8.1 The problem: in physical and MFM terms

(a) Description

Such layers have been much studied, both experimentally (eg by Reichardt, 1942) and theoretically (eg by Tollmien, 1926). It is known that the profile approximately fits the formula:

W' = [ 5 * Y' ** 2 - 2 * Y' ** 5 ] / 3

- where W' is the local velocity divided by the maximum velocity,
- and Y' is the distance from the low-velocity boundary divided by the width of the layer.

The selected MFM is characterized by:-

- a one-dimensional population, with the longitudinal velocity as the population-distinguishing attribute;
- a uniform division of the velocity range into 40 equal intervals;
- the presumption that all the 40 fluids have the same turbulent diffusivity, this being equal to: 0.5 * LEN * MNSQ;
- the presumption that the mixing-rate constant equals: 5.0 * MNSQ / LEN;
- the presumption the local length scale LEN obeys a transport
equation having a source term equal to: 0.015 * MNSQ , and a
turbulent Prandtl number equal to 0.1 .
MNSQ is defined as the local root-mean-square velocity fluctuation.

Its cross-stream width increased from nearly zero to 0.725 m at a distance of 3.12 m from the start.

The flow-direction velocity of the entering stream, designated as fluid 1, was 10 m/s; the reservoir fluid, designated as fluid 40, had a flow-direction velocity of zero.

These fluids entered the wedge-shaped computational domain, through its two inclined boundary surface, at the rates necessary to ensure conservation of momentum, total mass, and the satisfaction of the transport equations of all 40 fluids. They were outcomes of the calculation rather than bondary conditions.

The calculations were performed by the PHOENICS code, working in parabolic mode. The calculation took 642 seconds on a Pentium 200 PC.

(a) Some details of the computations

The last lines of the computer (200 MHz Pentium) output are:

- RUN COMPLETED AT 12:53:57 ON SATURDAY, 01 JUNE 1996
- MACHINE-CLOCK TIME OF RUN = 642 SECONDS.

So one does not have to wait long for results, some of which now follow. [The date may be noteworthy; it is that of the first- ever hydrodynamic-flow calculation based on MFM]

First, some velocity-vector concentration-contour plots will be shown. These will be followed by some computed PDFs.

The above-mentioned results are those of the 1996 calculation. Nowadays such calculations are routine, there being a mixing-layer example in the latest PHOENICS Input-File library.

The cross-stream profiles shown below come from the library case.

The velocity vectors are as expected.

The grid is an expanding one, with 40 intervals across and 100 along the layer.

Here are the velocity vectors

This is the distribution of the computed length scale

and this is

the computed effective-viscosity distribution

Agreement with experiment is good.

The following picture shows contours of the average concentration (for all 40 fluids) of material emanating from the upper stream.

It is just as a conventional turbulence model (eg mixing-length or k-epsilon) would reveal; but no transport equations for statistical quantities are being solved.

average concentration of material from the upper stream

But how much upper-stream fluid (fluid 40) remains in its pure state? Only this.

concentration of upper-stream fluid (fluid 40)

Where has the rest gone?

What about pure lower-stream fluid?

It also occupies very little space.

concentration of lower-stream fluid

So what lies in between? 38 other fluids, variously intermingled.

The contours for fluid 10

The contours for fluid 20

The contours for fluid 30

The results can be processed and displayed in many different ways.

Particularly relevant to the theme of the present paper are the discretized probability-density functions (PDFs), which may also be called:

Fluid-Population Distributions (ie FPDs) .

The PDFs are on the left in the next pictures.

The displays on the right are reminders of how the population of fluids might be distributed in a single computational cell, namely at random.

All the PDFs to be shown relate to a downstream section of the mixing layer, where the profiles are fully developed.

The first relates to a location near the lower (higher-velocity) edge of the layer.

The population consists almost entirely of higher-velocity fluid

higher-velocity edge

At 1/6 of the width from the higher-velocity edge

At 2/6 of the width from the higher-velocity edge

At 3/6 of the width from the higher-velocity edge

At 4/6 of the width from the higher-velocity edge

At 5/6 of the width from the higher-velocity edge

At a location very near the lower-velocity edge

All that has been used so far is the root-mean-square velocity fluctuation, as an input to the formulae for:-

- length-scale growth,
- effective viscosity,
- micro-mixing between fluids (ie coupling and splitting).

As expected, a self-similar distribution is found.

The value along the "spine" of the layer is approxinately 20 % which is in order-of-magnitude agreement with experimental data.

contour plot of MNSQ

- Fig.8.2.1 shows the population-mean velocity profile, while Fig.
8.2.2 shows the corresponding profile of root-mean-square velocity
fluctuations
Fig 8.2.1 The population-mean longitudinal velocity

Fig 8.2.2 The root-mean-square velocity fluctuation

- Figs.8.2.3 and 8.2.4 show the profiles of length scale and effective
kinematic viscosity which correspond to the same conditions.
Fig 8.2.3 The profile of the length scale

Fig 8.2.4 The profile of effective viscosity

- Fig.8.2.5 accordingly displays the profile of the multiplier in the
source term for the fluid mass fractions, T(i,j) (see Appendix).
Fig 8.2.5 The profile of the coupling-splitting rate

- Finally, Figs. 8.2.6 and 8.2.7 show the profile of some of the 40
fluids of which the population is supposed to consist. Fluids 1 and
40 have finite concentrations only near the appropriate boundaries;
and the intermediate fluids have finite concentrations at the
correspondingly intermediate locations.
Fig 8.2.6 Concentration profiles of entering and reservoir fluids

Fig 8.2.7 Concentration profiles of some intermediate fluids

The width is here defined as the distance across the layer between the points where the mean velocity is 0.1 and 0.9 times the maximum velocity. With this definition. the rate of-spread ratio is found to be: 0.153 .

Other numerical results of interest are, for the fully-developed region, the following dimensionles numbers:

- maximum MNSQ / maximum velocity = 0.14
- maximum LEN / layer width = 0.17
- maximum effective viscosity / (maximum velocity * width) = 0.0105
- maximum T(i,j) * width / maximum velocity = 0.927

8.3 Comparison with experimental data

(a) Rate of spread, etc

- the rate-of-spread ratio is approximately equal to 0.16, which is close to the 0.153 of the present calculations;
- the square root of the turbulence energy divided by the maximum velocity has a maximum of 0.18, which accords well enough with the above figure for MNSQ, which contains only the longitudinal-velocity fluctuations;
- the average dimensionless effective viscosity is around 0.0074, which is somewhat smaller than the above maximum value;
- when compared with mixing-length theory predictions, the data imply a mixing-length of about 0.12 times the width, which is of the order of the value for LEN, as might be expected.

Passive-scalar profiles are not exactly the same as longitudinal- velocity profiles in plane mixing layers, but have the less-curved shape which corresponds to an effective Prandtl number of less than 1.0.

Nevertheless, it is gratifying to note that the shapes of the PDFs reported by Batt are in qualitative accordance with those presented above, varying from near-symmetry in the centre of the layer to marked "lopsidedness" near the edges of the layer.

It may also be of interest to note that Fueyo et al (1995) reported a numerical study of the mixing layer by means of a Monte-Carlo- based method. Their reported PDFs were similar to those of the present paper; but the computer time, with 600 computational cells compared with the present 4000, was 45 minutes on a Convex C220 .

In particular, the multi-fluid model predicts the overall flow characteristics as well as a conventional model; and, because it computes the PDFs, as a conventional model cannot, it provides much more detailed insight into (what may be) the physics of the flow.

Nevertheless, no investigation has yet been made as to whether the constants which were found appropriate for the mixing layer will serve also for other flow situations.

Further, it needs to be established whether the discretization was adequately fine, in respect of the finenesses of the both the geometric and population grids. The PHOENICS library case facilitates this.