Variable Density Fluid Mixing.

Scott Reckinger

University of Colorado, Boulder

scott.reckinger@colorado.edu

Abstract. Variable density flows are investigated in this project. Particular

interest is focused on the mixing processes between two fluids of different

molecular weights. A recent study of buoyancy driven turbulent mixing for

variable density flows discusses an asymmetry in the mixing as the heavy fluid

mixes at a slower rate than that of the light fluid. In this study, the mass

fraction scalar field is initialized as random blobs of the two fluids. This results

in a nearly-double-delta probability density function field initially. The fluid is

given an incompressible velocity field. The only density variation is then due to

the variable density natural of the problem (two fluids of different molecular

weights). The fields are then subjected to the fully-compressible Navier-stokes

equations. The mixing occurs, and at large density differences (high Atwood

numbers), the PDF of the scalar field does in fact skew towards the heavy fluid.

However, there are two drawbacks to this study. One is that the mass fraction

field is not conserved and a change in the mean value may effect the symmetry

of the PDF. The second drawback is that the cases were run in regimes that

most likely will not develop into turbulence (Re=10).

1 Introduction

Turbulence is evident everywhere. Almost all fluid systems of practical relevance

have some turbulence component. Of these, many systems involve several fluids with

different molar weights (or densities) and turbulence assisted molecular mixing of

initially segregated material regions. These types of flow, in which density variations

can arise (in addition to other effects) due to the different molar weights of the fluids

participating in the mixing, are called variable density (VD) flows. VD flows can

involve incompressible or compressible fluids. A detailed understanding of the

mixing processes that occur in VD turbulent fluid systems has important

consequences to many scientific and engineering fields.

One important example of systems with VD turbulent mixing is Rayleigh-Taylor

instability, which is a buoyantly driven instability. As the light fluid penetrates into

the heavy fluid (and vice-versa), a mixing layer develops while the rest of the domain

remains quiescent. Rayleigh-Taylor instability occurs in many systems of interest,

such as inertial confinement fusion, atmosphere and oceans, or supernovae

explosions. At larger Reynolds numbers, the range of relevant temporal and spatial

scales increases significantly. Recent results indicate that current estimates may even

be too optimistic, as intermittency further increases the range of relevant scales,

2 Scott Reckinger

especially at high Reynolds numbers. Direct Numerical Simulations (DNS) requires

that all relevant scales of motion are accurately solved. DNS for very high Reynolds

number turbulent mixing may be impractical on fixed meshes, which could require

the use of adaptive mesh refinement (AMR) techniques. This is particularly attractive

in general, as intermittency is highly localized and most turbulent flows develop

coherent structures with decreased range of scales, and especially for flows like

Rayleigh-Taylor instability which contain large quiescent regions.

Turbulence has been studied for centuries. Variable density mixing is a topic

related to turbulence that yet has much to be discovered. In 1992, a group at Los

Alamos National Laboratory performed simulations for a multiple material system, in

which the mixing of two miscible fluids having different microscopic densities was

studied [1]. This was extended in a recent paper submitted to the Journal of Fluid

Mechanics, which discusses variable density mixing under buoyantly driven turbulent

conditions [2]. The variable density and diffusivity effects are the main focuses in this

study. As both effects are increased independently, the mixing becomes asymmetric.

That is, the pure light fluid disappears much faster than the pure heavy fluid in the

cases of high density differences and low mass diffusivity, via high Atwood numbers

and high Schmidt numbers, respectively. This study assumed incompressible fluids

with density variations only due to mixing.

In this paper, some of the results from the buoyantly driven turbulent mixing case

are investigated. A few modifications are made to make the problem well-suited for a

course project.

2 Problem Description

The problem of interest is variable density homogeneous turbulent mixing. The

problem is assumed triply-periodic with no gravity. Therefore, there is no buoyant

driving force. Instead, an initial random velocity field is applied. The results will then

shed light on the true cause for the asymmetric mixing observed in the buoyantly

driven turbulent mixing case. An asymmetry without any forcing would support the

notion that inertial effects are causing the irregularity, not gravitational effects.

The governing equations are the compressible Navier-Stokes equations with an

additional scalar equation representing the heavy fluid mass fraction.

Variable Density Fluid Mixing 3

∂ρ + ∂ (ρui )= 0

∂t ∂xi

( )∂ (ρui ) ∂ ∂P 1 ∂τ ij

∂x ∂xi Re ∂x j

∂t

+ ρuiu j = − +

j

( ) ( ) ( )∂(ρE) ∂ ∂ 1 ∂

∂x ∂x Re ∂x j

∂t

+ ρEu j = − ujP + ukτ jk

j j

1 ∂ ⎝⎜⎛⎜κ ∂T ⎞⎟⎠⎟ 1 ∂ ⎜⎛⎜⎝ Ns ρhl D ∂Yl ⎟⎟⎞⎠

Re Pr ∂x j ∂x j Re Sc ∂x j l =1 ∂x j

∑+ +

( )∂ (ρYl ) ∂ 1 ∂ ⎛⎜ ρD ∂Yl ⎟⎞

∂x Re Sc ∂x j ⎜⎝ ∂x j ⎟⎠

∂t

+ ρYl u j =

j

For the purposes of this course project, compressibility effects are negligible.

However, due to the variable density nature of the problem, the full Navier-Stokes

equations are solved. The nondimensionalization was completed with L as a length

scale (domain size) and a2=P0/ρ0 as a velocity scale, where P0 and ρ0 are the values

of pressure and density, respectively, at the interface between the two fluids (taken to

be an average of the values in the pure fluid). D is the mass diffusivity coefficient.

Energy is calculated as follows:

∑E = u ju j + Ns − P,

2 ρ

hl Yl

l =1

with the following equations of state:

P = ρRT

hl = c plT

with,

∑R = ℜ Ns Yl .

l =1 Wl

Here, W corresponds to the molecular weight for one of the fluids. For the course

project, the number of species is limited to two. Since the sum of the Y values must

be unity, all equations involving Y are made simpler. For example, the gas constant

simplifies to

4 Scott Reckinger

R = ℜ⎢⎡Y ⎜⎜⎛⎝ 1 − 1 ⎟⎟⎞⎠ + 1 ⎤

⎣ W1 W2 W2 ⎥

⎦

where Y is the mass fraction for species 1. It is assumed that species 1 is the heavy

fluid. The gas constant formula is similar to the formulas for conductivity, viscosity,

and specific heat.

κ = Y (κ1 − κ 2 ) + κ 2

μ = Y (μ1 − μ2 ) + μ2

( )c p = Y c p1 − c p 2 + c p 2

These equations assume a linear relationship as the fluids mix. This assumption may

not be the most valid, but it is easiest to work with. For the course project, a further

assumption is made that the conductivity, viscosity, specific heats, and mass

diffusivity are equal for the two fluids. Therefore, all of those properties are constant.

The fluids are also assumed to be Newtonian so that the stress τ can be easily

calculated.

It was found that the most important parameters affecting the asymmetric mixing

was the Atwood number (a density ratio) and the Schmidt number.

At = W1 − W2

W1 + W2

Sc = υ

D

It was also hypothesized that the Mach number could affect the mixing process as

well.

M= KE .

a0

The focus for this project is on the variable density effects. Therefore, only the

Atwood number will be investigated.

There are a few options for the initialization of the system. For the buoyantly

driven case, the density field was initialized with a near-double-delta probability

density function (PDF). However, they used transport equations instead of the scalar

mass fraction. Since the mass fraction gives a good measure of the total mass

encompassed by a pure fluid, it was chosen to initialize the scalar field with a near-

double-delta PDF. The result is that the scalar field initially contains random blobs of

pure fluid. This was achieved by taking Gaussian random numbers with a top-hat

spectrum between wave numbers 3 and 5, corresponding to large scale structures, and

Variable Density Fluid Mixing 5

creating a true double-delta PDF field by setting all negative values to 0 and all

positive values to 1. The field is then slightly diffused so that the interfaces are well-

resolved. Since DNS is the method of choice for this project, it is important to ensure

that all scales are fully resolved. Therefore, the following filter was applied, which is

the exact solution to the diffusion equation in Fourier space:

Yˆdiffused = Yˆsharp exp⎝⎜⎛ − α kr 2 ⎞⎠⎟

The α value was chosen just large enough so that the interface can be fully resolved.

An example of the initial mass fraction field is shown in Fig. 1. The thin diffusion

layers are visible in the figure.

At this point, many other fields could be initialized as well. For example, to match

the initialization of the buoyantly driven case, density would need to be initialized.

This is done by simply rescaling the near-double-delta PDF field. Suppose f is the

near-double-delta PDF field with peaks at 0 and 1. The pure fluid densities are

defined as

ρ1 = W1 = 1 + At

ρ2 = W2 = 1 − At

which comes from the incompressible limit relation

1 = Y ⎡1 − 1⎤ 1 ,

ρ ⎣⎢W1 ⎥ +

W2 ⎦ W2

so, when Y=0,

ρ = W2 ,

and when Y=1,

ρ = W1 .

Therefore, ρ can be initialized as a near-double-delta PDF field by simply rescaling:

ρ = (W1 − W2 ) f + W2 .

6 Scott Reckinger

Fig. 1. Initial mass fraction (scalar) field. A near-double-delta PDF field with thin diffusion layers

Similarly, ρY is initialized by

ρY = W1 f

and molar fraction,

Yα Wα Y

Yj = W1

∑X α = ⇒ X ⎡ 1 1 ⎤ 1 ,

j Wj ⎣⎢ W1 W2 ⎥⎦ W2

Y − +

Variable Density Fluid Mixing 7

is initialized by

X=f

The velocity fields are initialized with a Gaussian random distribution with the

following spectrum:

E(k ) = k4 exp⎜⎜⎝⎛ − 2 k2 ⎠⎞⎟⎟

k 5 k 2

0 0

The initial dilatational velocity is assumed zero to follow the incompressible

assumption. This random initial velocity field will play a role in the mixing process.

The start conditions for the thermodynamic variables correspond to uniform

temperature and mean density equal to one. For the mass-fraction-symmetric-PDF

initialization, the density fluctuations are initialized by solving the following equation,

which was derived by substituting the ideal gas equation of state p=ρRT into the

pressure Poisson equation:

( )∂ ∂2 uiu j

∂xi∂x j

∂xi

⎡1 ∂(ρR)⎤ = − 1 ,

⎢ ⎥ T

⎣ ρ ∂xi ⎦

where the gas constant for the mixture is a function of Y as described above. The

solution to this modified Poisson equation is not trivial. For this project, it was solved

using Fast Fourier Transforms. The equation is a little easier to deal with in Fourier

space. For the other initializations, the same equation is solved, but typically for other

variables than density. In the ρY-symmetric initialization, a nonlinear solver is

necessary.

3 Numerical Method

The original intention was to use the Adaptive Wavelet Collocation Method (AWCM)

for the DNS of the variable density mixing. The AWCM is an AMR technique that

utilizes the localized nature of wavelet to efficiently develop an adaptive grid over

which the solutions are solved while maintaining controllable accuracy [3,4].

However, due to the difficulties encountered in setting this problem up, the AWCM

was run in the non-adaptive limit. Therefore, the numerical method used was a

central-based finite difference in space with a krylov time integration scheme. The

method is 4th order in space and 2nd order with time. The simulations were performed

for various Atwood numbers. The domain size was limited to 643, again due to

difficulties in setting up the case. At this resolution, the maximum allowable

8 Scott Reckinger

Reynolds number to ensure that all scales are resolved is only about 10. This

restriction is to ensure that the kolmogrov microscale is resolved. At a Re of only 10,

it is likely that turbulence may not develop.

4 Results

Runs for the Y-symmetric (near-double-delta) were completed for various Atwood

numbers at a Mach number of 0.1 using the AWCM in the nonadaptive limit on a

fixed mesh of size 643. The low Mach number keeps compressibility issues minimal.

The resulting PDFs of the scalar field are shown in Fig. 2, Fig. 3, Fig. 4, and Fig. 5. The

plots show that, for high Atwood numbers, the species scalar PDF becomes skewed

towards the heavy fluid as the system develops and the two fluids mix. In the 4

figures, the PDF of the scalar field and the PDF of the spatial derivative of the scalar

field are given for varying At. The sharp, two-peaked, black line represents the initial

conditions. Then, as time progresses, the peaks collapse and mold together in the

center as the mixing processes take place. For the case when At=0.001 in Fig. 2, there

is almost no asymmetry present. So, at low density differences, the mixing occurs

evenly (at equal rates) for both fluids. There is no tendency for one fluid to remain

pure longer than the other fluid. As At increases, the asymmetry becomes ever more

evident. The red and green lines in the At=0.5, At=0.6, and At=0.8 cases really show

the asymmetry the best. The derivative plots show an initial state with a relatively

wide range of values, yet with the majority of the points clumped near zero. This

represents the pure fluid state. The tails of this initial PDF of the spatial derivative

gives information about the initial mixing layer. Since the tails of the derivative PDFs

decay, the large gradients have properly been diffused by the filter introduced above.

5 Discussion and Conclusions

There is one major flaw with the results found in this project. Since the scalar field, Y,

is not conserved by the governing equations, the mean value for Y shifts toward the

heavy fluid as the fluids mix. One could easily make an argument that the asymmetry

is simply due to the fact that the Y=1 initial peak does not have to travel as far on the

PDF as the Y=0 peak. One method to fix this issue is to begin with a different

initialization. The ρY-symmetric initialization would be favorable since ρY is a

conserved variable. Also, ρY is an extremely good indicator of the volume

encompassed by a pure fluid, whereas Y only indicates the total mass encompassed by

a fluid. The runs have been started for the ρY-symmetric initialization, but did not

finish in time for the due date. The molar-fraction-symmetric initialization would also

be a good option. The mole fraction, X, gives information about the total amount of a

fluid in the system. Also, the X-symmetric initialization reduces to the ρY-symmetric

initialization in the incompressible limit. The third option would be to initialize

density, similar to what was done for the buoyancy driven case. An attempt was made

at this initialization, but the results were unphysical (Y less than 0 or greater than 1).

Variable Density Fluid Mixing 9

Qualitatively, the pictures shown from these results closely match those found in the

buoyantly driven turbulent mixing case. That is, the pure heavy fluid in a variable

density setup, mixes more slowly than the pure light fluid. The initial nearly-double-

delta PDF is quickly skewed as the light fluid vanishes rapidly. Only at long times

does the pure heavy fluid disappear resulting in a Guassian distributed mixture of the

two pure fluids. For a Bousinesq fluid, the density PDF remains symmetric

throughout the mixing process. Both fluids, light and heavy, mix at the same rate. If

these results are valid, then it could be tentatively concluded that the asymmetry is

associated with inertial effects and not due to buoyancy. This result is one difference

between the mixing processes in a variable density fluid flow and that of a Boussinesq

fluid. This result has important consequences for all simulations for systems involving

variable density mixing. It appears that the molecular mixing processes act differently

on opposite sides of the mixing layer. Models for the mixing rate between two fluids

of varying density must take this result into consideration.

References

1. Besnard, D., Harlow, F.H., Rauenzahn, R.M., and Zemach, C. Turbulence transport

equations for variable-density turbulence and ther relationship to the two field models.

Technical Reports. 1992. Los Alamos National Laboratory.

2. Livescu, Daniel and Ristorcelli, J.R. Variable density mixing in buoyancy driven

turbulence. submitted to J. Fluid Mech., 2007, LA-UR-07-3399.

3. Vasilyev, Oleg V. Solving multi-dimensional evolution problems with localized structures

using second generation wavelets. Int. J. Comp. Fluid Dyn. special issue on high-

resolution methods in computational fluid dynamics. 17(2), 151-168. 2003.

4. Vasilyev, Oleg V. and Bowman, Christopher. Second-generation wavelet collocation

method for the solution of partial differential equations. Journal of Computational Physics

165 (2000). 660-693.

10 Scott Reckinger

Scalar PDF with At=0.001

9

t/tr=0

8 t/tr=0.40205

t/tr=1.0851

7 t/tr=2.8069

t/tr=5.356

6 t/tr=12.2543

5

PDF

4

3

2

1

0 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Y

x-Derivative of Scalar PDF with At=0.001

101

t/tr=0

t/tr=0.40205

100 t/tr=1.0851

t/tr=2.8069

t/tr=5.356

10-1 t/tr=12.2543

PDF 10-2

10-3

10-4

10-5 2 3

-3 -2 -1 0 1

dY/dx

Fig. 2. Variable Density Mixing Results with At=0.001.

Variable Density Fluid Mixing 11

Scalar PDF with At=0.5

9

t/tr=0

8 t/tr=0.52655

t/tr=0.97852

7 t/tr=2.7151

t/tr=5.3368

6 t/tr=12.969

5

PDF

4

3

2

1

0 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Y

x-Derivative of Scalar PDF with At=0.5

101

t/tr=0

t/tr=0.52655

t/tr=0.97852

100 t/tr=2.7151

t/tr=5.3368

t/tr=12.969

10-1

PDF

10-2

10-3

10-4 2 3

-3 -2 -1 0 1

dY/dx

Fig. 3. Variable Density Mixing Results with At=0.5.

12 Scott Reckinger

Scalar PDF with At=0.6

9

t/tr=0

8 t/tr=0.51456

t/tr=1.1296

7 t/tr=2.7666

t/tr=5.478

6 t/tr=13.6857

5

PDF

4

3

2

1

0 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Y

x-Derivative of Scalar PDF with At=0.6

101

t/tr=0

t/tr=0.51456

100 t/tr=1.1296

t/tr=2.7666

t/tr=5.478

10-1 t/tr=13.6857

PDF 10-2

10-3

10-4

10-5

-3 -2 -1 0 1 2 3

dY/dx

Fig. 4. Variable Density Mixing Results with At=0.6.

Variable Density Fluid Mixing 13

Scalar PDF with At=0.8

9

t/tr=0

t/tr=0.74613

8

t/tr=1.4002

t/tr=3.2429

7

t/tr=8.1468

t/tr=15.6403

6

5

PDF

4

3

2

1

0 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Y

x-Derivative of Scalar PDF with At=0.8

101

t/tr=0

t/tr=0.74613

t/tr=1.4002

100 t/tr=3.2429

t/tr=8.1468

t/tr=15.6403

10-1

PDF

10-2

10-3

10-4

-3 -2 -1 0 1 2 3

dY/dx

Fig. 5. Variable Density Mixing Results with At=0.8.