Fluid-Fluid Drag Coefficients

Simcenter STAR-CCM+ provides various drag coefficient methods.

Schiller-Naumann Drag Coefficient for Spherical Particles

Simcenter STAR-CCM+ selects the Schiller-Naumann drag law internally according to the type of continuous fluid.

For Newtonian fluids, the drag coefficient of spherical, rigid particles is computed based on the correlation of Schiller and Naumann [543]:

$$C_D=\{\begin{array}{c}\frac{24}{{\text{Re}}_d}(1+0.15{\text{Re}}_d^{0.687})\text{ 0}<{\text{Re}}_d\le 1000\\ 0.44{\text{Re}}_d>1000\end{array}$$

(1946)

The dispersed phase Reynolds number, ${\text{Re}}_d$ is defined as:

$${\text{Re}}_d=\frac{{\rho }_c\left|{\mathbf{\text{v}}}_r\right|l}{{\mu }_c}$$

(1947)

where:

• ${\rho }_c$ is the density of the continuous phase
• ${\mu }_c$ is the dynamic viscosity of the continuous phase
• $l$ is interaction length scale or bubble size.

Schiller-Naumann with Non-Newtonian Herschel–Bulkley Continuous Phase

If the continuous phase is non-Newtonian, the Schiller-Naumann drag is modified. From, [429], the drag coefficient of spherical, rigid particles in a Herschel–Bulkley (or Power Law) continuous phase is:

$$C_D=\frac{24}{{\text{Re}}_{HB}}(1+0.15{\text{Re}}_{HB}^{0.687})$$

(1948)

The dispersed phase Reynolds number, ${\text{Re}}_{HB}$ is defined as:

$${\text{Re}}_{HB}=\frac{{\text{Re}}_{PL}}{1+\frac{7\pi }{24}{\text{Bi}}_{HB}}$$

(1949)

The particle Bingham number, ${\text{Bi}}_{HB}$ , is defined as:

$${\text{Bi}}_{HB}=\frac{{\tau }_0}{K}{\left(\frac{d}{\left|{\mathbf{\text{v}}}_r\right|}\right)}^n$$

(1950)

The Power-Law Reynolds number is:

$${\text{Re}}_{PL}=\frac{{\rho }_c{\left|{\mathbf{\text{v}}}_r\right|}^{2-n}{d}^n}{K}$$

(1951)

where:

• ${\tau }_0$ is the yield stress
• $K$ is the consistency factor
• $n$ is the power-law component of the model
• $\left|{\mathbf{\text{v}}}_r\right|$ is the relative slip velocity between phases
• $d$ is the particle diameter.

Schiller-Naumann with Non-Newtonian Generalized Carreau-Yasuda Continuous Phase

If the continuous phase is non-Newtonian, the drag coefficient of spherical, rigid particles in a Carreau-Yasuda continuous phase is:

$$C_D=\frac{24}{{\text{Re}}_{HB}}(1+0.15{\text{Re}}_0^{0.687})(1+0.65\left(n-1\right){\lambda }^{0.20})$$

(1952)

where $n$ and $\lambda$ are parameters in the Carreau model [440].

${\text{Re}}_0$ is the zero-shear particle Reynolds number that uses the zero-shear viscosity of the fluid:

$${\text{Re}}_0=\frac{{\rho }_c\left|{\mathbf{\text{v}}}_r\right|d}{{\mu }_0}$$

(1953)

where:

• ${\mu }_0$ is the zero shear viscosity of the continuous phase
• $d$ is the interaction length scale or bubble size
• ${\rho }_c$ is the density of the continuous phase
• $\left|{\mathbf{\text{v}}}_r\right|$ is the magnitude of the relative velocity between the phases.

Rusche-Issa Drag Coefficient

The Rusche-Issa drag coefficient applies when the dispersed phase volume fraction becomes high. In order to account for the presence of a larger number of particles that increase drag, Rusche and Issa created a drag correction term that ranges from one at zero dispersed phase volume fraction and increases exponentially [537].

$$f\left({\alpha }_d\right)=e^{K_1{\alpha }_d}+{\alpha }_d^{K_2}$$

(1954)

The Rusche-Issa drag law uses a linearized Schiller-Naumann form combined with the Rusche-Issa correction.

Rusche [536] studied a series of experiments from academic literature for dispersed phases that were gas, liquid, and solid particles, and found correlations to fit the drag coefficient [538]. The averages of the experiments give the parameters that are summarized below:

Dispersed Phase	Gas Bubbles	Liquid Droplets	Solid Particles
K1	3.64	2.10	2.68
K2	0.864	0.249	0.430

Symmetric Drag Coefficient

The symmetric drag coefficient provides the linearized coefficient as:

$$A_{cd}^D=\frac{3}{4}{C}_D\frac{{\alpha }_c{\alpha }_d({\alpha }_c{\rho }_c+{\alpha }_d{\rho }_d)}{({\alpha }_c{l}_c+{\alpha }_d{l}_d)}\left|{\mathbf{\text{v}}}_r\right|$$

(1955)

where:

$l_c$ is the interaction length scale

$l_d$ is the inverted topology length scale

$$C_D=\{\begin{array}{cc}\frac{24}{R{e}_{cd}}(1+0.15R{e}_{cd}^{0.687})& 0<R{e}_{cd}\le 1000\\ 0.44& R{e}_{cd}>1000\end{array}$$

(1956)

with,

$$R{e}_{cd}=\frac{({\alpha }_c{\rho }_c+{\alpha }_d{\rho }_d)({\alpha }_c{l}_c+{\alpha }_d{l}_d)\left|{\mathbf{\text{v}}}_r\right|}{({\alpha }_c{\mu }_c+{\alpha }_d{\mu }_d)({\alpha }_c+{\alpha }_d)}$$

(1957)

Bozzano-Dente Drag Coefficient for Bubbles

This correlation covers a wide range of bubbly flow regimes and accounts for different bubble shapes with a single expression. It has been developed for the case of a single bubble rising in a liquid column under the action of gravity. It takes both Eotvos number and Morton number into account, so can be applied to highly viscous systems such as glycerol, molten glass and some oils.

It can also be used for low pressure and high pressure water systems.

Note	For small bubbles, this correlation gives a different asymptote (Re/48) for the drag coefficient when compared to the values normally associated with contaminated or pure water systems (Re/24 or Re/16).

For this model, from Bozzano and Dente [436], the drag coefficient is:

$$C_D=f{\left(\frac{a}{R_0}\right)}^2$$

(1958)

In which $f$ is a friction factor and ${(a/R_0)}^2$ is a deformation factor. These quantities are derived as

$$f=\left(\frac{48}{\text{Re}}\right)\frac{1+12{\text{Mo}}^{1/3}}{1+36{\text{Mo}}^{1/3}}+0.9\frac{{\text{Eo}}^{3/2}}{1.4(1+30{\text{Mo}}^{1/6})+{\text{Eo}}^{3/2}}$$

(1959)

and

$${\left(\frac{a}{R_0}\right)}^2\cong \frac{10(1+1.3{\text{Mo}}^{1/6})+3.1\text{Eo}}{10(1+1.3{\text{Mo}}^{1/6})+\text{Eo}}$$

(1960)

Where $\text{Re}$ is the Reynolds number, $\text{Mo}$ is the Morton number, and $\text{Eo}$ is the Eotvos number.

$$\text{Re}=\frac{{\rho }_c\left|{\mathbf{\text{v}}}_r\right|l}{{\mu }_c}$$

(1961)

$$\text{Eo}=\frac{\left|{\rho }_c-{\rho }_d\right|g{l}^2}{\sigma }$$

(1962)

$$\text{Mo}=\frac{\left|{\rho }_c-{\rho }_d\right|{g\mu }_c^4}{{\rho }_c{\rho }_d{\sigma }^3}$$

(1963)

• ${\rho }_c$ and ${\mu }_c$ are the density and dynamic viscosity of the continuous phase
• ${\rho }_d$ is the density of the dispersed phase
• $\left|{\mathbf{\text{v}}}_r\right|$ is the magnitude of the relative velocity between the phases
• $l$ is interaction length scale or bubble size
• $g$ is gravitational acceleration
• $\sigma$ is the surface tension.

Hamard and Rybczynski Drag Coefficient for Droplets

The Hamard and Rybczynski drag model is used for viscous Newtonian fluid droplets dispersed in a second immiscible viscous Newtonian fluid. From [440], the drag coefficient is:

$$C_D=\frac{24}{\text{Re}}Y$$

(1964)

where:

$\text{Re}$ is the particle Reynolds number

where:

$$Y=\frac{2+3X_E}{3+3X_E}$$

(1965)

$$X_E=\frac{{\mu }_d}{{\mu }_c}$$

(1966)

where ${\mu }_c$ is the dynamic viscosity of the continuous phase and ${\mu }_d$ is the dynamic viscosity of the dispersed phase.

For gas bubbles, $X_E«1$ and $Y\to 2/3$ . For solid particles $Y\to 1$ .

The Hamard and Rybczynski drag model is not valid for high Reynolds numbers, where the Schiller-Naumann model is expected to be more accurate.

Tomiyama Drag Coefficient for Bubbles

The Tomiyama correlation is suited to systems for a wide range of conditions, as shown by the range of Eotvos ( $\text{Eo}$ ), Morton ( $M\text{o}$ ), and Reynolds ( $\text{Re}$ ) numbers for which it is valid:

$$\begin{array}{c}{10}^{-2}<Eo<{10}^3\\ {10}^{-14}<Mo<{10}^7\\ {10}^{-3}<Re<{10}^5\end{array}$$

(1967)

The drag coefficient that is evaluated for a single bubble, which is based upon three contamination states, is from [557]:

• Pure:

$$C_D=\text{max}[\text{min}(\frac{16}{\text{Re}}(1+0.15{\text{Re}}^{0.687}),\frac{48}{\text{Re}}),\frac{8\text{Eo}}{3(\text{Eo}+4)}]$$

(1968)

• Moderate:

$$C_D=\text{max}[\text{min}(\frac{24}{\text{Re}}(1+0.15{\text{Re}}^{0.687}),\frac{72}{\text{Re}}),\frac{8\text{Eo}}{3(\text{Eo}+4)}]$$

(1969)

• Contaminated:

$$C_D=\text{max}[\left(\frac{24}{\text{Re}}(1+0.15{\text{Re}}^{0.687})\right),\frac{8\text{Eo}}{3(\text{Eo}+4)}]$$

(1970)

where $\text{Re}$ is the Reynolds number and $\text{Eo}$ is the Eotvos number.

$$\text{Re}=\frac{{\rho }_c\left|{\mathbf{\text{v}}}_r\right|l}{{\mu }_c}$$

(1971)

$$\text{Eo}=\frac{\left|{\rho }_c-{\rho }_d\right|g{l}^2}{\sigma }$$

(1972)

• ${\rho }_c$ is the density of the continuous phase
• ${\mu }_c$ is the dynamic viscosity of the continuous phase
• ${\rho }_d$ is the density of the dispersed phase
• $\left|{\mathbf{\text{v}}}_r\right|$ is the magnitude of the relative velocity between the phases
• $l$ is interaction length scale or bubble size
• $g$ is gravitational acceleration
• $\sigma$ is the surface tension.

The Tomiyama drag coefficient was used by Tomiyama et al. [558] while evaluating their lift coefficients. Therefore, use the Tomiyama lift coefficient with the Tomiyama drag coefficient.

Wang Drag Coefficient for Bubbles

This model is for concentrations of air bubbles in water at near atmospheric pressure. Use with care for other conditions.

Bubbles deform and depart from the spherical shape. Use the empirical correlation of Wang et al. [567], to obtain more realistic values. The correlation was derived using curve-fitting measurements for a single bubble rising in water:

$$C_D=e^{[a+b\ln {\text{Re}}_d+c{\left(\ln {\text{Re}}_d\right)}^2]}$$

(1973)

Where ${\text{Re}}_d$ is the dispersed phase Reynolds number.

The original coefficients are given in the table below.

${\mathbf{\text{Re}}}_d$	$\mathbf{\text{a}}$	$\mathbf{\text{b}}$	$\mathbf{\text{c}}$
${\mathbf{\text{Re}}}_d\le 1$	$\ln 24$	$-1$	$0$
$1<{\mathbf{\text{Re}}}_d\le 450$	$2.699467$	$-0.33581596$	$-0.07135617$
$450<{\mathbf{\text{Re}}}_d\le 4000$	$-51.77171$	$13.1670725$	$-0.8235592$
${\mathbf{\text{Re}}}_d>4000$	$\ln (8/3)$	$0$	$0$

The Simcenter STAR-CCM+ implementation has been modified from Wang’s original method [567] so that the drag coefficient is continuous as particle Reynolds number changes. This is important to minimize the residual achievable in applications with variable particle diameter, and the modified method gives reliable convergence with normal values of relaxation factors.

The modifications are:

• The breakpoint between the first two Re regimes in the above table has been moved from Re=1 to Re=2.197115.
• The last two Re regimes in the above table have been replaced using:

$$C_D=\frac{8}{3}\frac{{\text{Eo}}_{AW}}{{\text{Eo}}_{AW}+4}$$

(1974)

For close equivalence with the original Wang curve fit, the Eotvos number is here computed from bubble diameter $d$ using water and air properties that are fixed at a temperature of 293K.

$${\text{Eo}}_{AW}=\frac{\left({\rho }_W-{\rho }_A\right)g{d}^2}{{\sigma }_{AW}}$$

(1975)

• ${\rho }_W$ is 998.20 kg/m³.
• ${\rho }_A$ is 1.2069 kg/m³.
• $g$ is 9.81 m/s².
• ${\sigma }_{AW}$ is 0.072736 N/m.

Štrubelj and Tiselj Interface Drag Coefficient

For the intermediate regime, the method of Štrubelj and Tiselj [549] is adapted to calculate the interface linearized drag coefficient:

$$\begin{array}{ccc}{A}_{D,ir}=f_{ir}{\alpha }_p{\alpha }_s{\rho }_m& {\rho }_m={\alpha }_p{\rho }_p+{\alpha }_s{\rho }_s& f_{ir}=1/t_{sc}\end{array}$$

(1976)

• The default value of the relaxation time scale t_sc is taken as 0.01 seconds.
• Reduction of relaxation time scale helps in instantaneous equalizing of velocities of both of the phases as it leads to increased drag in the intermediate regime.
• By increasing the drag in the intermediate regime, the slip in that region can be reduced.

Blended Interface Drag Coefficient

The blended method calculates the intermediate regime drag by blending the drags of the dispersed regime on either side of the intermediate regime. This method is useful when a clear separated interface is not defined in the intermediate regime.

Within the intermediate regime, the Linearized drag ( $A_d$ ) is calculated as:

$$A_{d,ir}={\alpha }_p{A}_{d,fr}+{\alpha }_s{A}_{d,sr}$$

(1977)

where the subscripts $fr$ , $ir$ and $sr$ are for the first regime, the intermediate regime, and the second regime respectively.