Encyclopaedia Index

### CVAs, i.e. Continuously Varying Attributes of discretised fluids

#### 1. Introduction

CVA is a concept pertaining to the multi-fluid model of turbulence, MFM, which is a means of computing, for turbulent fluids, their temperatures, concentrations, velocities and other local properties, here referred to collectively as "attributes".

MFM allows for the fluctuations of attributes by treating a turbulent fluid as a population of distinct fluids, each possessing its own values of the said attributes.

MFM differentiates between:

• population-distinguishing attributes (PDAs), which are have values lying with a specified range for each fluid, and
• continuously-varying attributes attributes (CVAs), the values of which are limited only by physical considerations.

Typically, only one or two attributes are given PDA status; but CVAs are more numerous.

Examples are:

1. One-dimensional populations:
1. sea-surface-water, with
1. salinity (i.e. salt content)
as the one PDA and:
• temperature,
• density and
• vertical velocity
as the CVAs;

2. combustion of fuel and air, with
1. fuel/air ratio
as the one PDA and
• temperature,
• un-burned-fuel mass fraction and
• oxides-of-nitrogen concentration
• smoke concentration
as the CVAs.

2. Two-dimensional populations:
1. sea-surface-water, with
1. salinity (i.e. salt content) and
2. temperature
as the two PDAs and:
• density and
• vertical velocity
as the CVAs;

2. combustion of fuel and air, with
1. fuel/air ratio and
2. unburned-fuel mass fraction
as the two PDAs and
• temperature and
• oxides-of-nitrogen concentration
• smoke concentration
as the CVAs.

#### 2. Equations governing CVAs

The transport equation obeyed by CVAs is described in full in a lecture on the mathematical basis of MFM.

Here it suffices to say that the equations are the same as those which are encounterd in single-fluid computational fuid dynamics

except that:

1. in all places the dependent variable (i.e.the CVA) is multiplied by the mass fraction of the particular fluid in question; and
2. additional terms are present which represent fluid-to-fluid transfer.

#### 3.1 How to set up a CVA-solving simulation

In order to be recognised as a CVA, when a flow-simulation involving MFM is in progress, a solved-for variable must be given a name which:
• has a letter other than F as the first character (because F is reserved for the mass fractions of the various fluids),

• has digits 1, 2, 3, etcetera, up to 99, as the next one or two characters, which denote to which fluid the particular CVA applies, and

• has the letter C as the final character.

For example,the names H1C, H2C, H3C, H4C and H5C might represent the enthalpies of fluids 1, 2, 3, 4 and 5; however, the choice of H as the first character does not of itself determine this. Its significance as enthalpy must be given by way of the boundary conditions and source terms that are provided for each variable.

The source terms appropriate to CVAs are of two kinds, namely:

1. what might be called 'conventional' sources, such as:
• (for enthalpy) the energy gained or lost per unit mass by absorption or emission of radiation, or
• (for species concentration) the nett gain of species mass in unit time of unit mass of fluid;
and

2. 'MFM-related' sources, which express the loss and gain of the attribute in question by reason of micro-mixing.

The PHOENICS user is responsible for specifying the former; but the latter are computed and applied automatically by PHOENICS.

#### (a) Description and purpose

The example to be considered is L231 which concerns the generation of smoke in an idealised, indeed one-dimensional, combustor.

The smoke-generating reaction is also idealised, its rate being supposed to depend only on the unburned-fuel mass fraction and the temperature, as explained here.

Case L231 is a modification of case L230, in which the flame was supposed to be adiabatic: in case L231, by contrast, heat loss is allowed.

The absence of heat loss from case L230 made it unnecessary to employ any CVA at all, because:

• the unburned-fuel mass fraction and the temperature of each fluid remained constant;
• therefore the rate of smoke production within each fluid could be computed;
• consequently the total rate of production could be obtained by summing the individual rates.

In case L231, the heat loss reduces the temperature of each fluid; and that reduction lowers also, but more than proportionately, the production rate of smoke. Therefore, at least the temperature must appear as a CVA; and, if it influences the heat-loss rate, so must the smoke concentration also.

The effect of heat loss, it is true, is crudely represented in case L231: the temperature of each fluid is equal to the adiabatic temperature less a quantity whict depends upon the smoke concentration,

Specifically, the quantity is TEMPLOSS, which is computed from:

TEMPLOSS = RADFACT * SMOCONC * ADIATEMP**4
where:

• RADFACT is a factor communicated via SPEDAT.
• SMOCON is the smoke concentration in the fluid in question, and
• ADIATEMP is the adiabatic temperature of that fluid.

The ideas underlying this formula are that:

• the loss of heat from the fluid is by way of radiation, which accounts for the presence of the fourth power of; and
• its magnitude is proportional to the smoke concentration in the fluid.

Although crude, this presumption suffices to enable the use of CVAs to be illustrated and the effect of population-grid fineness to be explored.

#### 4. An alternative implementation: the use of GROUND coding created by PLANT

The use of this internal coding can be dispensed with if the user prefers to create his own, by way of GROUND.

Since this is however no light task, the user may wish to call in the assistance of PLANT, as has been done by S.V.Zhubrin in a report on the application of the multi-fluid model to turbulent combustion.

The MFM-and-PLANT-related seguences in three of the Q1 files which he employed, and be seen by clicking:

and here, for a two-phase example concerned with a wall-fired furnace.