Analysis and modelling of the effective reaction rate in a developing mixing layer using OpenFOAM R libraries

Analysis and modelling of the effective reaction rate in a developing mixing layer using OpenFOAM R libraries Karol Wedolowski 1,2, Konrad Bajer 1,2, Kamil Kwiatkowski 1,2 1 Faculty of Physics, University of Warsaw 2 Interdisciplnary Centre for Mathematical and Computational Modelling, University of Warsaw 6th OpenFOAM Workshop 13-16 June 2011 The University of Pennsylvania

Outline Problem Numerical model Results Conclusions

Outline Problem A mixing layer Reactions in a mixing layer Numerical model Results Conclusions

A one-dimensional mixing layer Flow in x direction depending on y One-dimensional diffusion equation for velocity and vorticity Analytical solution Unstable for a wide range of Reynolds numbers ( ) y U(y, t) = U 0 erf e x, 4ν(t0 + t) U 0 ω(y, t) = ( πν(t0 + t) exp y 2 ) e z 4ν(t 0 + t) U x [m/s] 0.6 0.4 0.2 0-0.2 t=0 s t=5 s t=10 s t=15 s t=20 s -ω [1/s] 10 9 8 7 6 5 4 3 t=0 s t=5 s t=10 s t=15 s t=20 s -0.4-0.6-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 2 1 0-0.3-0.2-0.1 0 0.1 0.2 0.3

A two-dimensional mixing layer Characteristic Kelvin-Helmholtz vortices Computational modelling needed but it can be simplified using a stream function approach No vortex stretching effect Unstable, however very important in the beginning of the instability growth Y T 1.2 1 0.8 0.6 t=0 s t=1 s t=5 s t=8 s t=10 s 0.4 0.2 0-0.3-0.2-0.1 0 0.1 0.2 0.3 y [m]

A three-dimensional mixing layer Three-dimensional velocity field evolves even from the two-dimensional initial conditions The vortex stretching concentrates the vorticity in small regions The mixing layer becomes turbulent

Reactions in a mixing layer We consider a simple passive reaction of the second order, A + B C The mass action law is assumed, w = ky A Y B Different values of Damköhler number are considered Da = ky R L 2 /D One can introduce the effective reaction constant as follows dȳc dt = k eff Ȳ A Ȳ B, where bar denotes averaging over the whole domain. Of course dȳc dt = ky A Y B, so k eff /k = Y AY B Ȳ A Ȳ B. QUESTION How the enhanced mixing will influence the effective reaction rate? Will it depend on the Damköhler number?

Reactions in a mixing layer We consider a simple passive reaction of the second order, A + B C The mass action law is assumed, w = ky A Y B Different values of Damköhler number are considered Da = ky R L 2 /D One can introduce the effective reaction constant as follows dȳc dt = k eff Ȳ A Ȳ B, where bar denotes averaging over the whole domain. Of course dȳc dt = ky A Y B, so k eff /k = Y AY B Ȳ A Ȳ B. QUESTION How the enhanced mixing will influence the effective reaction rate? Will it depend on the Damköhler number?

Outline Problem Numerical model Equations Schemes Initial and boundary conditions Results Conclusions

Vorticity equation The Vorticity equation is solved instead the classical Navier-Stokes equation ω t + (u )ω = (ω ) u + ν ω. It is often used in modelling transition phenomena, Rempfer (2003). Pressure no longer need to be considered but we must retrieve the velocity field from the vorticity distribution. Solution: u = ( u) ( u) u = ( u) We get three Poisson equations!

Vorticity equation The Vorticity equation is solved instead the classical Navier-Stokes equation ω t + (u )ω = (ω ) u + ν ω. It is often used in modelling transition phenomena, Rempfer (2003). Pressure no longer need to be considered but we must retrieve the velocity field from the vorticity distribution. Solution: u = ( u) ( u) u = ( u) We get three Poisson equations!

Vorticity equation - 2D When the flow is two-dimensional the useful stream function method could be applied A single Poisson equation! u = ψ y, v = ψ x, ω = ωez ψ = ω For comparison, in the PISO algorithm at least two Poisson equations need to be solved. Another simplification (ω ) u = 0 no vortex stretching effect Vorticity MUST be treated as a scalar field because it points out of the empty patch.

Vorticity equation - 2D When the flow is two-dimensional the useful stream function method could be applied A single Poisson equation! u = ψ y, v = ψ x, ω = ωez ψ = ω For comparison, in the PISO algorithm at least two Poisson equations need to be solved. Another simplification (ω ) u = 0 no vortex stretching effect Vorticity MUST be treated as a scalar field because it points out of the empty patch.

Vorticity equation - 2D When the flow is two-dimensional the useful stream function method could be applied A single Poisson equation! u = ψ y, v = ψ x, ω = ωez ψ = ω For comparison, in the PISO algorithm at least two Poisson equations need to be solved. Another simplification (ω ) u = 0 no vortex stretching effect Vorticity MUST be treated as a scalar field because it points out of the empty patch.

Transport equations Concentration Y i of the reaction species is governed by the advection-reaction-diffusion equations Y A t + (u )Y A = D Y A ky A Y B, Y B t Y C t + (u )Y B = D Y B ky A Y B, + (u )Y C = D Y C + ky A Y B. The most expensive operation is solving the elliptic equation for velocity thus each simulation includes reactions with different values of the rate constant.

Schemes Different interpolation schemes were tested in simulations. The special attention was focused on the advective terms. As a result: bounded variables (Y A, Y B, Y C ) were interpolated on the cell faces using the SUPERBEE scheme for the vorticity linearupwind scheme was used cubic scheme was applied to the dicretisation of the laplacian terms. The temporal discretisation of the vorticity equation appeared to be an important factor effecting the instability evolution. We considered: implicit discretisation of the 2 nd order of accuracy (backward, CrankNicholson) explicit Runge-Kutta schemes ( of the 2 nd and 4 th order)

Initial and boundary conditions Reference fields: erf function profiles of the streamwise velocity and the reactants concentration gaussian distribution of the spanwise vorticity Perturbations: ω z = 15A zu 0 k z 16[cosh(k zy) 0,25cos(k zx)] 2 ω x = A x sin(k x z) exp ( π y 2 δ 2 ), - Stewart s votrex Initial conditions are based on the work of Rogers (1992). Periodicity was assumed in streamwise and spanwise directions. Uniform irrational flow was imposed on the remaining patches. Re = 500 for majority of cases

Initial and boundary conditions

Initial and boundary conditions Substrates were initially seperated (actually, there was a thin region of the mixed reactants). The product concentration was set to 0 in the beginning.

Outline Problem Numerical model Results Kelvin-Helmholtz instability The effective reaction rate Conclusions

Evolution of the mixing layer Figure: Spanwise vorticity for t = 0, 1, and 3 s respectively. Initially a flat vortex sheet is rolled up and forms a single vortex. The vorticity is taken away from the region between neighbouring votrices.

Evolution of the mixing layer Figure: Streamlines starting from the z = δ plane at t = 5 and 7 s respectively.

Evolution of the mixing layer Figure: Vorticity magnitude for t = 5, 6.1, and 7.4 s respectively. When the three-dimensionality becomes significant the roller breaks up...

Evolution of the mixing layer Figure: Vorticity magnitude for t = 9, 10, 11, and 12 s respectively.... and afterwards the turbulence evolves.

Accelerated mixing 2D Y T 1.2 1 0.8 0.6 t=0 s t=1 s t=5 s t=8 s t=10 s 0.4 0.2 0-0.3-0.2-0.1 0 0.1 0.2 0.3 y [m] Figure: Averaged passive scalar profiles at different times. Initially the mixed layer grows rapidly due to forming of the roller. Then the layer maintains its width and only the molecular diffusion plays a role.

Accelerated mixing 3D 1.2 1 0.8 t=0 s t=2.5 s t=5 s t=7.5 s t=10 s 1.2 1 0.8 t=0 s t=2.5 s t=5 s t=7.5 s t=10 s Y A 0.6 Y A 0.6 0.4 0.4 0.2 0.2 0-0.3-0.2-0.1 0 0.1 0.2 0.3 y [m] 0-0.3-0.2-0.1 0 0.1 0.2 0.3 y [m] Figure: Averaged Y A profiles at different times for Da = 1 and 1000 respectively. For slow reactions substrates could be approximately treated as passive scalars. It is not the case when raction is faster.

Effective reaction rate yc 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 Da=1 Da=10 Da=100 Da=1000 0 0 2 4 6 8 10 12 14 16 18 20 t [s] yc 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Da=1 Da=10 Da=100 Da=1000 0 0 2 4 6 8 10 12 t [s] Figure: Evolution of ȲC for different values of the Damköhler number. Fast reactions appear to be very sensitive to the character of the mixing process. Enhanced mixing in the first stage of simulation leads to the development of product layer which separates substrates.

Effective reaction rate 1 0.1 Da=1 Da=10 Da=100 Da=1000 10 1 k eff /k k eff /k 0.1 0.01 0.001 0 2 4 6 8 10 12 14 16 18 20 t [s] 0.01 Da=1 Da=10 Da=100 Da=1000 0.001 0 2 4 6 8 10 12 14 t [s] Figure: Evolution of the effective reaction constant for different values of the Damköhler number. For Da = 100 or 1000 there is a significant drop in effective reaction constant value due to the stagnation in the mixing. This phenomenon does not appear in the regime of small Damköhler numbers.

Outline Problem Numerical model Results Conclusions

Conclusions Vorticity-velocity formulation of the Navier-Stokes equation proved to work well in a shear layer simulations. In two-dimensional flows speedup is possible but not thoroughly needed. In the developing mixing layer evolution three important stages could be distinguished. Kelvin-Helmholtz rollup stagnation turbulence growth Fast reactions are very sensitive to the character of mixing. Formation of the product layer between substrates seems to be an important phenomenon in the problem.

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