveequilibrium distribution function by such a way that its first-order moment provides the value of the mixture fluxinstead of the mixture velocity as in the previous model [7]. To correct the expression for strain tensor the additionalterm containing gradients of density is added to the expression for effective equilibrium distribution function andthe additional dependence on the temperature is introduced in the relaxation parameter in LBGK equations. Thisimproves further the stability properties of algorithm and simplifies the implementation of local 2nd order gridrefinement (embedded grids) [5] in the zones of high anisotropy of the flow.

To solve the governing equations of the flow, Eqs. (1)–(2), the lattice-BGK equation [3,4] is modified by thefollowing way:

fpi(t + δt , Er + Ecpiδt )= fpi(t, Er)+ω[f

eqpi (t, Er)− fpi(t, Er)

]p 6= 0, (3)

and

f0(t + δt , Er)= f0(t, Er)+ω[f

eq0 (t, Er)− f0(t, Er)

]−G(t, Er),G(t, Er)= ρmix(t, Er)− ρmix(t − δt , Er) (4)

with the following expression for the effective equilibrium distribution function

feqpi = tp

[P

ρ0c2s

+ ρmixuαcpiα

c2s

+ ρmixuαuβ

2c2s

(cpiαcpiβ

cs2− δαβ

)+ νuγ ∂δρmix

c2s

(cpiγ cpiδ

c2s

− δγ δ)],

where

P = p(1) + 23µ∂γ uγ + νρ0∂γ ρmixuγ = c2

s ρ0

∑p,i

fpi ,

ρmix Eu=∑p,i

fpi Ecpi. (5)


To correct the expression for strain tensor appearing in macrodynamical momentum equations the followingrelationship between kinematic viscosityν and the relaxation parameterω in Eqs. (3), (4) has to be introduced

ν = µ(T )ρ0

T a = δxc6

(2

ω− 1

),

µ(T = 1)

ρ0= UL

Re. (6)

Notice, that relaxation parameterω is now a field variable depending on the temperature even if no dependenceof dynamic viscosity on temperature is taken into account.

For the computations of steady-state problems termG(t, Er) in Eq. (4) can be omitted.Introduce Knudsen number of the flowε = δx/L as the ratio between lattice spacingδx and characteristic length

of the flowL. Here characteristic length is defined as the minimal distance on which hydrodynamic variables areessentially changed, as, for example, thickness of boundary layer. Introduce global Mach number of the flow asMg = U/c whereU is characteristic velocity of the flow and characteristic lengthL0 defined by geometry as, forexample, the size of the body in the flow. Below macrodynamical equations are obtained and their accuracy isestimated on some single grid on whichL∼ L0. The extension of this derivation to the case of strongly anisotropicflows resolved on the grids with different sizes is straightforward.

The general derivation of macrodynamical equations from LBGK equation is described, for example, in [3,4]. This derivation is essentially based on the use of two different characteristic times of the flow: characteristictime of acoustic waves propagation(T1 ∼ ε−1δt ) and characteristic time of low-frequency hydrodynamics(T2 ∼ ε−2δt ). However if external conditions are uniform in time then acoustic modes usually introduced in thenumerical solution with LBGK scheme by the initial conditions dissipate relatively fast and low-frequency regimewith characteristic timeT ∼ (Mgε)

−1δt is developed in the computational domain. These characteristic timescorrespond to the range of Strouhal numbers

Str= L0

UT= δt

TMgε∼O(1)

obtained experimentally for vortex streets. This behaviour was confirmed with the big amount of numericalexperiments [3–5]. Considering, for example, benchmark problem concerning incompressible flow around cylinderin the long rectangular channel with LBGK scheme [5] one can observe from the smooth curves for lift and dragcoefficients and pressure difference on the cylinder obtained on the late stages of computation that there is no moremodes characterized by the acoustic time scale.

For this limiting regime analysis of macrodynamical equations obtained from Eq. (3) can be done in the usualmanner used in the numerical methods using only one time-scale. Below we are dealing only with the solutionobtained with LBGK scheme in low-frequency limit. The intermediate part of numerical solution from the start ofcomputations up to obtaining of low-frequency limit is not analyzed.

To obtain macrodynamical equations Taylor expansion of the zeroth and first order moments of the modifiedlattice-BGK equations (3), (4) in a series ofδt = εL/c and expansion of distribution function around local effectiveequilibrium distribution function

fpi = f eqpi + δtf (1)pi

are performed.From the definition of functionP and mixture fluxρmixEu and expression for effective equilibrium distribution

function one can obtain∑p,i

f(1)pi = 0,

∑p,i

f(1)pi cpiα = 0.

From modified LBGK equation (Eq. (3)) forp 6= 0

f(1)pi =−

1

ω

(∂f

eqpi

∂t+ ∂f

eqpi

∂xαcpiα

)+O

(δtf

eqpi,t t , δxf

eqpi,tx , δxf

eqpi,xxc

).


Using symmetry properties of the lattice, the following system of macrodynamical equations for flows withcharacteristical timesT ∼ (Mgε)

−1δt if Mg ∼ ε is obtained

δt∂t

(P

ρ0c2s

+ ρmix

)+ δx∂αρmixuα/c+ 0.5δ2

x∂α∂β

(Pδαβ

ρ0c2+ ρmixuαuβ/c

2)

+O([ρmix]t ε4, [P/c2]t ε4, [ρmixu/c]sε3)= 0, (7)

δt∂tρmixuα/c+ δx∂β(Pδαβ

ρ0c2 + ρmixuαuβ/c2)

= δx/c∂β[ν(∂βρmixuα/c+ ∂αρmixuβ/c+ δαβ∂γ ρmixuγ /c)− νuγ /c∂δρmix(δαγ δβδ + δαδδβγ )

]+O

([ρmixu/c]t ε4, [ρmixuu/c2]sε3)= 0. (8)

Here[F ]t , [F ]s denote the deviations of variableF on the characteristic time and length accordingly,δt∂/∂t ∼ ε2

andδx∂/∂x ∼ ε. Substituting in Eqs. (7)–(8) the expression for functionP (Eq. (5)) andν (Eq. (6)) and taking intoaccount that time deviations of pressure and mixture flux are in the order of their spatial deviations, one can obtainwith second order accuracy in Knudsen numberε the following system of macrodynamical equations (written herein dimensional form)

∂tρmix + ∂αρmixuα = 0, (9)

(∂tρmixuα + ∂βρmixuαuβ)=−∂αp(1) + ∂βµ(∂βuα + ∂αuβ − 2

3δαβ∂γ uγ). (10)

It corresponds to the continuity and momentum equations of LMNA (Eqs. (1)–(2)).Numerical algorithm is based on time-splitting between LBGK part of the scheme and convective-diffusion

solver for equations of temperature and species. If at timet the values of distribution functions and allhydrodynamic variables are known then LBGK equations (Eq. (3)) are solved and functionsP and ρmixEu asmoments of distribution functions are obtained on the new time-levelt + δt . Convective-diffusion equations fortemperatureT and mass fractions of speciesni are solved within the levelst andt+δt . Using algebraic relationshipρmixT a = ρ0T0a0 one can obtain on the new time level the value of density of the mixtureρmix and relaxationparameter as the field variableω(T ). Then the expression for effective equilibrium distribution function on thenew time-level is completely defined and procedure repeats again.

4. Numerical examples

Although several numerical approaches were used for computation of low Mach number combustion [8] up tonow there is no complex benchmark computation for analyzing and validation of different codes. Therefore thenew scheme was validated on results obtained for a common test problem solved by a finite difference method inour numerical group. The finite difference method used for comparison was based on a pressure relaxation method,in which the Poisson equation of pressure was solved with an overrelaxated Gauss–Seidel iteration scheme.

The common test problem describes the flow of a hot oxidizer around a periodical grid of porous burners, throughwhich the cold fuel is injected into the flow with the velocity of injection of 10% of the velocity of the oxidizer atthe entrance. The Reynolds number of the flow is Re= 80 related to the velocity of the oxidizer at the entrance, tothe diameter of the burner and to the viscosity in the vicinity of the burner. Combustion is described by an one-stepglobal reaction

αFuel+ βOxidizer→ Product

with α = n/(n+1),β = 1/(n+1), the production ratewp/ρmix = nF ·nO ·exp(−TA/T ) ·Da, where the parametersareTA = 12, Da= 620,n= 10 and heat production rate of 4wp/ρmix.


Fig. 1. Streamlines and temperature of the flow through periodical grid of porous burners at Re= 80 (one-step global reaction), computed withpressure relaxation method (a) and LBGK method (b). Velocity of injection is equal to 10% of velocity of the oxidizer at the entrance.

Other parameters are Pr= 0.7148, Sc= 1, µ∼√T . The same molecular weights of all species is assumed inthis case. In Fig. 1(a) one can see the field of temperature and streamlines of the mixture computed with the pressurerelaxation method and in Fig. 1(b) – with the LBGK method. The lowest and highest values of the temperature areshown in the brackets. The agreement between the solutions obtained with different numerical schemes was foundto be excellent.

The next computational example is performed only with a novel numerical scheme inasmuch as it is essentiallyconnected with the use of embedded grids [5] due to the choice of relatively high Re and Da numbers. The flowof a hot oxidizer through the periodical grid of porous burners is considered. From the surfaces of the burnersdiffusional mass flux of the cold fuel is assumed. The Reynolds number of the flow is Re= 300 related to thevelocity of the oxidizer at the entrance, to the diameter of the burner and to the viscosity in the vicinity of theburner. The chemistry is described with an one-step global reaction

Fuel+Oxidizer→ 2Product

with the production rate

wF =−ωR ·WF , wO =−ωR ·WO, wP = 2ωRWP

and

ωR = T 2 · ρmixnF

WF

· ρmixnO

WO

· exp(−TA/T ) ·Da,

where the parameters areTA = 12 and heat production rate is 4ωR/ρmix. The other parameters are Pr=0.7148, Sc= 1, µ ∼ √T , Da= 15000. In Fig. 2 the flow obtained with the ratio of molecular weightsWO :WP :WF = 2:1.5:1 is shown: the field distribution of molar density of the mixturea – in Fig. 2(a), the field oftemperature – in Fig. 2(b), the mass fraction of product – in Fig. 2(c). As one can see for these values of Re and Danumbers the low-frequency oscillating regime is developed. The strong anisotropy of the flow and the necessity oflocal grid refinement in the zones of reactions and in the boundary layer becomes clear from Fig. 2.


Fig. 2. Flow through periodical grid of porous burners at Re= 300. Diffusional mass flux of the fuel from the surfaces of the burners. One-stepglobal reaction, the ratio of molecular weightsWO :WP :WF = 2:1.5:1. (a) molar density, (b) temperature, (c) mass fraction of the product.

5. Conclusions

A novel scheme including the extended LBGK model and the conventional convective-diffusion solver isproposed for the numerical solution of low Mach number combustion. Together with local 2nd order gridrefinement (embedded grids) and boundary-fitting formulation proposed earlier by the authors for incompressibleflows the scheme allows to consider low Mach number reactive flows in complex geometry and in wide region ofReynolds and Damköhler numbers on the simplest Cartesian grids.

Acknowledgement

We would like to thank Irenäus Wlokas for his assistance in producing the finite difference simulation used asreference solutions in this paper.

O.F. would like to thank Prof. S. Succi, Prof. V.S. Ryaben’kii, Dr. A. Povitsky and Dr. L.-S. Luo for the usefuldiscussions.


References

[1] G.J. Sivashinsky, Hydrodynamics theory of flame propagation in an enclosed volume, Acta Astronaut. 6 (1979) 631;G.R. Rehm, H.R. Baum, The equations of motion for thermally driven flows, J. Res. Natl. Bur. Standards 83 (3) (1978) 297;P.A. McMurtry, W.H. Jou, J.J. Riley, R.W. Metcalfe, Direct numerical simulations of a reacting mixing layer with chemical heat release,AIAA J. 24 (1986).

[2] A.J. Chorin, Numerical solution of the Navier–Stokes equations, Math. of Comput. 23 (1968) 745.[3] Y.H. Qian, D. d’Humieres, P. Lallemand, Lattice BGK models for Navier–Stokes equation, Europhys. Lett. 17 (6) (1992) 479;

S. Chen, Z. Wang, X. Shan, G.D. Doolen, Lattice BGK models for Navier–Stokes equation, J. Stat. Phys. 68 (1992) 379;Q. Zou, S. Hou, S. Chen, G. Doolen, An improved incompressible lattice Boltzmann model for time-independent flows, J. Stat. Phys. 81(1995) 35;X. He, L.-S. Luo, Lattice Boltzmann model for the incompressible Navier–Stokes equation, J. Stat. Phys. 88 (1997) 927;X. He, G. Doolen, Lattice Boltzmann method on curvilinear coordinate system; Vortex shedding behind a circular cylinder, Phys. Rev. E 56(1) (1997) 434.

[4] S. Chen, G.D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech. 30 (1998) 329.[5] O. Filippova, D. Haenel, Grid refinement for lattice-BGK models, J. Comput. Phys. 147 (1998) 219.[6] S. Succi, G. Bella, F. Papetti, Lattice kinetic theory for numerical combustion, J. Sci. Comput. 12 (4) (1997) 395.[7] O. Filippova, D. Haenel, Lattice-BGK model for low Mach number combustion, Int. J. Mod. Phys. C 9 (8) (1998).[8] E.S. Oran, J.P. Boris, Numerical simulation of reactive flow, Elsevier Science Publ. Co. (1987);

A.G. Tomboulides, J.C.Y. Lee, S.A. Orszag, Numerical simulation of low Mach number reactive flows, J. Sci. Comput. 12 (2) (1997) 139;A.G. Tomboulides, S.A. Orszag, A quasi-two-dimensional benchmark problem for low Mach number compressible codes, J. Comput. Phys.146 (1998) 691;M.D. Smooke, R.E. Mitchell, D.E. Keues, Numerical solution of two-dimensional axisymmetric laminar diffusion flames, Combust. Sci.and Techn. 67 (1989) 85.

a novel numerical scheme for reactive flows at low mach numbers

Documents