Fluid Film Turbulence

The Simcenter STAR-CCM+ fluid film turbulence model is based on the universal velocity profile approach.

The Fluid Film Turbulence Model in Simcenter STAR-CCM+ solves for the flow variables such as wall shear stress and the film surface velocity. Additionally, the optional Film Turbulent Viscosity model accounts for the turbulence viscosity in the liquid film, which is used for the calculation of thermal conductivity and mass diffusivity.

To compute the wall shear stress and film surface velocity, the non-dimensional velocity, $u^{+}$ is obtained as:

where:

• $k$ is a constant.
• $C$ is a constant.
• $\delta$ is the fluid film thickness.
• $h_{pm}$ is the film thickness corresponding to the point of intersection between the two parts of the equation.
• $y$ is the coordinate in the direction normal to film flow direction.
• $y^{+}$ is the non-dimensional form of $y$ .

  $$y^{+}=\frac{y\times u^{*}}{\nu }$$

  (2835)
  $u^{*}$ is the friction velocity, calculated as:

  $$u^{*}=\sqrt{\frac{{\tau }_{wall}}{\rho }}$$

  (2836)

  where

  • $\nu$ is the kinematic viscosity. ${\tau }_{wall}$ is the wall shear stress.
  • $\rho$ is the density of the fluid film.

Generally, annular films flows include scenarios where the film is either driven by shear or by gravity. Consequently, there are two turbulent viscosity models available in Simcenter STAR-CCM+ to account for both annular film flows options.

Cioncolini Model

The Cioncolini et al [617] model is adopted to simulate the shear driven fluid film flows. The model assumes a linear dependence of eddy diffusivity with the dimensionless film thickness, and is computed as:

$$1+\frac{{\mu }_t}{\mu }=\sqrt{1+0.9{\times 10}^{-3}\times {\left(y^{+}\right)}^2}$$

(2837)

where

• $\mu$ is the viscosity. ${\mu }_t$ is the turbulent viscosity of the film.

Mudawwar Model

This model, formulated by Mudawwar and El-Masri [627], accounts for fluid films falling under the effect of gravity. In Mudawwar's model a profile for $\frac{{\mu }_t}{\mu }$ is fitted to the bulk region of the film data of Ueda et al [630] and modified with the Van Driest damping function. The final equation is as follows:

$$\frac{{\mu }_t}{\mu }=-\frac{1}{2}+\frac{1}{2}\sqrt{1+4K^2{y}^{+2}\times (1-\frac{y^{+}}{{\delta }^{+}}{)}^2\times [1-\exp (-\frac{y^{+}}{26}\times (1-\frac{y^{+}}{{\delta }^{+}}{)}^{1/2}\times (1-\frac{0.865{\mathrm{Re}}_{crit}^{0.5}}{{\delta }^{+}})){]}^2}$$

(2838)

${\mathrm{Re}}_{crit}$ is the critical Reynolds number, obtained as:

$${\mathrm{Re}}_{crit}^{0.5}=\{\begin{array}{c}\frac{97}{K_a^{0.1}}Heating\\ \frac{0.04}{K_a^{0.37}}Evaporating\end{array}$$

(2839)

where:

• K=40 is a modelling constant.
• $K_a$ is the Kapitza number, obtained as:

  $$K_a=\frac{{\mu }^{4}g}{\rho {\sigma }^3}$$

  (2840)

  where:

  • g is the gravitational acceleration.
  • $\sigma$ is the surface tension of the fluid film.