Assessing Model Assumptions for Turbulent Premixed Combustion at High Karlovitz Number

Impact of Ka and Re on Vortex Evolution Through a Premixed FlameChris Bradley Brock Bobbitt Guillaume BlanquartCalifornia Institute of Technology

APS DFD November 24, 2014

Undergrad summer work1

Introduction

Previous research in this field has shown that interaction with turbulence affects a flame.It can also be shown that interaction with a flame affects turbulence.

Theory: eddies of turbulent flow bear resemblance to laminar vortices interacting with a flame.By studying laminar burning of methane plus vortex we can gain insight to the turbulent caseAdvantagesKeeps flow laminarMakes simulations cheap(in computer time)Provides insight to the turbulent case

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IntroductionMotivationVortex flame interaction provides case study for turbulent combustionGoalsSimulate the 2-D interaction between vortices and a methane flameStudy the effect of the vortex-flame interaction on the vortexFind patterns for different sizes and speeds of vortexDetermine physical mechanisms describing evolution of vorticity.Show which terms (diffusion, dilatation, baroclinic torque) are important in the evolution of different vortices.

Depiction of a methane flame interacting with a vortex moving at 50 times its velocity and 4 times its size

With the motivation that the vortex flame interaction provides a case study for turbulent combustion, I had these goals.

I wanted to simulate the 2-D interaction between vortices and a methane flame with the intent of studying the effect of the interaction on the vortex.

We noticed that if we varied the size and speed of the vortex, then the interaction affected it differently, so I wanted to see if there was any pattern to these effects.

From those patterns I wanted to determine what exactly was effecting the evolution of vorticity.

Finally, I wanted to be able to show which terms in the vorticity equation are important in the evolution of different vortices.3

Physical SetupFuel: methaneEquivalence ratio: 0.7Temperature: 298 K Pressure: 1 atmCentral refined Grid

SL - speed of the flame relative to the vortexlF - width of the flamelv - length of the vortexuv - speed of the rotation of the vortex.

The physical setup for our simulations are shown here. We used methane at an equivalence ration of 0.7, a temp of 298 K and a Pressure of 1atm.For the DNS, we were able to use a centrally refined grid to cut down on the cost of each simulation while eliminating boundary effects. As you can see from the image of a wrapped flame, all the important stuff happens in the refined region anyway, so we dont need as many grid points on the edges.

The code I used took these values and gave me the width and speed of the flame. From there, I ran simulation with different uv and lv values, which, as this image shows, correspond to the speed and size of the vortex

IF ASKED we studied how many points we need for the flame and the vortex4

Tabulated ChemistryTransport a progress variable with tabulated source term

Defined as the sum of 4 mass fractions

Lean methane has little effects of differential diffusion*

*J.F. Driscoll, Progress in Energy and Combustion Science 34 (2008)

We modeled the chemistry using tabulated chemistry where we transport a single progress variable; through which we can relate relevant quantities.5

Vorticity Equation

Goal: analyze simulations to find which terms are important in which regionsRequired many simulations to see patterns.

Dilatation

Baroclinic Torque

Viscous diffusion

Heres the vorticity equation for 2D velocity, notice it has three distinct, recognizable terms:Dilatation (reduces viscosity through density change)Baroclinic torqueViscous diffusion

Remember one of our goals was to see which of these terms most affected the evolution of the vorticity of the vortex for vortices of different sizes and speeds,So we ran many simulations varying those values.6

Parameter Space

Simulations I ran

Previous Work

Explain the axes

Looked for patterns in their shape during and after the vortex-flame interaction depending on location in grid.Figures: Plots of the grid developed running NGA. Each point represents a different size/speed vortex.

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Small VortexViscous term dominates in this regionWe only see vortices surviving with reduced viscosity

Vortex flame interaction for:lv /lF = 0.6, uv/SL = 0.1 (normal viscosity)Data for:lv /lF = 0.2, uv/SL = 1

For vortices that are smaller than the width of the flame, We noticed that the vortex decayed before it could even reach the flame.

We plotted the vorticity as a function of time [1/s] for a case with normal viscosity and one where we forced it to be reduced by a factor of 100.As you can see, the normal case [in blue] decays almost immediately (as shown in the video), while in the reduced case, we see the vortex survive much longer.From this, we can conclude that viscous diffusion dominates the other terms for small vortices.

Explain video8

Large-Slow VortexLarge Slow regionIn case shown: lv / lF = 10 uv / SL = 0.1Baroclinic torque dominates hereThis results in vortex decay into streaks of vorticity

Baroclinic torque Vortex flame interaction in Large slow region.

For large vortices viscous diffusion didnt have much of an affect on the vortex.For large, slow vortices however, we can see that baroclinic torque has a huge affect.

As you can see in the image where we plot positive BT as red and negative BT as blue, there are huge values of BT in the plane of the flame during interaction with the vortex. Notice that if we were to integrate the BT over space at this time, it would be about zero, though it still has an large positive and negetive values.

From the video, you can see the vortex is destroyed into distinct streaks of vorticity that match up with the plot. From this we see that BT results in the destruction of the vortex.9

Large-Fast Vortex

Vorticity [z component] lv / lF = 10 uv / SL = 50

Now onto the final region, that of a large fast vortex. Again, viscous diffusion doesnt have much of an effect.

Notice here that, though the vortex is initially broken up by the interaction with the flame, it appears to come back together. This new vortex is larger, but weaker, which is what you would expect if dilatation were the dominant term in this region.10

Large-Fast VortexBaroclinic torque is large, but cancels itself outDialitation dominates

Baroclinic Torque Dilatation Dominates

What about Baroclinic tourque?

We can see from the plot of BT shown on the left that there are large BT values, however in every case of large positive BT we see, there is a negative value right across from it. When we integrate the three terms over space and plot them vs time, we can see that this effect causes the BT to basically cancel itself out.

Thus, we conclude Dilatation dominates for large-fast vortices.

Remember this for next slide11

Results of integrating in time and space Viscous Diffusion

Baroclinic TorqueDilatation

To confirm these observations, we integrate the three terms over time and space for every simulation we ran. In these plots, the size of the circle corresponds value of the term. In other words, the large the circle, the more important the term.

Integrating in this way doesnt tell the whole story. When we look at the large, slow region, BT again averages to nearly zero, but in this case it is not produced right next to each other to cancel itself out spatially as it does in the large fast.

Analyzing these plots corroborate our conclusions. Dilatation dominates for large-fast vortices, Baroclinic Torque dominates for large-slow ones, and Viscous diffusion dominates for vortices under a certain size.

How does this relate to our equation for vorticity?12

Normalized Vorticity equationWe normalize pressure separately based on the magnitude of either the vortex or the flameHigh Velocity

Low Velocity

From simulations:Viscous diffusion dominates for small vorticesBaroclinic torque dominates for large and slow vorticesDilatation dominates for large-fast vortices

Well, first we have to normalize it, and we do that with respect to the vortex and the flame, each by themselves before they interact.

For the sake of time, Ill explain how we normalized through dilatation. The variable coefficients for dialitation are vorticity and density. We know the only vorticity is due to the vortex, so we normalize with that quantity. The only density change is due to the flame, so we normalize with that quantity. However, there are two pressure gradients. For High velocity vortices, the vortex pressure is what matters, and for low velocity, the flame pressure is more important.

So these are our equations, one for high velocity vortices, and one for low velocity vortices. Each of our original three terms has a coefficient which prescribes the magnitude of that term.

Analize by Ka and ReStart at High velocity: High Ka = viscous diffusion, High Re = low viscous diffusion, high BT(but it cancels out)Low Velocity: Low Re = High viscous diffusion, Low Ka = low vd, high BT

The only region with no defining value is upper right, so we assign Dilatation.

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Conclusion

viscositybaroclinic torquedilatation

So here is our final plot. As you can see we have clearly defined our three main regions. We saw from that data where each term dominates (point to each region), and I showed that our vorticity equation confirms these results. END14