JOURNAL OF TUR.BULENC E. http ://jot.iop.org / Averaged dynamics in the phase plane s of a scalar field using DNS datat

Transcription

1 JoT JOURNAL OF TUR.BULENC E http ://jot.iop.org / Averaged dynamics in the phase plane s of a scalar field using DNS datat Jesús Martín, César Dopazo and Luis Valiño Laboratorio de Investigación en Tecnologías de la Combustión (LITEC), CSIC/DGA, Zaragoza, Spai n j jmartin@ p osta.unizar. e s Received 29 November 2000; online 2 February Abstract. Scalar mixing in turbulence is studied in this work from a dynamical point of view. The joint evolution of different pairs of variables is investigated in their 2D phase planes. Using DNS data and taking conditional averages in th e transport equations at each point of a particular phase plane, a dynamical syste m is obtained. The result indicates the most probable evolution of a pair of variables, given the initial condition. This method, recently introduced for velocity gradient studies, is applied here to investigate the diffusion and dissipation processes o f an inert scalar field in isotropic turbulence. Some results describing new insight s into the effects of vorticity and strain on the local scalar variance and on the loca l scalar dissipation rate, using this technique, are also presented. PACS numbers: 47.27Gs, Qb, Content s 1 Introduction 2 2 Mathematical formulation. Averaged dynamical systems 2 3 Numerical experiments and results 3 4 Phase planes for scalar field variables : scalar diffusion and scalar dissipation rate _ 3 5 Vorticity and strain effects on mixing 5 6 Conclusions 7 t This article was chosen from selected Proceedings of the Eighth European Turbulence Conference (Advances in Turbulence VIII (Barcelona, June 2000) (Barcelona : CIMNE) ed C Dopazo. ISBN : ) IOP Publishing Ltd PI 1: (01) /01/ $

2 Averaged dynamics in the phase planes of a scalar field using DNS dat a 1. Introductio n Scalar mixing in a turbulent flow is characterized by a wide range of time and length scales, over which a variety of physical mechanisms take place [1]. The scalar evolution is driven by th e velocity field and involves convection, random straining and rotation and molecular transport. Much effort has been invested in recent years to describe and understand the scala r phenomena in turbulent flows using data from direct numerical simulation (DNS) [2]-[5]. Such research has been generally focused on the scaling of the scalar intermittency, on the scala r gradient and the scalar dissipation rate, on the study of statistical correlations between scalar and velocity fields and, also, on the description of the effect of the velocity field coherent structuresof strain and vorticity upon the scalar gradient spatial distributions. Joint statistics of scala r fluctuations and magnitudes such as the scalar diffusion or scalar dissipation have been widel y investigated. From this work, results on conditional diffusion, conditional dissipation, join t probability density functions (pdfs) or scatter plots describing turbulent scalar mixing hav e often been reported in the literature. The present work constitutes a further step in the study of a scalar in a turbulence field fro m a dynamical point of view. The objective here is to investigate the joint evolution of pairs of variables, pertaining either to the scalar or to the velocity field, in their associated phase spaces. The phase space for two scalar variables will describe a dynamical correlation of two, somehow connected, scalar mixing processes evolving with time ; if one of the variables is associated with the scalar field while the other is related to the velocity field, the result will display the dynamica l effect of a turbulence property on the scalar mixing process. The phase space dynamics presented in this work are not calculated from instantaneous, real trajectories, but correspond to averaged joint evolutions obtained from conditional averag e DNS data. Hence, the resulting averaged dynamics must be interpreted as the most probabl e patterns or motions in each of the considered phase planes. 2. Mathematical formulation. Averaged dynamical system s Given pairs of variables, related either to the scalar or to the velocity field, the method consists i n conditionally averaging their transport equations to obtain a dynamical system in the associate d two-dimensional (2D) phase space. This idea, introduced by Martín et al [6] for velocity gradient invariants, allows one to describe the joint evolution of any pair of variables in a simple and direc t manner. The technique can also be useful to isolate the effect of a particular term of the transpor t equation on the global dynamics, enabling the understanding of its role in the correspondin g physical mechanism. For a pair of variables X and Y, their transport equations are written a s and dx _ DX DX _ dt at + u3 Dx, Tlx + T2x +.. (1 ) dy _ DY 8Y dt at + u 3 = ny + T2y +.. (2 ) Dx where Tlx, T2x, T1y, etc represent the different terms on the right-hand sides of the equations. If the 2D phase plane XY is considered, conditional time derivatives can be defined at eac h point (X, Y) of that phase space as dx X (X, Y) X,Y) (3 ) dt Journal of Turbulence 2 (2001) 002 ( h ttp ://jot.iop.org/ ) 2

3 and (X,Y) (X,Y. Taking conditional averages in (1) and (2), one obtain s X (X, Y ) = IX, Y) + (T2xI X, Y ) +... Y(X, Y) = (T1Y IX, Y) + (T2y IX, Y) +..., The conditional time derivatives of X and Y, at each point of the plane (X, Y), result simply in the summation of the conditional averages of the terms in the transport equations. The vector (X (X, Y), Y(X, Y)) is the velocity at each point of the phase space. Equations (5) and (6) constitute a dynamical system, since this velocity depends solely o n the coordinates (X, Y). It must be noted at this point that this dynamical system is, for scalar field related variables, time dependent, since their statistics evolves with time. This fact raises the peed of further investigation into how the resulting scalar dynamics changes with time an d also suggests exploring the self-similarity and universal behaviour of these systems. Although this work is restricted to 2D phase spaces, this methodology can be easily extended to the joint evolution of more than two variables. Then averages conditional upon thre e variables can be considered in a 3D phase space XYZ, where X (X, Y, Z) (dx/dtix, Y, Z), Y(X, Y, Z) (dy/dtix, Y, Z) and Z(X, Y, Z) (dz/dt X, Y, Z). 3. Numerical experiments and result s Data fields from statistically isotropic and homogeneous DNS runs have been used for thi s work. The flow and the scalar field are solved in a cube with periodic boundary conditions usin g a pseudo-spectral code. The velocity field is forced and a Reynolds number Rea 47 is reached. This low Rey, has been taken to guarantee proper numerical resolution of the high-order scala r derivatives required for the calculations. The value of the Schmidt number, Sc, is 1.0. The initial scalar field is a ` blob' in the middle of the cube with value C = 1. The rest of the cube has n o scalar (C = 0). The mean is close to 0.5. This corresponds to a scalar pdf close to a double Dirac delta with peaks at zero and one. As time goes on, this `blob ' is stretched by convectio n and smeared out by diffusion. The results presented in this work correspond to a time when th e scalar pdf has relaxed to a nearly uniform distribution, with a remnant of the two peaks nea r the extreme values, zero and one. Each figure in this paper presents both the joint pdf and the averaged `velocity ' for severa l 2D phase planes. Conditional mean trajectories [7] can be calculated from the vector fields. These trajectories, not shown in the present work, represent the most probable evolution of th e variables for prescribed initial values. The phase planes were discretized into square bins, in order to numerically calculate th e required joint pdfs and conditional averages. The size of the bins was chosen to be large enoug h to include a significant number of samples, in order to reach stable statistics, and small enoug h to capture the variations of the averages. Only points corresponding to bins containing mor e than 100 samples are displayed in the results. The isocontours in the joint pdfs are represente d in logarithmic scale. All the numerical values correspond to the DNS units. 4. Phase planes for scalar field variables : scalar diffusion and scalar dissipation rat e The main variables used in this study to describe the scalar field and its evolution along th e mixing process are the scalar fluctuation c = C (C), its square c 2, the scalar diffusion ter m DV 2c and the scalar dissipation rate e, = Dc,ic,i. Journal of Turbulence 2 (2001) 002 ( h ttp ://jot.iop.org/ ) 3

4 Averaged dynamics in the phase planes of a scalar field using DNS dat a "ay, -0' *1 'Ó'' scalar fluctuation Figure 1. Joint pdf and averaged dynamics in the phase plane (c, DV 2 c). and The transport equations of these magnitudes ar e =DV2 c cit+uj, axj 2 2 2c2 le o at + uj áxj DV at -2Dc + uj axc,i c,jui + DV 2eo 2D 2 c,i a( á 2c) + u, a( 2c) = D2V (V c) 2Dui,7c,i7 Dc.v2uj. (10 ) a From these equations, results for different pairs of variables are obtained using the techniqu e described in the previous section. Figure 1 displays the dynamical system resulting for the phas e plane scalar fluctuation diffusion term, (c, DV 2c). A stable focus at the origin is apparent, wit h points moving clockwise spirally until they reach the centre ; it can be observed that point s with large values of diffusion change the sign of the scalar fluctuation before it becomes zero. Figure 2 shows the result for the square scalar fluctuation scalar dissipation rate phase plane, (c2, ea), where points with large initial fluctuations strongly increase a moderately small valu e of the dissipation; the larger the values of the scalar fluctuation and of the dissipation rate, th e larger that increment. In contrast, for small values of c 2 the scalar dissipation is a monotonically decreasing magnitude. The average of -2Dc,i c,jui,j = -2DSijc,ic,j in the transport equation of eo is positive i n turbulence and it is known to be a production term responsible for the amplification of the loca l scalar dissipation rate. This fact is due [3] to the alignment of the scalar gradient vector wit h the principal axis of the strain rate Sil corresponding to its negative eigenvalue ry. Similarly to the stretching rate a- = Sijwiw7/w2, used in the vorticity dynamics by Jiménez et al [8], a `scalar gradient stretching rate' is defined here as Sil e,ic,7 0-oc = ( 11 ) c,kc,k Journal of Turbulence 2 (2001) 002 ( http ://jot.iop.org/ ) 4 (7) ( 8), v7 (9)

5 J Martín et a l o , square scalar fluctuatio n Figure 2. Joint pdf and averaged dynamics in the phase plane (c2, E,). This variable indicates the rate of change (positive or negative) of the local scalar dissipatio n due to the turbulence straining action. Negative values of o -ve yield amplification of E e, while positive ones imply a reduction. An evolution equation for o-v, can be found, obtaining its time rate of change. However, thi s procedure is cumbersome and many of the resulting terms are difficult to evaluate numerically. Alternatively, the derivation rules for the stretching rate definition, equation (11), can be used ; namely, do y _ d Sij c,ic,i (12 ) dt dt c,k c, k Applying the material derivative, d/dt, to the quotients and products the time rate o f change of the scalar stretching is finally expressed a s do _ 1 dsii dcj d(c kc k) dt [c,kc,k (c,kc,k)2 (dt + 2Si*ci + Si,c,ic,i (13 ) dt) dt The value of do- /dt can be then expressed in terms of `basi c' variables as c,i or Sij, which are easier to obtain from DNS data. In general, this strategy is useful to calculate time derivatives of variables having a complicated definition. Figure 3 shows the dynamics in the plane (ave, Ec ). The results indicate that under almost any circumstance the magnitude of o-v, increases, this growth being larger for negative values than for positive ones. The scalar dissipation increases in a slow but systematic way for all point s in the semiplane to the left of the vertical line Uv, -6.0 (this value is approximately thre e times the average value of uvc) ; for points to the right of that line the dissipation decreases, thi s decay being slower as the dissipation becomes smaller. 5. Vorticity and strain effects on mixing The effects of vorticity and strain on scalar mixing were studied by considering the phase plane s (Q, c 2) and (Q, c c). The second invariant of the velocity gradient tensor, Q = Qw + Qs, provides a quantitative estimation of the local balance between rotation (Qw = w 2/4) and strai n (Qs = -Sij Sij /2). Figure 4 for the plane (Q, c2) indicates that the square scalar fluctuation s Journal of Turbulence 2 (2001) 002 ( http ://jot.iop.org/ ) 5

6 Averaged dynamics in the phase planes of a scalar field using DNS dat a scalar gradient stretching rat e Figure 3. Joint pdf and averaged dynamics in the phase plane (vpc, ec). Q Figure 4. Joint pdf and averaged dynamics in the phase plane (Q, c 2 ). decrease for all values of Q, as it should ; however, this process seems to be more intense in th e left semiplane (Q < 0) corresponding to high strain and low vorticity regions in the flow. This behaviour is better explained through the results in figure 5, where s is found to increase strongl y for large strain values (Q < O), while it is progressively reduced as the strain diminishes and th e vorticity dominates the balance (Q > 0). Both processes, the increase of the scalar dissipation in the left semiplane and the reduction in the right, are enhanced for large values of e c. The correlation between the scalar gradient stretching rate and the local strain rate intensity clearly appears in figure 6, depicting the plane (v c, Qs). The joint pdf displays similar trend s to the results reported by Jiménez et al [8] for the joint pdf of the strain and the vorticit y stretching rate, where high values of the former are correlated with positive values of the latter ; here, however, the isocontours are extended towards larger values of the strain for negative value s of o-e. This fact indicates that high strain is associated with large positive values of -2DSzic tic?, Journal of Turbulence 2 (2001) 002 ( 6

7 J Martín et a l Q Figure 5. Joint pdf and averaged dynamics in the phase plane (Q, ea) l0' ' 1 0 scalar gradient stretching rate Figure 6. Joint pdf and averaged dynamics in the phase plane (Uva, Qs). which implies increasing the scalar dissipation. Therefore, the scalar dissipation is enhanced in strain dominated regions of the small-scale structures of turbulence, similarly to the enstrophy (squared vorticity). The averaged dynamics of v e and Qs in figure 6 also shows that nega,tive scalar stretching rates present higher time amplification with increasing strain, while positiv e ones do not seem to be affected by the magnitude of the strain. 6. Conclusions The main conclusions that can be drawn from this work are as follows. A method, previously introduced for the study of velocity gradients, has been applied here t o investigate scalar mixing in turbulence. Using DNS data, the averaged pairwise dynamic s Journal of Turbulence 2 (2001) 002 ( ) 7

8 Averaged dynamics in the phase planes of a scalar field using DNS dat a of scalar (or velocity) related variables is obtained from their transport equations usin g conditional averages. This technique can easily be extended to explore the joint dynamics of three or more variables. The method allows the analysis of the effect of different mechanisms on the scalar evolution. The influence of the local topology motion during the mixing process can be explicitl y described and visualized. The averaged dynamical systems obtained here are time dependent due to the nonstationary nature of the scalar field. Further work is needed in order to analyse th e implications of this time evolution. References Dopazo C 1994 Recent developments in pdf methods Turbulent Reacting Flows ed P A Libby and F A Williams (New York: Academic) ch 7, p 37 5 Kerr R M 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropi c numerical turbulence J. Fluid Mech Arhurst W T, Kerstein A R, Kerr R M and Gibson C H 1987 Alignment of vorticity and scalar gradient wit h strain rate in simulated Navier Stokes turbulence Phys. Fluids A Ruestch G R and Maxey M R 1991 Small-scale features of vorticity and passive scalar fields in homogeneou s isotropic turbulence Phys. Fluids A Pumir A 1994 A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient Phys. Fluids Martín J, Ooi A, Chong M S and Soria J 1998 Dynamics of the velocity gradient tensor invariants in isotropi c turbulence Phys. Fluids Ooi A, Martín J, Soria J and Chong M S 1999 A study of the evolution and characteristics of the invariant s of the velocity gradient tensor in isotropic turbulence J. Fluid Mech Jiménez J, Wray A, Saffman P and Rogallo R 1993 The structure of intense vorticity in homogeneou s turbulence J. Fluid Mech Journal of Turbulence 2 (2001) 002 ( 8