**2. Numerical model**

The present numerical model being proposed for the evaluation of the particle dispersion is the Eulerian approach, which applies both the 3D RANS modeling of the carrier flow and the particulate phase [24] and the statistical PDF approach focusing on the mathematical descrip‐ tion of the second moments of the particulate phase [9]. Within the Eulerian approach, the particulate phase is considered as the diluted medium; therefore, the effect of the particle collision is negligible that means the application of the one-way coupling.

The numerical simulation considered the turbulent dispersion of solid particles in horizontal channel uniform shear turbulent flow for two different cases: i) shear of the mean flow velocity is along the direction of gravity (**Figure 1a**) and ii) shear of the mean flow velocity is direct‐ ed normally to gravity (**Figure 1b**). Here *u* is the mean axial velocity of gas.

**Figure 1.** Shear of the mean flow velocity in horizontal channel flow. (a) shear of the mean flow velocity is along the direction of gravity; (b) shear of the mean flow velocity is directed normally to gravity.

The particles were brought into the uniform shear gas flow, which has been preliminarily computed to obtain the velocity flow field.

The system of the momentum and closure equations of the gas phase are identical for the unladen flows, while the particle-laden flows are under the impact of the viscous drag force. The Cartesian coordinates are used here.

#### **2.1. Governing equations for the particulate phase**

The 3D governing equations for the particulate phase are written as follows:

The particle mass conservation equation:

On the contrary, there are very less studies on the particle turbulent dispersion, which are based on the Eulerian approach. The most significant are the numerical investigations [9, 23], where the statistical PDF models were applied to the particle behavior in the turbulent flows.

The 3D RSTM approach presented here applies the closure of equations of motion of the particulate phase, which is carried out similarly to the closure of the carrier flow, i.e., the equations are written for the normal and shear components of the Reynolds stress. The Reynolds stress equations are derived from the PDF model [9] and presented in a general case. The advantage of the given model is in use of the same closure for both the carrier flow and

The given 3D RSTM model has been applied for the turbulent dispersion of solid particles in

The obtained numerical results have been verified and validated by comparison with the

The present numerical model being proposed for the evaluation of the particle dispersion is the Eulerian approach, which applies both the 3D RANS modeling of the carrier flow and the particulate phase [24] and the statistical PDF approach focusing on the mathematical descrip‐ tion of the second moments of the particulate phase [9]. Within the Eulerian approach, the particulate phase is considered as the diluted medium; therefore, the effect of the particle

The numerical simulation considered the turbulent dispersion of solid particles in horizontal channel uniform shear turbulent flow for two different cases: i) shear of the mean flow velocity is along the direction of gravity (**Figure 1a**) and ii) shear of the mean flow velocity is direct‐

collision is negligible that means the application of the one-way coupling.

ed normally to gravity (**Figure 1b**). Here *u* is the mean axial velocity of gas.

particulate phase, namely, the Reynolds differential equations.

342 Numerical Simulation - From Brain Imaging to Turbulent Flows

a turbulent horizontal channel flow imposed to uniform shear.

experimental data.

**2. Numerical model**

$$
\frac{
\partial \left(
\alpha \mathbf{u}\_s
\right)
}{
\partial \mathbf{x}
} + \frac{
\partial \left(
\alpha \mathbf{v}\_s
\right)
}{
\partial \mathbf{y}
} + \frac{
\partial \left(
\alpha \mathbf{w}\_s
\right)
}{
\partial \mathbf{z}
} = \frac{
\partial
}{
\partial \mathbf{x}
} D\_s \frac{
\partial \alpha
}{
\partial \mathbf{x}
} + \frac{
\partial
}{
\partial \mathbf{y}
} D\_s \frac{
\partial \alpha
}{
\partial \mathbf{y}
} + \frac{
\partial
}{
\partial \mathbf{z}
} D\_s \frac{
\partial \alpha
}{
\partial \mathbf{z}
},
\tag{1}
$$

*x*-component of the momentum equation:

$$\begin{split} \frac{\partial}{\partial \mathbf{x}} \alpha \Big[ \mathbf{u}\_{s}^{2} - \overline{\mathbf{u}\_{s}^{'2}} \Big] + \frac{\partial}{\partial \mathbf{y}} \alpha \Big[ \mathbf{u}\_{s} \mathbf{v}\_{s} - \overline{\mathbf{u}\_{s}^{'} \mathbf{v}\_{s}^{'}} \Big] + \frac{\partial}{\partial \mathbf{z}} \alpha \Big[ \mathbf{u}\_{s} \mathbf{w}\_{s} - \overline{\mathbf{u}\_{s}^{'} \mathbf{w}\_{s}^{'}} \Big] = \alpha \mathbf{C}\_{D}' \frac{\left( \mathbf{u} - \mathbf{u}\_{s} \right)}{\tau\_{p}} \\ + \alpha \operatorname{sgn} \left( \mathbf{g}\_{x} \right) \Big( 1 - \frac{\rho}{\rho\_{p}} \Big), \end{split} \tag{2}$$

*y*-component of the momentum equation:

$$\begin{split} \frac{\partial}{\partial \mathbf{x}} \alpha \Big[ u\_s \mathbf{v}\_s - \overline{u\_s' \mathbf{v}\_s'} \Big] + \frac{\partial}{\partial \mathbf{y}} \alpha \Big[ \mathbf{v}\_s^2 - \overline{\mathbf{v}\_s'^2} \Big] + \frac{\partial}{\partial \mathbf{z}} \alpha \Big[ \mathbf{v}\_s \mathbf{w}\_s - \overline{\mathbf{v}\_s' \mathbf{w}\_s'} \Big] = \alpha C\_D' \frac{\left( \mathbf{v} - \mathbf{v}\_s \right)}{\mathbf{r}\_p} \\ + \alpha \text{sgn} \left( \mathbf{g}\_\mathcal{V} \right) \Big( 1 - \frac{\rho}{\rho\_p} \Big), \end{split} \tag{3}$$

*z*-component of the momentum equation:

$$
\frac{\partial}{\partial \mathbf{x}} \alpha \Big[ \boldsymbol{u}\_{s} \boldsymbol{w}\_{s} - \overline{\boldsymbol{u}\_{s}^{\prime} \boldsymbol{w}\_{s}^{\prime}} \right] + \frac{\partial}{\partial \mathbf{y}} \alpha \Big[ \mathbf{v}\_{s} \boldsymbol{w}\_{s} - \overline{\boldsymbol{v}\_{s}^{\prime} \boldsymbol{w}\_{s}^{\prime}} \Big] + \frac{\partial}{\partial \boldsymbol{z}} \alpha \Big[ \mathbf{w}\_{s}^{2} - \overline{\boldsymbol{w}\_{s}^{\prime 2}} \Big] = \alpha \boldsymbol{C}\_{D}^{\prime} \frac{\left( \boldsymbol{w} - \boldsymbol{w}\_{s} \right)}{\tau\_{p}},\tag{4}
$$

where *u, v*, and *w* are the axial, transverse, and spanwise time-averaged velocity compo‐ nents of gas, respectively; *u*s, *v*s and *w*s are the axial, transverse, and spanwise time-averaged velocity components of particulate phase, respectively; *ρ* is the material density of gas; *ρp* is the material density of particles; α is the particle mass concentration; *gy* is the *y*-component of gravity.

The relative friction coefficient *C*<sup>D</sup> ' is calculated according to [25].

The closure model for the transport equations of the particulate phase was applied to the PDF model [26], where *D*s is the coefficient of the turbulent diffusion of the particulate phase.

The equations for the second-order moments of the fluctuating velocity (turbulent stresses) of the particulate phase are written based on the PDF approach in [9]. These equations describe convective and diffusive transfer, generation of particle velocity fluctuations due to the velocity gradient, generation of fluctuations resulting from particle entrainment into the fluctuating motion of carrier gas flow, and dissipation of turbulent stresses of the particulate phase caused by interfacial forces:

Equation of the *x*-normal component of the Reynolds stress:

$$\frac{\partial}{\partial \mathbf{x}} \alpha \left[ u\_s \overline{u\_s'^2} - \tau\_p \left( \overline{u\_s'^2} + \mathbf{g}\_u' \overline{u'^2} \right) \frac{\partial \overline{u\_s'^2}}{\partial \mathbf{x}} \right] + \frac{\partial}{\partial \mathbf{y}} \alpha \left[ v\_s \overline{u\_s'^2} - \frac{\tau\_p \left( \overline{v\_s'^2} + \mathbf{g}\_u'' \overline{v'^2} \right)}{\mathbf{3}} \frac{\partial \overline{u\_s'^2}}{\partial \mathbf{y}} \right]$$

$$\begin{split} &+\frac{\widehat{\mathcal{O}}}{\widehat{\mathcal{O}}\pi}\alpha \left[w\_{s}\overline{u\_{s}^{'2}}-\frac{\tau\_{p}}{3}\Big(\overline{w\_{s}^{'2}}+\mathsf{g}\_{\mu}^{k}\overline{\mathsf{w}^{'}}\Big)\frac{\partial\overline{u\_{s}^{'2}}}{\partial\overline{z}}\right] \\ &=\frac{\widehat{\mathcal{O}}}{\widehat{\mathcal{O}}\pi}\left\{\alpha\tau\_{p}\Big[\left(\overline{u\_{s}^{'}\mathsf{v}\_{s}^{'}}+\mathsf{g}\_{\mu}^{n}\overline{u^{\prime}\mathsf{v}^{'}}\right)\frac{\partial\overline{u\_{s}^{'2}}}{\partial\overline{\mathsf{v}}}+\left(\mathsf{u}\_{s}^{'}\mathsf{w}\_{s}^{'}+\mathsf{g}\_{\mu}^{k}\overline{u^{\prime}\mathsf{w}^{'}}\right)\frac{\partial\overline{u\_{s}^{'2}}}{\partial\overline{z}}\right]\bigg\} \end{split}$$

$$+\frac{\partial}{\partial \mathbf{y}} \left\{ \frac{\alpha \mathbf{r}\_p}{3} \left[ 2 \overline{\left( \mathbf{u}\_s'^2 + \mathbf{g}\_u' \overline{\mathbf{u}'^2} \right)} \frac{\partial \overline{\mathbf{u}\_s' \mathbf{v}\_s'}}{\partial \mathbf{x}} + 2 \overline{(\mathbf{u}\_s' \mathbf{v}\_s' + \mathbf{g}\_u'' \overline{\mathbf{u}'} \mathbf{v})} \frac{\partial \overline{\mathbf{u}\_s' \mathbf{v}\_s'}}{\partial \mathbf{y}} + 2 \left( \overline{\mathbf{u}\_s' \mathbf{w}\_s'} + \mathbf{g}\_u^k \overline{\mathbf{u}' \mathbf{w}'} \right) \frac{\partial \overline{\mathbf{u}\_s' \mathbf{v}\_s'}}{\partial \mathbf{z}} \right]$$

$$\begin{aligned} &+ \left( \overline{\overline{u\_s'} \overline{v\_s'}} + \mathbf{g}\_{\mu}^{\overline{l}} \overline{u' \overline{v'}} \right) \frac{\widehat{\mathcal{O}u\_s'^2}}{\widehat{\mathcal{O}x}} + \left( \overline{\overline{v\_s'} \overline{w\_s'}} + \mathbf{g}\_{\mu}^{\overline{k}} \overline{\overline{v'} \overline{w'}} \right) \frac{\widehat{\mathcal{O}u\_s'^2}}{\widehat{\mathcal{O}z}} \Bigg] \\ &+ \frac{\widehat{\mathcal{O}}}{\widehat{\mathcal{O}z}} \Bigg[ \frac{\alpha \tau\_p}{3} \Bigg] 2 \left( \overline{u\_s'^2} + \mathbf{g}\_{\mu}^{\overline{l}} \overline{u'^2} \right) \frac{\widehat{\mathcal{O}u\_s' \overline{w\_s'}}}{\widehat{\mathcal{O}x}} \end{aligned}$$

$$\begin{aligned} \left[ +2\left( \overline{\mathbf{u}\_{\boldsymbol{s}}^{\prime} \mathbf{v}\_{\boldsymbol{s}}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\prime} \overline{\mathbf{u}^{\prime} \mathbf{v}^{\prime}} \right) \frac{\partial \overline{u\_{\boldsymbol{s}}^{\prime} \mathbf{v}\_{\boldsymbol{s}}^{\prime}}}{\partial \mathbf{y}} + 2\left( \overline{\mathbf{u}\_{\boldsymbol{s}}^{\prime} \mathbf{w}\_{\boldsymbol{s}}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\boldsymbol{k}} \overline{\mathbf{u}^{\prime} \mathbf{v}^{\prime}} \right) \frac{\partial \overline{u\_{\boldsymbol{s}}^{\prime} \mathbf{w}\_{\boldsymbol{s}}^{\prime}}}{\partial \mathbf{z}} + \left( \overline{\mathbf{u}\_{\boldsymbol{s}}^{\prime} \mathbf{w}\_{\boldsymbol{s}}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\prime} \overline{\mathbf{u}^{\prime} \mathbf{w}^{\prime}} \right) \frac{\partial \overline{u\_{\boldsymbol{s}}^{\prime}}^{2}}{\partial \mathbf{x}} \\\\ + \left( \overline{\mathbf{v}\_{\boldsymbol{s}}^{\prime} \mathbf{w}\_{\boldsymbol{s}}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\ast} \overline{\mathbf{v}^{\prime} \mathbf{w}^{\prime}} \right) \frac{\partial \overline{u\_{\boldsymbol{s}}^{\prime}}^{2}}{\partial \mathbf{y}^{\prime}} \end{aligned} \tag{5}$$

Equation of the *y*-normal component of the Reynolds stress:

( )

 a

*p*

*p*

t

2 2

3

2 2

t

(3)

t

( )

*g*

The relative friction coefficient *C*<sup>D</sup>

phase caused by interfacial forces:

a

a

a

a

gravity.

*y*

+ - ç ÷

sgn 1 ,

344 Numerical Simulation - From Brain Imaging to Turbulent Flows

*z*-component of the momentum equation:

*p*

*xyz*

 a

'

Equation of the *x*-normal component of the Reynolds stress:

 t

2 22 2

*<sup>n</sup> ps u <sup>l</sup> s s ss p s u s s v gv u u uu u gu v u <sup>x</sup> x y <sup>y</sup>*

é ù æ ö é ù ç ÷ ¢ ¢ <sup>+</sup> ê ú ¶ ¶ ¶ ¶ ¢ ¢ æ ö è ø ê ú ¢ ¢¢ -+ + - ê ú ¢ ç ÷ ¶ ê ú è ø ¶ ¶ ê ú ¶ ë û ë û

a

r

æ ö

ç ÷ è ø

r

*x yz*

2 2

aa

¶ ¶¶ - é ù é ù - + -+ - = ¢ ¢ ¢ é ù ¢¢ ¢ ¶ ¶¶ ë û ê ú ë û ë û

*<sup>s</sup> ss ss s s ss ss D*

( ) 2 2 , *<sup>s</sup> ss ss ss ss ss D*

where *u, v*, and *w* are the axial, transverse, and spanwise time-averaged velocity compo‐ nents of gas, respectively; *u*s, *v*s and *w*s are the axial, transverse, and spanwise time-averaged velocity components of particulate phase, respectively; *ρ* is the material density of gas; *ρp* is the material density of particles; α is the particle mass concentration; *gy* is the *y*-component of

is calculated according to [25].

The closure model for the transport equations of the particulate phase was applied to the PDF

The equations for the second-order moments of the fluctuating velocity (turbulent stresses) of the particulate phase are written based on the PDF approach in [9]. These equations describe convective and diffusive transfer, generation of particle velocity fluctuations due to the velocity gradient, generation of fluctuations resulting from particle entrainment into the fluctuating motion of carrier gas flow, and dissipation of turbulent stresses of the particulate

model [26], where *D*s is the coefficient of the turbulent diffusion of the particulate phase.

¶ ¶¶ - é ù é ùé ù - + - + -= ¢ ¢ ¢ ¢ ¢ ¢ ë ûë û ê ú ¶¶¶ ë û (4)

aa

*w w uw uw vw vw w w C*

*v v uv uv v v vw vw C*

$$\begin{split} \frac{\partial}{\partial \mathbf{x}} \alpha \bigg[ u\_s \overline{v\_s'^2} - \frac{\tau\_p}{3} \left( \overline{u\_s'^2} + \mathbf{g}\_u^l \overline{u'^2} \right) \frac{\partial \overline{v\_s'^2}}{\partial \mathbf{x}} \bigg] + \frac{\partial}{\partial \mathbf{y}} \alpha \bigg[ \mathbf{v}\_s \overline{v\_s'^2} - \tau\_p \left( \overline{v\_s'^2} + \mathbf{g}\_u^n \overline{v'^2} \right) \frac{\partial \overline{v\_s'^2}}{\partial \mathbf{y}'} \bigg] \\ + \frac{\partial}{\partial \mathbf{z}} \alpha \bigg[ \mathbf{w}\_s \overline{v\_s'^2} - \frac{\tau\_p}{3} \left( \overline{w\_s'^2} + \mathbf{g}\_u^k \overline{w'} \right) \frac{\partial \overline{v\_s'^2}}{\partial \mathbf{z}} \bigg] \\ = \frac{\partial}{\partial \mathbf{x}} \bigg[ \frac{\alpha \tau\_p}{3} \bigg[ 2 \left( \overline{u\_s' v\_s'} + \mathbf{g}\_u^n \overline{u'} \overline{v'} \right) \frac{\partial \overline{u\_s' v\_s'}}{\partial \mathbf{x}} + 2 \left( \overline{v\_s'^2} + \mathbf{g}\_u^n \overline{v'^2} \right) \frac{\partial \overline{u\_s' v\_s'}}{\partial \mathbf{y}'} \bigg] \end{split}$$

( ) ( ) ( ) 2 2 <sup>2</sup> *k nk s s <sup>s</sup> <sup>s</sup> ss u ss u ss u u v <sup>v</sup> <sup>v</sup> v w g vw u v g uv u w g uw zy z* ùü ¶ ¶¶ ¢ ¢ ¢ ¢ <sup>ï</sup> + + ++ + + ¢ ¢ ¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ úý ¶¶ ¶ úïûþ ( ) ( ) 2 2 *l k s s p ss u ss u v v u v g uv v w g uw y xz* at ì ü é ù ¶ ï ï ¶ ¶ ¢ ¢ + + ++ í ý ê ú ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ ¶ ¶¶ ï ï î þ ë û ( ) 2 2 2 2 3 *<sup>p</sup> l n s s s s ss u s u v w v w u v g uv v gv z xy* ¶ ï ìat <sup>é</sup> ¶ ¶ ¢ ¢ ¢ ¢ æ ö + + ++ <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¢ ¢ ç ÷ ¶ ¶¶ è ø ïî êë ( ) ( ) ( ) 2 2 <sup>2</sup> *k ln s s <sup>s</sup> <sup>s</sup> ss u ss u ss u v w <sup>v</sup> <sup>v</sup> v w g vw u w g uw v w g vw z xy* ùü ¶ ¶¶ ¢ ¢ ¢ ¢ <sup>ï</sup> + + ++ ++ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ úý ¶ ¶¶ úïûþ (6)

Equation of the *z*-normal component of the Reynolds stress:

2 2 2 22 2 22 3 3 *p p l n s s ss s u ss s u w w uw u gu vw v g v <sup>x</sup> x y <sup>y</sup>* t t a a <sup>é</sup> ù é <sup>ù</sup> ¶ ¶ ¶ ¶ ¢ ¢ æ ö æ ö <sup>ê</sup> ¢ ¢¢ -+ + -+ ú ê ¢ ¢¢ <sup>ú</sup> ç ÷ ç ÷ ¶ <sup>ê</sup> è ø ¶ ¶ ú ê è ø ¶ <sup>ú</sup> <sup>ë</sup> û ë <sup>û</sup> ( ) <sup>2</sup> 2 2 2 3 *k n <sup>s</sup> <sup>p</sup> s s ss p s u ss u <sup>w</sup> u w ww w g w u w g uw <sup>z</sup> z x <sup>x</sup>* at a t ¶ ¶ ¶ ¶ ¢ <sup>ù</sup> <sup>ï</sup> <sup>ì</sup> <sup>é</sup> ¢ ¢ éæ ö + -+ = + ¢ ¢¢ <sup>ú</sup> <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ç ÷ <sup>ê</sup> ¶ ëè ø ¶ ¶ <sup>ú</sup> <sup>ê</sup> ¶ ïî <sup>ë</sup> <sup>û</sup> ( ) ( ) ( ) 2 2 2 2 2 2 *<sup>n</sup> s s ss u k nk s s <sup>s</sup> <sup>s</sup> s u ss u ss u u w v w g vw y u w <sup>w</sup> <sup>w</sup> w gw u v g uv u w g uw zy z* ¶ ¢ ¢ + + ¢ ¢ ¢¢ ¶ ùü æ ö ¶ ¶¶ ¢ ¢ ¢ ¢ <sup>ï</sup> + + ++ + + ¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ <sup>ú</sup> ç ÷ <sup>ý</sup> è ø ¶¶ ¶ úïûþ ( ) ( ) 2 2 2 3 2 2 *<sup>p</sup> s s <sup>l</sup> ss u n k s s s s ss u s u v w u w g uw y x v w v w v w g vw w gw y z* ¶ ì ïat <sup>é</sup> ¶ ¢ ¢ + + <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¶ ¶ ïî êë ¶ ¶ ¢ ¢ ¢ ¢ æ ö + + ++ ¢ ¢ ¢¢ ¢ ¢ ç ÷ ¶ ¶ è ø

Two-Fluid RANS-RSTM-PDF Model for Turbulent Particulate Flows http://dx.doi.org/10.5772/63338 347

(8)

$$\begin{aligned} &+ \left( \overline{\mathbf{u}\_s' \mathbf{v}\_s'} + \mathbf{g}\_u^I \overline{\mathbf{u}' \mathbf{v}'} \right) \frac{\widehat{\mathbf{c}} \overline{\mathbf{w}\_s'}^2}{\widehat{\mathbf{c}} \mathbf{x}} + \left( \overline{\mathbf{v}\_s' \mathbf{w}\_s'} + \mathbf{g}\_u^k \overline{\mathbf{v}' \mathbf{w}'} \right) \frac{\partial \overline{\mathbf{w}\_s'^2}}{\partial \mathbf{z}} \Bigg] \\\\ &+ \frac{\partial}{\partial \mathbf{z}} \Bigg[ \alpha \mathbf{r}\_p \left[ \left( \overline{\mathbf{u}\_s' \mathbf{v}\_s'} + \mathbf{g}\_u^I \overline{\mathbf{u}' \mathbf{w}'} \right) \frac{\partial \overline{\mathbf{w}\_s'^2}}{\partial \mathbf{x}} + \left( \overline{\mathbf{v}\_s' \mathbf{w}\_s'} + \mathbf{g}\_u^n \overline{\mathbf{v}' \mathbf{w}'} \right) \frac{\partial \overline{\mathbf{w}\_s'^2}}{\partial \mathbf{y}} \right] \Bigg] \end{aligned} \tag{7}$$

Equation of the *xy* shear stress component of the Reynolds stress:

( ) ( ) ( ) 2 2 <sup>2</sup> *k nk s s <sup>s</sup> <sup>s</sup> ss u ss u ss u u v <sup>v</sup> <sup>v</sup> v w g vw u v g uv u w g uw*

ùü ¶ ¶¶ ¢ ¢ ¢ ¢ <sup>ï</sup> + + ++ + + ¢ ¢ ¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ úý ¶¶ ¶ úïûþ

ì ü é ù ¶ ï ï ¶ ¶ ¢ ¢ + + ++ í ý ê ú ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ ¶ ¶¶ ï ï î þ ë û

( ) 2 2 2 2

*z xy*

2 22 2 22

<sup>é</sup> ù é <sup>ù</sup> ¶ ¶ ¶ ¶ ¢ ¢ æ ö æ ö <sup>ê</sup> ¢ ¢¢ -+ + -+ ú ê ¢ ¢¢ <sup>ú</sup> ç ÷ ç ÷ ¶ <sup>ê</sup> è ø ¶ ¶ ú ê è ø ¶ <sup>ú</sup> <sup>ë</sup> û ë <sup>û</sup>

*p p l n s s ss s u ss s u w w uw u gu vw v g v <sup>x</sup> x y <sup>y</sup>*

 a

*k n <sup>s</sup> <sup>p</sup> s s ss p s u ss u <sup>w</sup> u w ww w g w u w g uw <sup>z</sup> z x <sup>x</sup>*

3 3

¶ ¶ ¶ ¶ ¢ <sup>ù</sup> <sup>ï</sup> <sup>ì</sup> <sup>é</sup> ¢ ¢ éæ ö + -+ = + ¢ ¢¢ <sup>ú</sup> <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ç ÷ <sup>ê</sup> ¶ ëè ø ¶ ¶ <sup>ú</sup> <sup>ê</sup> ¶ ïî <sup>ë</sup> <sup>û</sup>

*k nk s s <sup>s</sup> <sup>s</sup> s u ss u ss u*

( )

*n k s s s s ss u s u*

*v w v w v w g vw w gw*

¶ ¶ ¢ ¢ ¢ ¢ æ ö + + ++ ¢ ¢ ¢¢ ¢ ¢ ç ÷ ¶ ¶ è ø

( ) 2 2

*<sup>p</sup> s s <sup>l</sup> ss u*

<sup>é</sup> ¶ ¢ ¢ + + <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¶ ¶ ïî êë

*u w <sup>w</sup> <sup>w</sup> w gw u v g uv u w g uw*

ùü æ ö ¶ ¶¶ ¢ ¢ ¢ ¢ <sup>ï</sup> + + ++ + + ¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ <sup>ú</sup> ç ÷ <sup>ý</sup> è ø ¶¶ ¶ úïûþ

*v w u w g uw*

2 2 2

<sup>é</sup> ¶ ¶ ¢ ¢ ¢ ¢ æ ö + + ++ <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¢ ¢ ç ÷ ¶ ¶¶ è ø ïî êë

( ) ( ) ( ) 2 2 <sup>2</sup> *k ln s s <sup>s</sup> <sup>s</sup> ss u ss u ss u v w <sup>v</sup> <sup>v</sup> v w g vw u w g uw v w g vw*

ùü ¶ ¶¶ ¢ ¢ ¢ ¢ <sup>ï</sup> + + ++ ++ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ úý ¶ ¶¶ úïûþ

*<sup>p</sup> l n s s s s ss u s u v w v w u v g uv v gv*

*z xy*

2 2

 t

( ) <sup>2</sup>

3

( ) ( ) 2 2

*zy z*

*y z*

at

(6)

at

3

Equation of the *z*-normal component of the Reynolds stress:

t

 t

*<sup>n</sup> s s ss u*

¶ ì ïat *y*

2 3

*y x*

2 2

*u w v w g vw*

¶ ¢ ¢ + + ¢ ¢ ¢¢ ¶

( )

2 2

a

a

2

2

¶ ì ïat

346 Numerical Simulation - From Brain Imaging to Turbulent Flows

*zy z*

( ) ( ) 2 2 *l k s s p ss u ss u v v u v g uv v w g uw y xz*

( ) 2 2 2 2 2 2 2 3 2 3 3 2 2 3 *<sup>p</sup> <sup>l</sup> s s s ss s u k ps u <sup>p</sup> <sup>n</sup> s s s s s ss s u s ss <sup>p</sup> n k s s ss u ss u u v uuv u g u x x w gw u v u v vuv v g v wuv <sup>y</sup> y z <sup>z</sup> u v u v g uv u w g u x y* t a t t a a at ¶ é ù ¶ ¢ ¢ æ ö ê ú ¢¢ ¢ ¢ - + ç ÷ ¶ ¶ è ø ë û æ ö <sup>ù</sup> ç ÷ ¢ ¢ <sup>+</sup> é ù <sup>ú</sup> ¶ ¶ ¶ ¶ ¢ ¢ ¢ ¢ æ ö è ø + -+ + - ê ú ¢¢ ¢ ¢ ç ÷ <sup>é</sup> ¢ ¢ <sup>ú</sup> ¶ ê ú è ø ¶ ¶ <sup>ë</sup> ¶ <sup>ú</sup> ë û <sup>ú</sup> û ¶ ¶ ¢ ¢ = + ++ ¢ ¢ ¢¢ ¢ ¢ ¢ ¶ ¶ ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 3 2 *ss s ln k s s ss u s u sw u <sup>p</sup> <sup>l</sup> s s s u n k <sup>s</sup> ss u s s u <sup>n</sup> ss u uv u <sup>w</sup> z x u u u v g uv v g v v w g vw y z v v u gu y x y <sup>v</sup> u v g uv u w g uw <sup>z</sup> uv g u* at <sup>ì</sup> <sup>é</sup> <sup>ï</sup> ¶ ¶ ¢¢ ¢ <sup>í</sup> <sup>ê</sup> ¢ <sup>+</sup> <sup>ê</sup> ¶ ¶ <sup>ï</sup> î ë ùü æ ö ¶ ¶ ¢ ¢ <sup>ï</sup> ´ + ++ + + ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ ¢ ¢ <sup>ú</sup> ç ÷ <sup>ý</sup> è ø ¶ ¶ úïûþ <sup>ì</sup> <sup>é</sup> ¶ <sup>ï</sup> ¶ ¶ ¢ ¢ æ ö + ++ <sup>í</sup> <sup>ê</sup> ¢ ¢ ç ÷ ¶ êè ø ¶ ¶ <sup>ï</sup> î ë ¶ ¢ ´+ + + ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ ¶ + + ¢ ¢ ¢¢ ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 3 *s s <sup>k</sup> s s ss u <sup>p</sup> l n s s s s s u ss u k l s s ss u ss u s s n k s s s u ss u u v u v v v w g vw x z v w v w u gu u v g uv z xy v w u w g uw u v g uv <sup>z</sup> u w u w v g v vw g v x y* at ùü ¶ ¶ ¢ ¢ ¢ ¢ <sup>ï</sup> + + ¢ ¢ ¢¢ úý ¶ ¶ úûïþ ¶ ï <sup>ì</sup> <sup>é</sup> ¶ ¶ ¢ ¢ ¢ ¢ æ ö + + ++ <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢ ¢ ¢¢ ç ÷ ¶ ¶¶ è ø ïî êë ¶ ¢ ¢ + + ++ ¢ ¢ ¢ ¢ ¢ ¢ ¢¢ ¶ ¶ ¶ ¢ ¢ ¢ ¢ æ ö ´ ++ + + ¢ ¢ ¢¢ ¢ ç ÷ ¶ ¶ è ø ( ) ( ) ( ) , *s s l n s s s s ss u ss u u w <sup>w</sup> <sup>z</sup> u v u v u w g uw v w g vw x y* ¶ ¢ ¢ ¢ ¶ ùü ¶ ¶ ¢ ¢ ¢ ¢ <sup>ï</sup> ++ ++ ¢ ¢ ¢¢ ¢ ¢ ¢¢ úý ¶ ¶ úûïþ

Equation of the *xz* shear stress component of the Reynolds stress:

$$\frac{\partial}{\partial \mathbf{x}} \alpha \left[ \mu\_s \overline{u\_s' w\_s'} - \frac{2 \tau\_p}{3} \left( \overline{u\_s'^2} + \mathbf{g}\_u' \overline{u'^2} \right) \frac{\partial \overline{u\_s' w\_s'}}{\partial \mathbf{x}} \right]$$

$$+ \frac{\partial}{\partial \mathbf{y}} \alpha \left[ \nu\_s \overline{u\_s' w\_s'} - \frac{\tau\_p}{3} \left( \overline{\nu\_s'^2} + \mathbf{g}\_u'' \overline{\nu'^2} \right) \frac{\partial \overline{u\_s' w\_s'}}{\partial \mathbf{y}} \right]$$

$$\begin{split} &+\frac{\partial}{\partial z}\alpha \left[ \left( w\_s \overline{u\_s' w\_s'} - \frac{2\pi\_p}{3} \left( \overline{w\_s'^2} + \mathbf{g}\_u^k \overline{w'^2} \right) \frac{\partial \overline{u\_s' w\_s'}}{\partial z} \right] \right] \\ &= \frac{\partial}{\partial \mathbf{x}} \left\{ \frac{\alpha \tau\_p}{3} \left[ 2 \left( \overline{u\_s' v\_s'} + \mathbf{g}\_u^n \overline{u'^2} \right) \frac{\partial \overline{u\_s' w\_s'}}{\partial \mathbf{y}} + 2 \frac{\partial \overline{u\_s' w\_s'}}{\partial z} \right] \right. \end{split}$$

$$\begin{split} & \left( \overline{\boldsymbol{\mu}\_{s}^{\prime} \boldsymbol{\nu}\_{s}^{\prime}} + \mathbf{g}\_{\boldsymbol{\mu}}^{k} \overline{\boldsymbol{\mu}^{\prime} \boldsymbol{\nu}^{\prime}} \right) + \left( \overline{\boldsymbol{\mu}\_{s}^{\prime} \boldsymbol{\nu}\_{s}^{\prime}} + \mathbf{g}\_{\boldsymbol{\mu}}^{l} \overline{\boldsymbol{\mu}^{\prime} \boldsymbol{\nu}^{\prime}} \right) \frac{\overline{\boldsymbol{\mathcal{O}} \boldsymbol{\mu}\_{s}^{\prime 2}}}{\widehat{\boldsymbol{\mathcal{O}} \boldsymbol{\nu}}} \\ & + \left( \overline{\boldsymbol{\nu}\_{s}^{\prime} \boldsymbol{\nu}\_{s}^{\prime}} + \mathbf{g}\_{\boldsymbol{\mu}}^{n} \overline{\boldsymbol{\nu}^{\prime} \boldsymbol{\nu}^{\prime}} \right) \frac{\overline{\boldsymbol{\mathcal{O}} \boldsymbol{\mu}\_{s}^{\prime 2}}}{\widehat{\boldsymbol{\mathcal{O}} \boldsymbol{\nu}}} + \left( \overline{\boldsymbol{\nu}\_{s}^{\prime 2}} + \mathbf{g}\_{\boldsymbol{\mu}}^{k} \overline{\boldsymbol{\nu}^{\prime 2}} \right) \frac{\partial \overline{\boldsymbol{\mu}\_{s}^{\prime 2}}}{\widehat{\boldsymbol{\mathcal{O}} \boldsymbol{z}}} \end{split}$$

$$\begin{split} &+\frac{\widehat{\mathcal{O}}}{\widehat{\mathcal{O}}\mathcal{V}} \Bigg\{ \frac{\alpha \tau\_{p}}{3} \Bigg[ \left( \overline{\boldsymbol{u}\_{s}^{\prime}\prime}^{2} + \mathbf{g}\_{\boldsymbol{u}}^{\prime}\overline{\boldsymbol{u}^{\prime}\prime}^{2} \right) \frac{\widehat{\mathcal{O}}\overline{\boldsymbol{v}\_{s}^{\prime}\prime\boldsymbol{w}\_{s}^{\prime}}}{\widehat{\mathcal{O}}\boldsymbol{x}} + \left( \overline{\boldsymbol{u}\_{s}^{\prime}\prime\prime}\_{s} + \mathbf{g}\_{\boldsymbol{u}}^{\prime\prime}\overline{\boldsymbol{u}^{\prime}\prime\prime} \right) \frac{\widehat{\mathcal{O}}\overline{\boldsymbol{v}\_{s}^{\prime}\prime\boldsymbol{w}\_{s}^{\prime}}}{\widehat{\mathcal{O}}\boldsymbol{v}^{\prime}} \\ &+ \left( \overline{\boldsymbol{u}\_{s}^{\prime}\prime\boldsymbol{w}\_{s}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\prime\prime}\overline{\boldsymbol{u}^{\prime}\prime\prime\prime} \right) \frac{\widehat{\mathcal{O}}\overline{\boldsymbol{v}\_{s}^{\prime}\prime\boldsymbol{w}\_{s}^{\prime}}}{\widehat{\mathcal{O}}\boldsymbol{z}} + \left( \overline{\boldsymbol{u}\_{s}^{\prime}\prime\boldsymbol{w}\_{s}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\prime\prime}\overline{\boldsymbol{u}^{\prime}\prime\prime\prime} \right) \end{split}$$

$$\begin{split} & \times \frac{\widehat{\boldsymbol{\alpha}} \overline{\boldsymbol{u}\_{s}^{\prime} \boldsymbol{v}\_{s}^{\prime}}}{\widehat{\boldsymbol{\alpha}} \boldsymbol{\alpha}} + \left( \overline{\boldsymbol{v}\_{s}^{\prime} \boldsymbol{w}\_{s}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\boldsymbol{n}} \overline{\boldsymbol{\nu}^{\prime} \boldsymbol{w}^{\prime}} \right) \frac{\overline{\boldsymbol{u}\_{s}^{\prime} \boldsymbol{v}\_{s}^{\prime}}}{\widehat{\boldsymbol{\alpha}} \boldsymbol{v}} + \left( \overline{\boldsymbol{w}\_{s}^{\prime 2}} + \mathbf{g}\_{\boldsymbol{u}}^{k} \overline{\boldsymbol{w}^{\prime 2}} \right) \frac{\partial \overline{\boldsymbol{u}\_{s}^{\prime} \boldsymbol{v}\_{s}^{\prime}}}{\widehat{\boldsymbol{\alpha}} \boldsymbol{z}} \\ & + \left( \overline{\boldsymbol{u}\_{s}^{\prime} \boldsymbol{v}\_{s}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\boldsymbol{l}} \overline{\boldsymbol{u}^{\prime} \boldsymbol{v}^{\prime}} \right) \frac{\partial \overline{\boldsymbol{u}\_{s}^{\prime} \boldsymbol{v}\_{s}^{\prime}}}{\widehat{\boldsymbol{\alpha}} \boldsymbol{v}} + \left( \overline{\boldsymbol{v}\_{s}^{\prime} \boldsymbol{w}\_{s}^{\prime}} + \mathbf{g}\_{\boldsymbol{u}}^{\boldsymbol{k}} \overline{\boldsymbol{v}^{\prime} \boldsymbol{w}^{\prime}} \right) \end{split}$$

$$\begin{split} \left| \times \frac{\widehat{\alpha} \overline{u\_{s}' {\boldsymbol{w}\_{s}'}}}{\widehat{\alpha} \overline{z}} \right| &+ \frac{\partial}{\partial \overline{z}} \left\{ \frac{\alpha \tau\_{p}}{3} \left[ \left( \overline{u\_{s}'^{2}} + \mathbf{g}\_{u}^{l} \overline{u'^{2}} \right) \frac{\widehat{\alpha} {\boldsymbol{w}\_{s}'}^{\boldsymbol{Z}}}{\widehat{\alpha} \mathbf{x}} \right. \right. \\ &+ \left( \overline{u\_{s}' {\boldsymbol{w}\_{s}'}} + \mathbf{g}\_{u}^{n} {\boldsymbol{u}' {\boldsymbol{w}\_{s}'}} \right) \frac{\widehat{\alpha} {\boldsymbol{w}\_{s}'}^{\boldsymbol{Z}}}{\widehat{\alpha} \mathbf{y}} + \left( \overline{u\_{s}' {\boldsymbol{w}\_{s}'}} + \mathbf{g}\_{u}^{k} {\boldsymbol{u}' {\boldsymbol{w}\_{s}'}} \right) \frac{\widehat{\alpha} {\boldsymbol{w}\_{s}'}^{\boldsymbol{Z}}}{\widehat{\alpha} \mathbf{z}} \\ &+ 2 \left( \overline{u\_{s}' {\boldsymbol{w}\_{s}'}} + \mathbf{g}\_{u}^{k} {\boldsymbol{u}' {\boldsymbol{w}\_{s}'}} \right) \frac{\widehat{\alpha} {\boldsymbol{w}\_{s}'}^{\boldsymbol{w}\_{s}'} + \mathbf{g}\_{u}^{n} {\boldsymbol{w}\_{s}'} \overline{\widehat{\alpha} {\boldsymbol{w}\_{s}'}^{\boldsymbol{W}}}} \left| \frac{\partial \overline{u\_{s}'} {\boldsymbol{w}\_{s}'}}{\partial {\boldsymbol{\mathcal{V}}}} \right| \right), \end{split} \tag{9}$$

Equation of the *yz* shear stress component of the Reynolds stress:

2 2

*<sup>p</sup> <sup>l</sup> s s sss s u*

*u w uuw u g u x x*

¶ é ù ¶ ¢ ¢ æ ö ê ú ¢¢ ¢ ¢ - + ç ÷ ¶ ¶ è ø ë û

¶ é ù ¶ ¢ ¢ æ ö + -+ ê ú ¢¢ ¢ ¢ ç ÷ ¶ ¶ è ø ë û

*<sup>p</sup> <sup>n</sup> s s sss s u*

2 2 2

*<sup>p</sup> <sup>k</sup> s s sss s u*

2 2

*uw uw u v g uv <sup>x</sup> y z*

*<sup>p</sup> <sup>n</sup> ss ss ss u*

2

( )

2 2 2 2

*y z*

*<sup>p</sup> l n s s s s s u ss u*

*y xy*

*s s n k s s* 2 2 *s s ss u s u*

*u v u v u v v w g vw w g w x yz*

¶ ¶ ¢ ¢ ¢ ¢ ¢ ¢ æ ö ´ + + ++ ¢ ¢ ¢¢ ¢ ¢ ç ÷ ¶ ¶¶ è ø

<sup>é</sup> ¶ ¶ ¢ ¢ ¢ ¢ æ ö + + ++ <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢ ¢ ¢¢ ç ÷ ¶ ¶¶ <sup>ï</sup> è ø <sup>î</sup> êë

*v w v w u gu u v g uv*

*u w vuw v g v y y*

2 3

a

348 Numerical Simulation - From Brain Imaging to Turbulent Flows

a

3

( )

3

¶ ìïat at

a

¶ ì

t

3

( )

<sup>ï</sup> <sup>é</sup> ¶ ¶ ¢¢ ¢¢ = ++ <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¶ <sup>ê</sup> ¶ ¶ ïî <sup>ë</sup>

*u w wuw w g w z z*

¶ é ù ¶ ¢ ¢ æ ö + -+ ê ú ¢¢ ¢ ¢ ç ÷ ¶ ¶ è ø ë û

3

( ) ( )

( ) ( )

*v w u w g uw u w g uw <sup>z</sup>*

*n n s s ss u ss u*

( )

( ) ( )

*u w u v g uv v w g vw <sup>x</sup>*

*l k s s ss u ss u*

¶ ¢ ¢ ++ + + ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ ¶

¶ ¢ ¢ ++ ++ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¶

2 2

¶ ¢ ´+ + + ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¶

*k l <sup>s</sup> ss u ss u*

*<sup>u</sup> u w g uw u w g uw <sup>x</sup>*

*n k s s ss u s u*

ùü ¶ ¶ ¢ ¢ æ ö <sup>ï</sup> + + ++ ¢ ¢ ¢¢ ¢ ¢ <sup>ú</sup> ç ÷ <sup>ý</sup> ¶ ¶ è ø úïûþ

*u u v w g vw w g w*

t

t

2 2

2 2 2 2 3 2 3 *<sup>p</sup> <sup>l</sup> s s ss s s u <sup>p</sup> <sup>n</sup> s s ss s s u v w uvw u g u x x v w vvw v g v y y* t a t a ¶ é ù ¶ ¢ ¢ æ ö ê ú ¢¢ ¢ ¢ - + ç ÷ ¶ ¶ è ø ë û ¶ é ù ¶ ¢ ¢ æ ö + -+ ê ú ¢¢ ¢ ¢ ç ÷ ¶ ¶ è ø ë û ( ) 2 2 2 2 2 3 3 *<sup>p</sup> <sup>k</sup> s s ss s s u <sup>p</sup> l n s s ss u s u v w wvw w g w z z u w u v g uv v g v x x* t a at ¶ é ù ¶ ¢ ¢ æ ö + -+ ê ú ¢¢ ¢ ¢ ç ÷ ¶ ¶ è ø ë û ¶ ì <sup>ï</sup> <sup>é</sup> ¶ ¢ ¢ æ ö = + ++ <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¢ ¢ ç ÷ ¶ ¶ è ø ïî êë ( ) ( ) ( ) ( ) 2 2 *s s k l s s s s ss u ss u n k s s ss u s u u w u w u v vw g vw uw g uw yzx u v vw g vw w g w y* ¶¶¶ ¢ ¢ ¢ ¢ ¢ ¢ ´ ++ ++ ¢¢ ¢¢ ¢¢ ¢¢ ¶¶¶ ¶ ¢ ¢ + + ++ ¢¢ ¢¢ ¢ ¢ ¶ ( ) ( ) <sup>2</sup>( ) <sup>3</sup> *s s n k s s s s ss u ss u <sup>p</sup> <sup>l</sup> s s ss u u v v w v w u v g uv u w g uw zy z v w u v g uv y x* at ùü ¶¶ ¶ ¢ ¢ ¢ ¢ ¢ ¢ <sup>ï</sup> ´ ++ + + ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ úý ¶¶ ¶ úûïþ ¶ ì <sup>ï</sup> <sup>é</sup> ¶ ¢ ¢ + + <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¶ ¶ ïî êë

( ) ( ) ( ) ( ) 2 2 2 2 2 2 *k l s s <sup>s</sup> ss u ss u n k s s ss u s u v w <sup>v</sup> vw g vw uw g uw z x v v vw g vw w g w y z* ¶ ¶ ¢ ¢ ¢ + + ++ ¢¢ ¢¢ ¢¢ ¢¢ ¶ ¶ ùü ¶ ¶ ¢ ¢ <sup>ï</sup> + + ++ ¢¢ ¢¢ ¢ ¢ úý ¶ ¶ úûïþ ( ) ( ) 2 2 2 2 3 *<sup>p</sup> <sup>l</sup> <sup>s</sup> ss u n k s s s u ss u <sup>w</sup> u v g uv z x w w v g v v w g vw y z* at <sup>ì</sup> <sup>é</sup> ¶ <sup>ï</sup> ¶ ¢ + + <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¶ ¶ <sup>ê</sup> <sup>ï</sup> î ë ¶ ¶ ¢ ¢ æ ö ++ + + ¢ ¢ ¢ ¢ ¢¢ ç ÷ è ø ¶ ¶ 22, ( ) ( ) *k n s s s s ss u ss u v w v w u w g uw v w g vw x y* ùü ¶ ¶ ¢ ¢ ¢ ¢ <sup>ï</sup> ++ ++ ¢ ¢ ¢¢ ¢ ¢ ¢¢ úý ¶ ¶ úûïþ (10)

where *g*<sup>u</sup> *l* , *g*<sup>u</sup> *<sup>n</sup>*, and *<sup>g</sup>*<sup>u</sup> *k* are the coefficients characterizing the entrainment of particles into the fluctuating motion of the flow [9] for *x, y*, and *z* directions, respectively.

#### **2.2. Boundary conditions**

The wall conditions are set for the gas at the side walls of the channel based on the control volume method by [27, 28] in a similar way as in the case of the coincidence of shear of the mean flow velocity and gravity.

The numerical simulation considers the turbulent dispersion of solid particles in horizontal channel uniform shear turbulent flow for two different cases: i) shear of the mean flow velocity is along the direction of gravity (**Figure 1a**) and ii) shear of the mean flow velocity is direct‐ ed normally to gravity (**Figure 1b**). Therefore, two sets of the boundary conditions are used for the calculations.

The boundary conditions for the particulate phase are set at the flow axis as follows:

Case 1 for *z* = 0:

$$\frac{\partial u\_s}{\partial \mathbf{z}} = \mathbf{v}\_s = \mathbf{w}\_s = \frac{\partial \overline{u\_s'^2}}{\partial \mathbf{z}} = \frac{\partial \overline{v\_s'^2}}{\partial \mathbf{z}} = \frac{\partial \overline{u\_s'^2}}{\partial \mathbf{z}} = \overline{u\_s' \mathbf{v}\_s'} = \overline{u\_s' \mathbf{w}\_s'} = \overline{\mathbf{v}\_s' \mathbf{w}\_s'} = \frac{\partial \alpha}{\partial \mathbf{z}} = \mathbf{0}.\tag{11}$$

Case 2 for *y* = 0:

Two-Fluid RANS-RSTM-PDF Model for Turbulent Particulate Flows http://dx.doi.org/10.5772/63338 351

$$\frac{\partial u\_s}{\partial \mathbf{y}} = \mathbf{v}\_s = \mathbf{w}\_s = \frac{\partial \overline{u\_s'^2}}{\partial \mathbf{y}} = \frac{\partial \overline{v\_s'^2}}{\partial \mathbf{y}} = \frac{\partial \overline{u\_s'^2}}{\partial \mathbf{y}} = \overline{u\_s' \mathbf{v}\_s'} = \overline{u\_s' \mathbf{w}\_s'} = \overline{\mathbf{v}\_s' \mathbf{w}\_s'} = \frac{\partial \alpha}{\partial \mathbf{y}} = \mathbf{0} \tag{12}$$

The boundary conditions for the particulate phase are set at the channel walls according to [9]: Case 1 for *y* = 0.5*hy*:

$$\frac{\partial u\_s}{\partial \mathbf{y}} = 2 \left( \frac{1 - \mathcal{X}e\_\mathbf{x}}{\mathbf{l} + \mathcal{X}e\_\mathbf{x}} - \frac{\mathbf{l} - \mathcal{X}}{\mathbf{l} + \mathcal{X}} \right) \left( \frac{\mathbf{2}}{\pi \overline{\mathbf{v}\_s'^2}} \right)^{0.5} \frac{u\_s}{\tau\_p}, \tag{13}$$

$$
\omega\_s = \omega\_s = \frac{\partial \alpha}{\partial \mathbf{y}} = \mathbf{0},
\tag{14}
$$

$$P\_{\rm uus} = -\overline{u\_s' v\_s'} \frac{\partial u\_s}{\partial \mathbf{y}} = P\_{\rm uus} = \eta\_x \overline{v\_s'^2} \frac{\partial u\_s}{\partial \mathbf{y}},\tag{15}$$

and applying the expression *u* ′ s*v* ′ s ¯ <sup>=</sup> <sup>−</sup>*ηx<sup>v</sup>* ′ s ¯2 , where *ηx* is the coefficient of friction between the particles and the wall,

$$P\_{\nu\nu s} = P\_{\nu\nu\nu s} = P\_{\mu\nu s} = P\_{\mu\nu s} = P\_{\nu\nu s} = 0\_{\ast}$$

Case 2 for z = 0.5*hz*:

( ) ( )

¶ ¶ ¢ ¢ ¢ + + ++ ¢¢ ¢¢ ¢¢ ¢¢ ¶ ¶

2 *k l s s <sup>s</sup> ss u ss u*

*v w <sup>v</sup> vw g vw uw g uw z x*

*n k s s*

2 2 2 2

( )

*y z*

*x y*

2 2

are the coefficients characterizing the entrainment of particles into the

a

2

*y z*

2

(10)

(11)

( ) ( )

*v v vw g vw w g w*

( )

*<sup>w</sup> u v g uv z x*

<sup>ì</sup> <sup>é</sup> ¶ <sup>ï</sup> ¶ ¢ + + <sup>í</sup> <sup>ê</sup> ¢ ¢ ¢¢ ¶ ¶ <sup>ê</sup> <sup>ï</sup>

*<sup>p</sup> <sup>l</sup> <sup>s</sup> ss u*

*n k s s s u ss u*

*w w v g v v w g vw*

¶ ¶ ¢ ¢ æ ö ++ + + ¢ ¢ ¢ ¢ ¢¢ ç ÷ è ø ¶ ¶

22, ( ) ( ) *k n s s s s ss u ss u v w v w u w g uw v w g vw*

The wall conditions are set for the gas at the side walls of the channel based on the control volume method by [27, 28] in a similar way as in the case of the coincidence of shear of the

The numerical simulation considers the turbulent dispersion of solid particles in horizontal channel uniform shear turbulent flow for two different cases: i) shear of the mean flow velocity is along the direction of gravity (**Figure 1a**) and ii) shear of the mean flow velocity is direct‐ ed normally to gravity (**Figure 1b**). Therefore, two sets of the boundary conditions are used

The boundary conditions for the particulate phase are set at the flow axis as follows:

0. *s ss s s s ss s s s s u uvw v w uv uw vw z zzz <sup>z</sup>* ¶ ¶¶¶ ¢¢ ¢ ¶

== = = = = = = = = ¢¢ ¢ ¢ ¢ ¢ ¶ ¶¶¶ ¶

22 2

ùü ¶ ¶ ¢ ¢ ¢ ¢ <sup>ï</sup> ++ ++ ¢ ¢ ¢¢ ¢ ¢ ¢¢ úý ¶ ¶ úûïþ

2

350 Numerical Simulation - From Brain Imaging to Turbulent Flows

where *g*<sup>u</sup> *l* , *g*<sup>u</sup>

*<sup>n</sup>*, and *<sup>g</sup>*<sup>u</sup> *k*

mean flow velocity and gravity.

**2.2. Boundary conditions**

for the calculations.

Case 1 for *z* = 0:

Case 2 for *y* = 0:

3

î ë

fluctuating motion of the flow [9] for *x, y*, and *z* directions, respectively.

at

ùü ¶ ¶ ¢ ¢ <sup>ï</sup> + + ++ ¢¢ ¢¢ ¢ ¢ úý ¶ ¶ úûïþ

*ss u s u*

$$\frac{\partial u\_s}{\partial \mathbf{y}} = 2 \left( \frac{1 - \mathcal{X}e\_\times}{1 + \mathcal{X}e\_\times} - \frac{1 - \mathcal{X}}{1 + \mathcal{X}} \right) \left( \frac{2}{\pi \overline{\mathbf{w}\_s'^2}} \right)^{0.5} \frac{u\_s}{\tau\_p} \tag{16}$$

$$
\omega\_s = \omega\_s = \frac{\partial \alpha}{\partial \mathbf{z}} = \mathbf{0},
\tag{17}
$$

$$P\_{\rm uus} = -\overline{u\_s' \mathcal{W}\_s'} \frac{\partial u\_s}{\partial \mathbf{z}} = \eta\_\chi \overline{\mathcal{W}\_s'^2} \frac{\partial u\_s}{\partial \mathbf{z}},\tag{18}$$

$$P\_{\nu\nu s} = P\_{\nu\nu s3} = P\_{\mu\nu s} = P\_{\mu\nu s} = P\_{\nu\nu s} = 0. \tag{19}$$

*ex* is the coefficient of restitution in the axial direction, which is modeled as:

$$e\_{\mathbf{x}} = \begin{cases} 1 - \xi\_{\mathbf{x}}, & 0 \le \xi\_{\mathbf{x}} \le \frac{2}{7} \\\\ \frac{\mathfrak{g}}{7}, & \xi\_{\mathbf{x}} > \frac{2}{7} \end{cases} \tag{20}$$

Here, the parameter *ξ<sup>x</sup>* =*ηx*(1 + *e*r)*tanθx*, where *e*<sup>r</sup> is the coefficient of restitution of the particle velocity normal to the wall; *θ<sup>x</sup>* =tan−<sup>1</sup> (*v*<sup>s</sup> / *u*s) is the angle of attack between the trajectory of the particle and the wall; *χ* is the reflection coefficient, which is the probability of the particles recoiling off the boundaries and back to the flow. The coefficient of restitution reflects the loss of the particle momentum as the particle hits the walls. In the given model, *χ*= 1/3, *e*<sup>r</sup> = 1 and *η<sup>x</sup>* = 0.39 [29].

The conditions for the transverse and spanwise components of the gas velocity are set at the channel walls in terms of impenetrability and no-slip.

The set of boundary conditions for gas and particulate phase at the exit of the channel is written, respectively, as follows:

$$\frac{\partial u\_s}{\partial \mathbf{x}} = \frac{\partial v\_s}{\partial \mathbf{x}} = \frac{\partial w\_s}{\partial \mathbf{x}} = \frac{\partial \alpha}{\partial \mathbf{x}} = \frac{\partial u\_s'^2}{\partial \mathbf{x}} = \frac{\partial v\_s'^2}{\partial \mathbf{x}} = \frac{\partial w\_s'^2}{\partial \mathbf{x}} = \frac{\partial \overline{u\_s' \mathbf{v}\_s'}}{\partial \mathbf{x}} = \frac{\partial \overline{u\_s' \mathbf{w}\_s'}}{\partial \mathbf{x}} = \frac{\partial \overline{v\_s' \mathbf{w}\_s'}}{\partial \mathbf{x}} = 0. \tag{21}$$

#### **2.3. Computational method**

The control volume method was applied to solve the 3D partial differential equations written for the unladen flow and the particulate phase (Eqs. (1)–(11)), taking into account the boun‐ dary conditions (Eqs. (12)–(21)). The governing equations were solved using the implicit lower and upper (ILU) matrix decomposition method with the flux-blending-differed correction and upwind-differencing schemes by [27]. This method is utilized for the calculations of the particulate turbulent flows in channels of the rectangular and square cross sections. The calculations were performed in the dimensional form for all the flow conditions. The num‐ ber of the control volumes was 1120000.

### **3. Laboratory experiments**

The obtained numerical results have been verified and validated in comparison with the data obtained by the experimental facility of Tallinn University of Technology.

The experimental method for the determination of the particle dispersion was based on recording the particle trajectories by means of a high-speed video camera on separate re‐ gions of a flow that locate at various distances from a point source of particles, and the subsequent processing of the frames [30].

The experimental setup for the investigations of particle dispersion (**Figure 2**) allowed to generate the shear flow similarly to [31] by means of flat plates installed with a varied pitch. The test section was 2 m long with 400 × 200 mm cross section.

**Figure 2.** Experimental setup.

<sup>2</sup> 1 , 0 <sup>7</sup>

*x x*

particle and the wall; *χ* is the reflection coefficient, which is the probability of the particles recoiling off the boundaries and back to the flow. The coefficient of restitution reflects the loss

The conditions for the transverse and spanwise components of the gas velocity are set at the

The set of boundary conditions for gas and particulate phase at the exit of the channel is written,

The control volume method was applied to solve the 3D partial differential equations written for the unladen flow and the particulate phase (Eqs. (1)–(11)), taking into account the boun‐ dary conditions (Eqs. (12)–(21)). The governing equations were solved using the implicit lower and upper (ILU) matrix decomposition method with the flux-blending-differed correction and upwind-differencing schemes by [27]. This method is utilized for the calculations of the particulate turbulent flows in channels of the rectangular and square cross sections. The calculations were performed in the dimensional form for all the flow conditions. The num‐

The obtained numerical results have been verified and validated in comparison with the

data obtained by the experimental facility of Tallinn University of Technology.

 x

(20)

= 1 and *η<sup>x</sup>*

(21)

is the coefficient of restitution of the particle

(*v*<sup>s</sup> / *u*s) is the angle of attack between the trajectory of the

5 2 , 7 7

x

of the particle momentum as the particle hits the walls. In the given model, *χ*= 1/3, *e*<sup>r</sup>

22 2 0. *ss s s s s ss s s s s u v w u v w uv uw vw xxxxx x x x x x* ¶¶¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ = = == = = = = = = ¶¶¶¶¶ ¶ ¶ ¶ ¶ ¶

<sup>ï</sup> <sup>&</sup>gt; ïî

x

*x*

<sup>ì</sup> - ££ ïï <sup>=</sup> <sup>í</sup>

*x*

*e*

Here, the parameter *ξ<sup>x</sup>* =*ηx*(1 + *e*r)*tanθx*, where *e*<sup>r</sup>

352 Numerical Simulation - From Brain Imaging to Turbulent Flows

channel walls in terms of impenetrability and no-slip.

a

velocity normal to the wall; *θ<sup>x</sup>* =tan−<sup>1</sup>

= 0.39 [29].

respectively, as follows:

**2.3. Computational method**

ber of the control volumes was 1120000.

**3. Laboratory experiments**

Two cases of spatial orientation of shear of the mean flow velocity were investigated. **Figure 2** shows the top view of the setup for the case when shear is along the direction of gravity (**Figure 1a**). For investigations of the particle dispersion when shear is directed normally to gravity (**Figure 1b**), the setup was turned sideways as a whole at an angle of *90°* around the axis of the flow.

The mean flow velocity was 5.1 m/s. Glass spherical particles (physical density of 2500 kg/m3 ) with an average diameter of 55 μm were used in the experiment runs. The root-mean-square deviation of the diameter of particles did not exceed 0.1. The particles were entered into the flow through the source point which was the L-shaped tubule of 200 μm inner diameter.

All measurements and data processing were carried out at the flow location *x* = 1212 mm.

The data processing technique [30] was applied to determine the particle spatial displace‐ ment along the *y*-axis, namely *Dy*, which characterizes quantitatively the particle turbulent dispersion. *Dy* is calculated as the axial displacement of the maximum value of distribution of the particle mass concentration determined at the location *x* = 1212 mm relative to the initial flow location that disposes near the exit of the source point.
