Advective and Diffusive Turbulent Mixing

Transcription

1 Advective and Diffusive Turulent Mixing J. J. Monaghan School of Mathematical Sciences Monash University, Clayton Melourne, Australia J. B. Kajtar School of Mathematical Sciences Monash University, Clayton Melourne, Australia Astract Recent studies of a turulence model for SPH [], [2] show that, for two dimensional turulence in a square container with no slip walls, it gives results that are in good agreement with those from a high accuracy spectral method. A key feature of the model is that the fluid is advected with a velocity v otained y smoothing the velocity v. The equations of motion are otained from a Lagrangian in which the kinetic energy per unit mass is 2 v v, and the movement of an SPH particle is given y dr/ = v. The linear and angular momentum and a discrete form of the circulation are conserved. The simulation of turulent flow in a square container shows that the velocity correlation function for low resolution is close to that calculated using a resolution twice as fine. The reason for the improvement is that the noise in the velocity field is greatly reduced y the smoothing. However, the smoothing of the velocity field affects the mixing ecause the velocity variations on a short length scale are removed, and there are clear differences in the flow. Some of the difference can e attriuted to the fact that the turulent flow is chaotic and the smoothing makes initial small changes to the advection and these changes are amplified. However, part of the difference is due to the effect of smoothing on the small length scale motion. In this paper we analyse the time variation of the mixing at different length scales and we show the results otained y including a diffusion equation which descries the diffusion of material due to small velocity fluctuations. This equation takes the same form as that given y Monaghan [3] in connection with the freezing of inary solutions. A key feature of the formulation is that the difference etween v and v determines the velocity fluctuation, and this determines the coefficient of diffusion. I. INTRODUCTION When a laminar flow ecomes turulent mixing occurs. In particular the linear and angular momentum are mixed and this changes the stresses in the fluid. Any scalar quantity the fluid carries is also mixed. The mixing occurs on all scales ut it varies with the length scale with the small scales eing less easily mixed than large scales. If turulence is simulated using high resolution, the turulent flow is found to eventually involve thin sheets and rions that decay y viscosity. To simulate these structures correctly it is necessary to resolve down to the Reynolds length l R which is given in 2D y l R = L/R /2 where L is the macroscopic scale and the Reynolds numer is R = V L/ν where V is the large scale velocity and ν the kinematic viscosity coefficient. These thin structures are seen in the direct SPH simulations of 2D turulence [], [2], [4]. Unfortunately, in many simulations, the resolution required to descrie the turulence correctly cannot e afforded, even with large parallel clusters. In the case of finite difference calculations the way of escape is to use a turulence model where the original fluid equations are smoothed in space. The method is called Large Eddy Simulation LES and the grid scale is larger than grid spacing of l R. The use of a coarse grid is partly compensated y including a su-grid stress that must e guessed. It has een shown that a recently proposed turulence model for SPH has a numer of desirale properties []. These include conservation of linear and angular momentum and circulation approximately together with the aility to recover the correlation functions of direct numerical simulations while using a much coarser resolution. The equations of motion are derived in a similar way to that used in a continuum Lagrangian turulence model called the Lagrangian Averaged Navier-Stokes alpha model LANS-α due to Holm and his colleagues [5], [6], and for further references see Lunasin et al. [7]. LANS was originally developed from a lengthy consideration of turulent fluctuations. However, the end result is simple. A smoothed, or regularised, velocity v is calculated y a linear operation on the un-smoothed velocity v, and then the Eulerian equations of motion are determined from a Lagrangian with kinetic energy per unit mass 2v v. The particles are moved according to dr/ = v, and the comination of v and v guarantees that circulation is conserved in the asence of dissipation. The overall result is that, in driven turulence in the asence of viscosity, the energy in small length scales is redistriuted to larger scales. Comined with a viscous term, the equations give a very satisfactory description of turulence in periodic domains [6], gyres relevant to oceanography [8], and mixing [9]. II. GOVERNING EQUATIONS The smoothed velocity v a and the un-smoothed velocity v a for an SPH liquid particle a are related y []: v a = v a + ɛ m M v v K r a r, l. Here m is the mass of particle and M is a mass closely related to the typical mass of a particle and 0 ɛ <. The function K is a smoothing function with typical length scale l. It is similar to a Gaussian. The integral of K over the space of the simulation is equal to l d where d is the numer of dimensions. The reader familiar with SPH will recognise this smoothing as the XSPH variant of SPH [0]. It can e shown that, if the velocities are expanded in a Fourier series, the

2 7 th international SPHERIC workshop Prato, Italy, May 29-3, 202 values of the coefficients for wavenumers greater than /l are reduced y a factor ɛ y the smoothing []. Typical calculations use ɛ = 0.8 in which case the coefficients of high wave numer modes are reduced y a factor 5 every time step. The Lagrangian leading to the Euler equations is the particle equivalent of Eckart s Lagrangian []. Taking into account the smoothing, it has the form L = m 2 v v uρ, s, 2 where ρ is the density of particle, u is the thermal energy/mass and s assumed constant is its entropy. Particle moves with the velocity v. By writing L in terms of v and r for each particle it is straightforward, though lengthy, to show that the Lagrange equations for particle c take the form dv c = m Pc ρ 2 c + ɛ 2 + P ρ 2 c W c m M v c v 2 c K c, 3 where P is the pressure and the equation is the SPH equivalent of the Euler equations []. Taking the limit where the numer of particles goes to infinity the SPH equation ecomes v t + v v = ρ P + ɛ ρ vr vr 2 Kr r, ldr, 4 2M where dr denotes a volume element. The last term on the right hand side is the extra stress that appears in the acceleration equation ecause of the smoothing. It is the equivalent of the Reynolds stress. If the integral involving K is expanded in a Taylor series it can e shown [] that it reduces to the form appearing in the LANS model [5]. It is straightforward to show that this equation conserves energy E in the form E = m 2 v v + uρ, s, 5 and the equations conserve linear and angular momentum when they should. A simple analysis, ased on the invariance of the system to the exchange of neighouring particles around a loop [], [3], shows that a discrete form of the circulation is conserved with an accuracy that depends on the scale of the velocity field relative to the loop over which the circulation is calculated. To model the Navier-Stokes equations an SPH viscous term is added. Because the strength of the smoothing and the new stress terms depend on the parameter ɛ we call this the SPH-ɛ turulence model. This SPH turulence model has een applied to twodimensional turulence which, though different to turulence in three dimensions, has the same features of eing non-linear and disordered []. The results of the SPH turulence model for the decaying energy and enstrophy are in good agreement with those from the experiments and other numerical simulations [], [2], [3]. In this paper we focus on the mixing that occurs in turulent flow. We will explore whether the diffusion of a material associated with different parts of the fluid in a low resolution calculation can reproduce the mixing in a high resolution calculation. We refer to this scalar quantity as the concentration associated with an element of fluid. An analogy would e to consider the concentration as a mass fraction of salt in a ody of fresh water, ut in the present case the concentration does not influence the dynamics. We egin y considering a general case, where there are more than one sustances in the fluid. The concentration of sustance j is denoted C j, where the mass of the sustance in a mass M j of fluid is CM j. The rate of change of the concentration takes a form similar to a heat conduction equation [3] such that dc j = ρ Dj C j, 6 where D j is the diffusion coefficient for sustance j. The SPH form of this equation for the rate of change of concentration of sustance j for particle a, C a j, is given y dc a j = m D a j + D j C a j C j F a, 7 ρ a ρ 2 where r a F a = a W a. This SPH form of the equation ensures that the total mass of the sustances in question is conserved. In this paper we consider only the case where there is one sustance in the fluid, and hence the superscript j can e dropped. The diffusion coefficient for particle a, D a, must have dimensions ML T. We chose a coefficient ased on the differences etween the smoothed and unsmoothed velocities of a so that D a = βρ a ṽh, 8 where β is dimensionless parameter chosen y experimentation, and we typically use β = 5. For these calculations we have used a speed ṽ = v a v a. 9 Other choices are possile, for example, a speed ased on the smoothed differences such as m ṽ = v v ρ 2 W a. 0 III. TIME STEPPING We use the same time stepping as that descried y [] ut we use a different technique to calculate the smoothed velocity field. The direct use of requires an iteration since it depends on the particle coordinates which in turn depend on v. In almost all cases that we have examined the iteration converges quickly 3 or 5 iterations. However in some cases it fails to converge. As an alternative, which is faster, and stale

3 7 th international SPHERIC workshop Prato, Italy, May 29-3, 202 at least for 0 ɛ 0.85 we convert into a differential equation. The time derivative of is d v a = f a + ɛ m M [f ak a + v a v a a K a ], where, f a = dv a / and, for any vector A, the notation A a = A a A has een used. This differential equation is integrated together with the other differential equations. IV. STUDY OF DECAYING TURBULENCE Here we consider a square container of fluid with no-slip walls. The domain is similar to that considered y Monaghan [], in which it was shown that the turulence model for SPH gives results that are good agreement with a high accuracy spectral method. The velocity field is initialised y a 4 4 set of Gaussian vortices. The vortices were initially centred on a grid of squares with separation 0.2, ut then randomly shifted in oth x and y y 0.02η, where η is a uniformly distriuted random numer in the range < η <. A given initial seed value generates a set of random numers such that the simulations are exactly reproducile. The sign of the rotation was positive or negative, and followed a checkeroard pattern. The velocity at position r due to a vortex with centre at R is vr = Ωe z r R, 2 where e z is a unit vector normal to the fluid, and Ω is given y d Ω = 2π r R 2 e r R 2 /d 2, 3 and d = The no-slip condition requires that the velocity field goes to zero at the oundary. This is achieved y smoothing the velocity field calculated from 2 []. The Reynolds numer was R = V L/ν = 000, where the characteristic velocity is V = 0., which is maximum speed at the instant the vortices are placed, and the characteristic length is L = 0.5, or half the wih of the ox. We ran the calculations at two resolutions. For the high resolution case, there were fluid particles in the domain. It has een shown [] that at this resolution, an SPH simulation using a standard algorithm reproduces the results otained y other researchers for a similar prolem. For the low resolution there were fluid particles. The turulence model plus concentration diffusion were used for the low resolution simulation. The particles in a square region initially are coloured light lue, and the outer particles are dark lue. The colouring is used simply to indicate how the advection would mix a contaminant. Fig. shows the fluid in similar states for the two resolutions at a short period of time after the vortices are applied. There is a difference in the elapsed time etween the two ecause the kinetic energy decays more rapidly for the low resolution case. Fig. a is the high resolution case without the turulence model, and Fig. is the low resolution case with the turulence model. Although there are some differences, it can e seen that the low resolution case captures many of the same features. Fig. 2 shows the low resolution case with concentration diffusion. The fluid particles in Fig. 2 and Fig. are in identical positions. The central square region initially had concentration C = dark lue, and the outer region had C = 0 yellow. As the fluid moves, the concentration diffuses and the intermediate colours are generated. The diffused concentration traces the features of the high resolution calculation more accurately than the particles alone in the low resolution calculation. As an example, consider the plume in the top left corner that curls around in an anti-clockwise direction. In Fig. a, the plume is represented y small ut steady stream of light lue particles. For the equivalent structure in Fig., it is represented y significantly few particles. However, in Fig. 2, it can e seen that the same structure is represented more accurately y the light green stream. Ultimately we would like to understand whether a lower resolution calculation with the turulence model and concentration diffusion can reproduce the same mixing properties as a direct numerical simulation. As an attempt investigate this, we divided the domain into 6 6 square cells and computed the mean concentration in each cell. For example, if a cell has ten particles of which nine particles have concentration C = 0 and one particle has C =, then the mean concentration for that cell is C = 0.. Initially, 25% of the cells have mean concentration C = the central square, and the remaining 75% have C = 0. We ran each resolution with four different sets of initial random shifts for the centres of the Gaussian vortices. This amounts to simply choosing four different initial random numer seeds. With the four sets, we then produced an ensemle average of the mean concentration level against the percentage of cells with that concentration. Fig. 3 shows the histogram as a result of this procedure. The curves are given for the high resolution case, and for the low resolution, oth with and without the consideration of the concentration diffusion. The graph shows, for example, that approximately 40% of the cells contain mean concentration 0 C < 0. in each case. If the entire domain were perfectly mixed, then 00% of the cells would contain mean concentration C = 0.25 since that was the initial distriution on the domain scale. Since all three curves are very similar, it is difficult to determine whether diffusion scheme improves the mixing measure for lower resolution. However it is interesting to note that the mixing for the lower resolution cases is as good as the high resolution case. Other measures for mixing will e explored in future, such as the more general measure considered y Roinson et al. [4]. V. CONCLUSION We have presented a scheme of turulence modelling and concentration diffusion that attempts to capture the mixing properties of a turulent flow. The turulence model for SPH is the same as that presented y Monaghan [], ut here the

4 7th international SPHERIC workshop Prato, Italy, May 29-3, 202 a Fig. 2. Concentration map for the particle case, with the turulence model and concentration diffusion. Dark lue corresponds to concentration C = and yellow corresponds to C = Percentage of cells Mean concentration Fig.. Particle positions 5s after the Gaussian vortices are applied. The particles in a central square region initially are coloured light lue, and the outer particles are dark lue. a is with fluid particles and no turulence model, is with particles and the turulence model. smoothed velocity is computed y integrating a differential equation, rather than iterating. As a result, the computation time is reduced y 25%. A concentration diffusion equation was introduced as a way of modelling the small-scale mixing features of the turulent flow. It was shown that the diffusion effectively captures the structures seen in a high resolution calculation despite the significantly smaller numer of particles in those regions with coarser resolution. The level of mixing on the scale we chose was the same for all three cases that were considered: low resolution oth with and without diffusion and high resolution. R EFERENCES [] J.J. Monaghan, A turulence model for Smoothed Particle Hydrodynamics. Eur. J. Mechanics B/Fluids. 30, , 20. Fig. 3. Histogram of mean concentration for a given cell against the percentage of cells in the domain with that concentration after t 30. Blue denotes the high resolution case, green denotes the low resolution case with diffusion, and red denotes the low resolution case without diffusion. [2] A. Valizadeh, J.J. Monaghan, Smoothed particle hydrodynamics simulations of turulence in fixed and rotating oxes in two dimensions with no-slip oundaries. Phys. Fluids, 24, 03507, 202. [3] J.J. Monaghan, Smoothed particle hydrodynamics. Rep. Prog. Phys., 68, , [4] M. Roinson, J.J. Monaghan, Direct numerical simulation of decaying two-dimensional turulence in a no-slip square ox using Smoothed Particle Hydrodynamics. Int. J. Numer. Meth. Eng., DOI: 0.002/fld.2677, 20. [5] D.D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion. Physica, 33, , 999. [6] S. Chen, D.D. Holm, L.G. Margolin, R. Zhang, Direct numerical simulations of the Navier Stokes alpha model. Physica D., 33, 66-83, 999. [7] E. Lunasin, S. Kurien, M.A. Taylor, E.S. Titi, A study of the Navier Stokes model for two dimensional turulence. J. Tur., 830, -2, [8] D.D. Holm, B.T. Nagida, Modeling mesocale turulence in the arotropic domain. J. Phys. Oceangr., 333, [9] B.J. Geurts, D.D. Holm, Leray and LANS-α modelling of turulent mixing. J. Tur., 70, -33, [0] J.J. Monaghan, On the prolem of penetration in particle methods. J. Computat. Phys., 82, -5, 989.

5 7 th international SPHERIC workshop Prato, Italy, May 29-3, 202 [] C. Eckart, Variation principles of hydrodynamics. Phys. Fluids, 3, [2] H.J.H. Clercx, S.R. Maassen, G.J.F. van Heijst, Decaying twodimensional turulence in square containers with no-slip or stress-free oundaries. Phys. of Fluids, 3, 6-626, 999. [3] G.J.F. van Heijst, H.J.H. Clercx, and D. Molenaar, The effects of solid oundaries on confined two-dimensional turulence. J. Fluid Mech., 554, 4-43, [4] M. Roinson, P. Cleary, J.J. Monaghan, Analysis of mixing in a Twin- Cam mixer using Smoothed Particle Hydrodynamics. AIChe Journal, 548, 2008.