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  • Particles in a Periodic Double Shear Flow - Pre-Analysis & Start-Up

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Pre-Analysis & Start-Up

In the Pre-Analysis & Start-Up step, we'll review the following:

  • Mathematical Model: We will look at the governing equations, boundary conditions, initial field velocity function of the jet as well as the formula for calculating Stokes number in this case.
  • Expected Results: We will discuss the expected results from this simulation and compare the numerical results with the expected results.

Mathematical Model 

Governing Equations:

In almost any fluid dynamics problem, the most important governing equation has to be the Navier-Stokes equation and continuity equation. The two equations govern the fluid flow. Here we will list the Navier-Stokes equations but we will not go into further details.

Continuity Equation:

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\begin

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\frac

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+ \nabla \cdot (\rho \textbf

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\Rightarrow \nabla \cdot \textbf

= 0
\end

Navier-Stokes Equation:

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\begin

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\rho (\frac{d \textbf{v}}

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+\textbf

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)=- \nabla p + \mu \nabla ^2 \textbf

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+ \textbf

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\end

In this case, however, we have discrete particles in the flow. Since we are using a one-way coupled scheme, the fluid imposes a force on the particles, but not vice versa. By using the particle force balance equation listed below, the particle movement can be calculated by integrating the acceleration term.

Particle Force Balance:

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\begin

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&\frac

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= F_D(\overrightarrow

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-\overrightarrow

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)+\frac{\overrightarrow

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(\rho_p-\rho)}

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+\overrightarrow

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\end

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$\overrightarrow

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$

 is an additional acceleration (force per unit particle mass) term. 
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$F_D(\overrightarrow

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-\overrightarrow

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)$

 is the drag force per unit particle mass.

 

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 can be calculated using the formula below:

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\begin

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F_D = \frac

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\frac

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\end

Here, 

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$\overrightarrow

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$

 is the fluid phase velocity, 
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$\overrightarrow

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$

 is the particle velocity, 
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 is the molecular viscosity of the fluid, 
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 is the fluid density, 
Unknown macro: {latex} $\rho_p$
 is the density of the particle, and 
Unknown macro: {latex} $d_p$
 is the particle diameter. Re is the relative Reynolds, number, which is defined as 

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\begin

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Re \equiv \frac{\rho d_p |\overrightarrow

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-\overrightarrow

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|}

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\end

Initial Field Velocity Function

The initial field setup of a jet is essentially two shear layers. The two shear layers are symmetrical to the centerline. A hyperbolic tangent velocity function is used to initialize velocity field of a shear layer. The hyperbolic tangent velocity functions are preferred over discontinuous piecewise functions because hyperbolic tangent functions are more physical. The initial field velocity function for the entire jet is given below:

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[
U_

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=
\left{
\begin

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tanh(32y-8) & : 0 \leq x \leq 0.5
-tanh(32y-24) & : 0.5n < n \leq 1
\end

\right.
]

     

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\begin

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&V_

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= 0
&\text


&tanh=\frac{e^x-e^{-x}}{e^x+e^{-x}}
\end

The function is plotted below (x velocity versus y) using simple MATLAB codes:

Stokes Number

The Stokes number is a dimensionless number that corresponds to the behavior of particles suspended in a fluid flow. The Stokes number is defined as below:

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\begin

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Stk = \frac

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(1)
\end

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 is the relaxation time of the particle, 
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 is the fluid velocity well away from the particle, and 
Unknown macro: {latex} $d_c$
 is the characteristic length scale of the obstacle, in this case it is the diameter of the particle.

Thus: 

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In the case of Stokes flow, where the Reynolds number is sufficiently small the relaxation time can be expressed using the equation below:

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\begin

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\tau = \frac

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(3)
\end

Plug (2) and (3) into (1), we get the formula for calculating the Stokes Number in our case below:

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\begin

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\boxed{Stk = \frac

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{18\mu_g}}
\end

Expected Results

Under Construction


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