Self-similar solutions of the G-equation - analytic description of the flame surface
aa r X i v : . [ phy s i c s . f l u - dyn ] O c t Self-similar solutions of the G-equation - analytic description of the flame surface
I. F. Barna
KFKI Atomic Energy Research Institute of the Hungarian Academy of Sciences,(KFKI-AEKI), H-1525 Budapest, P.O. Box 49, Hungary (Dated: November 10, 2018)The main feature of the flame kinematics can be desribed with the G-equation. We investigatethe solutions of the G-equation with the well-known self-similar Ansatz. The results are discussedand the method how to get self-similar solutions is briefly mentioned.
To understand the dynamics of flame fronts is a general interest for combusion engineering. One way to gain suchknowledge is the investigation of the G-equation model. Since the G-equation was introduced by Markstein in 1964[1] numerous analytical [2] and numerical approaches [3–5] have been used to find the solutions. The study of [2] usesthe method of characteristics for nonlinear first order partial-differential equations(PDA).Dekena [5] uses a computaional fluid dynamics(CFD) code Fire to investigate turbulent flame propagation inGasoline Direct Injection engines. Further references for application of the G-equation is given therein.In the G-equation model the flame itself is treated as a surface (flame front) that separates the burnt from theunburnt gas. A detailed derivation of the model can be found in [2]. The flame front is described here by the relationof G ( r, z, t ) = z − ζ ( r, t ) = 0 . (1)The most general form of the G-equation is the following: ∂ζ ( r, t ) ∂t + u ∂ζ ( r, t ) ∂r − v + S l s (cid:18) ∂ζ ( r, t ) ∂r (cid:19) = 0 . (2)Where u ( r, z, t ) and v ( r, z, t ) are the radial and the axial components of the gas velocity, and S L is the constantmodulus of the laminar burning velocity, respectively.In the following we use a well-known method, which is very popular to investigate nonlinear PDAs, namely searchingfor self-similar solutions. We are looking for solution of (2) in the form of ζ ( r, t ) = t − α f (cid:16) rt β (cid:17) := t − α f ( η ) . (3)The similarity exponents α and β are of primary physical importance since α represents the rate of decay of themagnitude ζ ( x, t ), while β is the rate of spread (or contraction if β < − αt − α − f ( η ) − βt − α − β − f ′ ( η ) r − u ( r, z, t ) t − α − β f ′ ( η ) − v ( r, z, t ) + (4) S l q t − α − β [ f ′ ( η )] = 0where prime denotes differentiation with respect to η. After setting set u = S l = 1 and v = 0, one can see that this is a non-linear ordinary differential equation(ODE) ifand only if α = − β = 1 ( the universality relation ). The corresponding ODE we shall deal with is f ( η ) − ηf ′ ( η ) − f ′ ( η ) + p f ′ ( η ) ] = 0 . (5)The solutions can be easily obtained and read: f ( η ) = c η − c − q c ; f ( η ) = c p η ( η −
2) (6)where c is the real integration constant. The solutions can be seen in Fig 1. The corresponding self-similar solutionsare ζ ( r, t ) = c r − c − q c ; ζ ( r, t ) = c t s(cid:18) r ( r − t ) t (cid:19) . (7)Both solutions can be prooven with direct derivation. The first solution is trivial however the non-trivial second oneis presented on Fig. 2.We may consider more sophysticated flow systems where thew radaial and axial componenst of the gas velocity andthe S L are functions of time and radial position. First let’s consider that the velocities and the constant modulus ofthe laminar burning velocity are real numbers ( u, v, S l ǫ ℜ .) Now the ODE has the following form f ( η ) − ηf ′ ( η ) − uf ′ ( η ) − v + S l p f ′ ( η ) ] = 0 . (8)The solutions became a bit more complicated f ( η ) = c η + uc + v − S l q c ; f ( η ) = c q − u − η + S L − ηu + v. (9)The function f ( η ) is presented in Fig. 3. with the following fixed set of parameters ( u = 2 , v = − , c = 1 , S L = 3) . Note, that the solution has now a compact support with a non-vanishing first derivatives at the border. This meansthat there is no flux-conservation at the boundaries. There are more analytical solutions for special u ( x, t ) , v ( x, t ) and S l ( x, t ) functions. The only remaining task is to give reasonable physical interpretation for u ( x, t ) , v ( x, t ) and S l ( x, t ).Another generaly interesting question is the dispersion relation and the attenuation distances. It can be examinedhow wave equations or other nonlinear evolutionary equations propagate plain waves in time and in space. Insertingthe usual plain wave approximation ζ ( r, t ) = e i ( kr + ωt ) into (2) the dispersion relation and the attenuation distancecan be obtained. These are the followings v p = ωRe ( k ) = ω p | − k | (1 − signum [ − k ]) ; (10) α = 1 Im ( k ) = 1 ω + k + p | − k | (1 − signum [ − k ]) (11)inserting the relation of ω = kc where c is the propagation velocity of the signal the formulas only depend on thewavenumber vector k. Considering c = 1 propagation speed Fig. 4. shows the phase velocity as the function of thewavenumber. The (1 − signum [ − k ]) in the formula is responsible for the compact support of the function. Fig.5. shows the attenuation distance of the various waves. At k = 1 the α ( k ) function is non-analytic.In this short study we just wanted to present that self-similar solution can be easily used to generate analyticsolutions for the G-equation. With more general, and more physical relations for the radial and axial gas flow velocityhopefully more physical solutions can be obtained. Anyhow, any kind of analytical solution of a nonlinear PDA canbe usefull giving a solid basis for testing complex and sophisticated two or three dimensional numerical finite-elementcomputational fluid dynamics codes. [1] G. H. Markstein, Nonsteady flame propagation
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Accoustically Perturbed Bunsen Flames: Modelling, Analytical Investigations and Numerical Simulations , PhD thesis , Eindhoven University of Technology 2007, ISBN 978-90-386-1118-1.[3] H. Pitsch, ” A G-equation formulation for large-eddy simulation of premixed turbulent combustion”
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Progress in Nonlinear Differ-ential Equations and Their Applications, Birkh¨auser Verlag, Basel-Boston-Berlin, 2004, ISBN 3-7643-7071-8.[7] I.F. Barna and R. Kersner ”Heat conduction: a telegraph-type model with self-similar behavior of solutions” J. Phys. A:Math. Theor. , (2010) 375210. FIG. 1: Solutions of Eq. (5), red line presents f ( η ) and the green one shows f ( η ) .FIG. 2: The self-similar solution ζ ( r, t ) , from Eq. (7) in the range r = 0 .. 15, t = -2 .. 7 for c = 1. FIG. 3: The solution f ( η ) , u = 2 , v = − , c = 1 , S L = 3 . FIG. 4: The dispersion relation v p ( k ) for Eq. (2). FIG. 5: The attenuation function α ( kk