Math 690N - Final Report

Transcription

1 Math 690N - Final Report Yuanhong Li May 05, 008 Accurate tracking of a discontinuous, thin and evolving turbulent flame front has been a challenging subject in modelling a premixed turbulent combustion. Level set method can be applied to improve the premixed turbulent combustion simulation by providing a better capturing of this highly trasient flame front []. This report presents the application of the level set method derived from the G-equation [] to simulate a flame propagation in premixed internal combustion engine combustions. Problem description In a premixed engine combustion, it is well accepted that a corrugated flame front divides the flow field into a burnt region and an unburnt region []. In the level set method, this flame front corresponds to an iso-scalar surface or zero-level set surface as G(x, t) = G 0. () As shown in Figure, G > G 0 is the region of burnt gas and G < G 0 is that of the unburnt mixture. The choice of G 0 is arbitary but fixed for a particular combustion event. The level set model will be derived from the above equation.

2 Algorithm description The well-known level-set model (G-equation) was initially introduced by Markstein [] in 964. In 985, Williams [3] introduced the method to describe flame front evolution. Later, Peters [] made significant contributions for modelling turbulent combustion using this approach. In this section, a transport equation for an instantaneous flame location G is first derived from equation (), followed by the derivation of the transport equation for the mean G.. Equation of Motion Differentiating equation () with respect to t results in a transport equation for the scalar G as G t + G d x f = 0, () dt where d x f is the propagation velocity and equals the sum of the flow velocity v dt f at the front and the burning velocity in the normal direction d x f = v f + ns L, (3) dt and the vector normal to the front in the direction of the unburnt gas n is defined by n = G G. (4) Substitution of equation (3) and (4) into equation () gives equation of motion as G t + v f G = s L G. (5). Reformulation The burning velocity s L is usually not well defined since it can be complicated by many factors such as flame stretch and curvature. By taking into account the effects of flame stretch and curvature, the motion equation is reformulated as G t + v f G = s T G Dκ G, (6) where s T is the turbulent turning velocity, D is Markstein diffusivity, and κ is the flame curvature defined by κ = n = ( G ). (7) G.3 Level set description Mean quantities are usually solved in CFD applications, i.e., mean pressure, mean velocities and mean temperature are solved in the averaged Navier-Stokes governing equations. To be consistent with CFD convention, a mean G is sought as well. According to Peters [], the equation for the mean location of turbulent flame front can be written as G t + ( v f v v ) G = ρ u ρ G D t κ G, (8) where v v is the velocity of the moving vertex, ρ u is the unburnt gas density and ρ is the gas density at the mean location of the turbulent flame which is defined by G( x, t) = 0, and κ is mean flame front curvature.

3 .4 Needed level set techniques The level set equation (8) will not be solved until the fully developed turbulent flame is established as dictated by an ignition kernel length [4]. At this moment, an initial value, G( x, 0), is required, which is computed by using the signed distance from each vertex to the kernel front as follows. When the cell vertex i4 is outside the ignition kernel, G( x, t) = (x(i4) x p (i4)) + (y(i4) y p (i4)) + (z(i4) z p (i4)), (9) and when the cell vertex i4 is inside the ignition kernel, G( x, t) = (x(i4) x p (i4)) + (y(i4) y p (i4)) + (z(i4) z p (i4)), (0) where x(i4),y(i4),and z(i4) are the coordinates of vertex i4, and x p (i4),y p (i4),and z p (i4) are the positions of the ignition marker particles. In addition, reinitialization of the level set function is needed in order to reduce the numerical errors caused by steepening and flattening effects [5]. Also the so-called narrow band concept [6] is employed to reduce computation time where only the vertices near the flame surface is updated and reinitialized. 3 Discretization Solving the level set equation involves the discretizations of a temporal term and several spatial terms. With physics in mind the convection term and diffusion terms must be discretized using different schemes. 3. Discretization schemes Here a second-order ENO [7] is used to discretize the spatial gradients involved in the terms of G and ρū G. Central differences are used to discretize each term of D ρ t κ G. A first-order forward Euler scheme is used for the temporal term. Thus equation (8) can be discretized as follows. G n+ G n t = ( ρ u) n ρ n [max(s 0n t, 0) + + min(s 0n ( D t ) n ( κ) n D 0x + D0y + D0z [max(u n un vertex, 0)A + min(un un vertex, 0)B + max(v n v n vertex, 0)C + min(v n v n vertex, 0)D + max(w n w n vertex, 0)E + min(w n w n vertex, 0)F ], t, 0) ] () 3

4 where A, B, C, D, E, F are the terms resulting from the nd order ENO scheme as A = D x + x l B = D +x + x r C = D y + y f D = D +y + y d E = D z + z b F = D +z + z t min mod(d x x min mod(d +x+x min mod(d y y min mod(d +y+y min mod(d z z min mod(d +z+z, D+x x, D+x x ), () ), (3), D+y y ), (4), D+y y ), (5), D+z z ), (6), D+z z ), (7) where x l, x r, y f, y d, z b, z t are the distances between vertices i and i,i and i +, j and j, j and j +, k and k, k and k +, respectively, the switch function min mod is given by { sign(x) min( x, y ) if xy > 0, min mod(x, y) = 0 otherwise. and the differencing operators in x coordinate direction are defined by D +x x D x x D +x+x D x = G G i,j,k x l, (8) D +x = G i+,j,k G x r, (9) = G i+,j,k x l G ( x l + x r ) + G i,j,k x r ( x l x r + x l x r ), (0) = G i,j,k ( x l + x l ) G i,j,k x l G x l [ x l ( x, () l + x l ) ( x l + x l ) x l ] = G i+,j,k ( x r + x r ) G i+,j,k x r G x r [ x r ( x. () r + x r ) ( x r + x r ) x r ] Other terms such as D y, D+y, D z, D+z, D+y y, D y y, D+y+y defined similarly. The gradient terms +, are differenced as follows, D+z z, D z z, D+z+z can be + = [min(a, 0) + max(b, 0) + min(c, 0) + max(d, 0) + (3) min(e, 0) + max(f, 0) ], = [min(b, 0) + max(a, 0) + min(d, 0) + max(c, 0) + (4) min(f, 0) + max(e, 0) ]. 4

5 3. Re-initialization The re-initialization of the level set function is required to reduce the numerical errors caused by steepening and flattening gradient effects [5]. The condition G = is applied to the level sets for G( x, t) 0. The algorithm proposed by Sussman [8] is used to ensure that this condition is satisfied by the level sets for G( x, t) 0. In the algorithm a new level-set functin G ( x, t) is constructed in such a way that its zero level set is the same as the level set function G 0 ( x, t) at the flame front while it is set to be the signed normal distance to the flame front away from the zero level set. Specifically, G ( x, t) is defined as where S( G 0 ) is the sign function designed as G ( x, t) = S( G 0 )( G ( x, t) ), (5) S( G 0 ) = G 0 ( x, t), (6) G 0 ( x, t) + ε where ε is a small number. And at t = 0, G ( x, 0) = G 0 ( x, 0). 4 Test Problems and expected results A simple sphere growth problem [4] is considered to test the numerical schemes used for the level set method. In this experiment the sphere with an initial radius of 0mm grows radially at a prescribed speed of s grow = 0m/s. And no flow convection is considered. So the level set equation reduces to G t = s grow G. (7) To measure the shape change of the sphere during its grow, an effective radius error parameter,r rms,is defined as (r rtheory ) R rms =, (8) n where n is the total number of piercing points of the sphere s surface with cell edges, r is the calculated radius, and r theory is the theoretical radius of the growing sphere. 4. Computational setup The reduced equation is discretized by the schemes described above. To test the effects of grid size on the scheme prediction accuracy, three different meshes with average cell sizes of 6., 3. and.3mm are used. The initial sphere radius of 0mm specifies both the initial and boundary conditions for the calculations. 4. Expected results Since only sphere growth is considered here, the calculated surface would be a nearly perfect sphere surface at any later time if the schemes were accurate. Figure shows the comparison between the calculated radius and the theoretical radius. Only slight differences are seen in the radius. As the sphere grows, the error increases. But the relative error R rms /r theory is still relatively small, as can be seen in Figure 3. The predicted radius for the different grids are shown in Figure 4. For the test case, the numerical schemes are not sensitive to the grid size. 5

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7 References [] Peters, N., Turbulent Combustion, Cambridge university press, 000. [] Markstein, G., Nonsteady flame propagation, Pergamon Press,Oxford, 964. [3] Williams, F.A., The Mathematics of Combustion, SIAM, Philadelphia, PA, pp.97-3,985. [4] Tan, Z. and Reitz, R.D., An ignition and combustion model based on the level-set method for spark ignition engine multidimensional modeling, Combustion and Flame, 45, pp.-5,006. [5] Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag New York, Inc, 003. [6] Chopp, D.L., Computing minimal-surfaces via level set curvature flow, Journal of Computational Physics, 06, pp.77-9,993. [7] Harten, A., Engquist, B. Osher, S. and Chakravarthy, S., Uniformly high-order accurate essentially nonoscillatory schemes,3, Journal of Computational Physics, 7, pp.3-303,987. [8] Sussman, M., Smereka, and S. Osher, A level set approach for computing solutions to incompressible -phase flow, Journal of Computational Physics, 4, pp.46-59,994. 7