(averaged_quantities)=
# Averaged quantities

```{admonition} References
- {cite:t}`Zuber1965-nm`: Drift-flux model
```

Flow structures in two-phase flows show a significant variety. At a relatively small gas volume flux, the gas phase injected into a vertical pipe filled with still liquid may form rising bubbles. Bubble flows are termed bubbly flows. With increasing gas flow rate, two-phase co-current flows in a vertical pipe may include large bubbles in a bullet-like shape followed by small bubbles in liquid slug. This flow structure is called a slug flow. A very high gas flow rate allows the gas phase to form a gas core in the center region of a pipe, and the liquid phase pushed out toward the pipe wall takes a ring-like shape; therefore, this flow pattern is referred to as annular flow. In between the slug and annular flows, we may observe a more complicated flow characteristics; referred to as a churn flow. {numref}`OneDimensional_Mishima_Ishii` shows a flow pattern map drawn by using Mishima-Ishii's criteria {cite:p}`Mishima1984-fv`. 

```{figure} ../python/fig-Mishima-Ishii.png
:name: OneDimensional_Mishima_Ishii
Flow pattern map drawn by Mishima-Ishii criteria for air-water system in 20 mm pipe. The horizontal and vertical axes are the gas and liquid volumetric fluxes. 
```

It is of course very difficult to understand everything about complex two-phase dynamics; however, for engineering purposes, simplified models are of great use for the design of industrial devices. We discuss the drift-flux model, which is a general approach to estimate the volume fraction of the two phases only with two model parameters. 

Estimating the volume fraction of each phase in the two-phase system is important in the design and operation of two-phase flow devices. The drift-flux model {cite:p}`Zuber1965-nm` is one of the simplified approaches for the estimation of the volume fraction. The simple principle realizes wide-range applicability, and the model has been utilized in many applications. The drift-flux model uses averaging to extract only two model parameters that represent the two-phase flow characteristics, that is, the distribution parameter $C_{0}$ and the drift velocity $v_{kj}$. A brief description of the model derivation is given in the following. 

The phase indicator defined by the following equation is a numerical tool to represent the state of the position $\mathbf{x}$ at time $t$; which phase occupies that space: 
```{math}
:label: eq:VoidFraction_eq_phase_indicator
    X_{k} (\mathbf{x}, t)
    =
    \left\{
    \begin{array}{ll}
        1 &\text{if}~~\mathbf{x} \in \text{phase}~k \\
        0 &\text{otherwise}
    \end{array}
    \right.
```
The local volume fraction is defined by 
```{math}
:label: eq:DriftFlux_nonref_0
    \alpha_{k} (\mathbf{x}, t)
    =
    \frac{1}{T} \int_{t - T/2}^{t + T/2} X_{k} (\mathbf{x}, t) dt
```
The time duration $T$ may be taken as small as possible such that $\alpha_{k}$ is an instantaneous quantity, but should be set finite to make $\alpha_{k}$ statistically meaningful. For the gas phase $k = G$, $\alpha_{G}$ is called the void fraction. See {numref}`OneDimensional_volume_fraction` for a schematic description of the definitions. 
```{figure} ../fig/volume_fraction.png
:name: OneDimensional_volume_fraction
Volume fraction
```

The velocity field of the two-phase flow is described by the local instantaneous velocity: $\mathbf{v} (\mathbf{x}, t)$. The local volumetric flux is defined as the $X$-weighted time-average of the fluid velocity: 
```{math}
:label: eq:DriftFlux_nonref_1
    \mathbf{j}_{k} (\mathbf{x}, t) = \frac{1}{T} \int_{t-T/2}^{t+T/2} X_{k} \mathbf{v} dt
```
This operation extracts the local instantaneous velocity of the $k$th phase. The total volumetric flux is defined as the sum of the fluxes of each phase: 
```{math}
:label: eq:DriftFlux_nonref_2
    \mathbf{j} = \sum_{k=1,2} \mathbf{j}_{k}
```
The phase-averaged (local) quantities are generally written as 
```{math}
:label: eq:DriftFlux_nonref_3
    f_{k} (\mathbf{x}, t)
    = \frac{\frac{1}{T} \int X_{k} f dt}{\frac{1}{T} \int X_{k} dt} 
    = \frac{\frac{1}{T} \int X_{k} fdt}{\alpha_{k}}
```
Taking $f = \mathbf{v}$ yields the phase-averaged velocity: 
```{math}
:label: eq:DriftFlux_nonref_4
    \mathbf{v}_{k} (\mathbf{x}, t)
    = \frac{\frac{1}{T} \int_{t-T/2}^{t+T/2} X_{k} \mathbf{v} dt}{\frac{1}{T} \int_{t-T/2}^{t+T/2} X_{k} dt} 
    = \frac{\mathbf{j}_{k}}{\alpha_{k}}
```
Therefore, 
```{math}
:label: eq:Introduction_eq_local_volumetric_flux
    \mathbf{j}_{k} = \alpha_{k} \mathbf{v}_{k}
```
The local relative velocity between the two phases is defined by 
```{math}
:label: eq:DriftFlux_nonref_5
    \mathbf{v}_{R} = \mathbf{v}_{2} - \mathbf{v}_{1}
```
An alternative representation of the velocity difference is the local drift velocity defined by 
```{math}
:label: eq:Introduction_eq_local_drift_velocity
    \mathbf{v}_{kj} = \mathbf{v}_{k} - \mathbf{j}
```
If we assume that the two fluids form a homogeneous two-phase mixture, there is no relative motion between the two phases, i.e. $\mathbf{v}_{R} = 0$, and therefore $\mathbf{v}_{k} = \mathbf{j}$ and $\mathbf{v}_{kj} = 0$, corresponding to the homogeneous model.