Simple Particle Relaxation Modeling in One-Dimensional Oscillating Flows

Abstract

The relaxation of a rigid particle suspended in a one-dimensional oscillating flow is calculated according to different drag models and the results are compared. Conditions are derived under which relaxation can be neglected or drag models can be substituted by simpler ones. This investigation is conducted analytically and graphically via the plane defined by the Reynolds number and amplitude parameter. This work matches various, mostly analytic drag models together to consider simple particle relaxation with a few, broad range input parameters and cover large parts of the plane spanned by Reynolds number and amplitude parameter.

1. Introduction

A rigid, spherical particle suspended in a harmonically oscillating one-dimensional flow is considered. This configuration can be found, for example, with ultrasonic levitators [1,2] or pulsation reactors [3,4,5]. In case the particle is fixed in position (or the particle executes harmonic oscillations in a fluid at rest), the interaction between the fluid and the particle as well as the resulting flow state is defined by two dimensionless numbers: the particle Reynolds number $Re$ and the amplitude parameter $ϵ$. The particle Reynolds number $Re=U{\rho }_{f}d/\eta$ expresses the ratio between inertial and viscous forces with the slip velocity amplitude U, the density of the fluid ${\rho }_f$, the particle diameter d, and the dynamic viscosity of the fluid $\eta$. The amplitude parameter $ϵ=A/d$ sets the particle displacement A in relation to the particle diameter, while the particle displacement $A=U/\omega$ itself is the ratio of slip velocity amplitude to the angular frequency $\omega$ of the oscillating fluid. Reynolds number and amplitude parameter span a plane in which all flow states can be pinpointed, as shown in Figure 1. Note that the origin of Figure 1 has the coordinates $ϵ=1,Re=1$. In case the particle is fixed in place, the slip velocity amplitude equals the far-field fluid velocity amplitude $U_f$. Within this work, the far-field fluid velocity will be referred to as fluid velocity and can be described as $u_f\left(t\right)=U_{f}cos\left(\omega t\right)$. In case the particle is not considered fixed in place, the slip velocity $u\left(t\right)$ differs from the flow velocity due to inertia and drag forces acting on the particle (relaxation). The particle will perform oscillations with the same frequency as the fluid, which is also true for the slip velocity. If harmonic behavior is assumed, the only unknown quantities, which are necessary to describe the slip velocity, are the slip velocity amplitude and the phase shift $\varphi$ with respect to the fluid velocity [6]. The basic geometrical relations of these central quantities are displayed in Figure 2. In Section 2.1, several drag models are presented with their respective ranges of validity to calculate relaxation in large parts of the $ϵ$-$Re$ plane. Afterward, in Section 2.2, the slip velocity is calculated and discussed. Distinguishing between cases where the particle can be considered fixed and where this is not the case is discussed in Section 2.3. In the last section (Section 3), criteria are derived under which the simple Stokes model can be applied in the validity ranges of the other models.

2. Method

2.1. Drag Models

If all quantities (${\rho }_p,{\rho }_f,U_f,\omega ,\eta ,\mathrm{and}d$) but U are known and constant, U can be determined by equating the inertia of the particle (here treated as a mass point) with its drag when omitting all other forces. The slip velocity is not only central in working with the $ϵ$-$Re$ plane (since it is part of $ϵ$ and $Re$), but it is also crucial for further considerations like flow patterns and heat and mass transfer, which are not subjects of this work. While the validity of the Navier-Stokes equations (NSE) is assumed, the complex process of solving them can be circumvented by applying various simplified drag models in their respected ranges of validity. Several drag models are necessary to cover large parts of the $ϵ$-$Re$ plane, while the relations in Figure 2 holds true, independent of the applied drag model. It will now be investigated how these models behave and relate to each other. The expressions for the drag force of each drag model can be found in

Table 1. Amplitude ratio of slip to fluid velocity $U/U_f$ for several drag models dependent on the Womersley number $W{o}^2$ and the density ratio $\gamma$.

Name	Drag Force ${\mathit{F}}_{\mathit{D}}$	Slip Velocity Amplitude Ratio $\mathit{U}/{\mathit{U}}_{\mathit{f}}=$
Stokes	$3\pi \eta du$	${\left[1+{\left(\frac{18}{W{o}^2\gamma }\right)}^2\right]}^{-1/2}$
Schiller & Naumann	$3\pi \eta duSN$	${\left[1+{\left(\frac{18SN}{W{o}^2\gamma }\right)}^2\right]}^{-1/2}$	$SN=1+0.158R{e}^{2/3}$
Basset	$\begin{array}{l}3\pi \eta du-\frac{\pi }{6}{d}^3\frac{\partial p}{\partial x}+\frac{\pi }{12}{d}^3{\rho }_f\frac{du}{dt}\\ +\frac{3}{2}{d}^2\sqrt{\pi {\rho }_f\eta }{\int }_{t_0}^t\frac{du/dt}{\sqrt{t-t^{\prime }}}d{t}^{\prime }\end{array}$	${\left[1-{(1+f_1)}^2-f_2^2\right]}^{-1/2}$	$f_1=\frac{\left(1-\gamma \right)\left(0.5+\gamma +\frac{9}{\sqrt{2}Wo}\right)}{{\left(\frac{18}{W{o}^2}+\frac{9}{\sqrt{2}Wo}\right)}^2+{\left(0.5+\gamma +\frac{9}{\sqrt{2}Wo}\right)}^2}$ $f_2=\frac{\left(1-\gamma \right)\left(\frac{18}{W{o}^2}+\frac{9}{\sqrt{2}Wo}\right)}{{\left(\frac{18}{W{o}^2}+\frac{9}{\sqrt{2}Wo}\right)}^2+{\left(0.5+\gamma +\frac{9}{\sqrt{2}Wo}\right)}^2}$
Landau & Lifshitz	$\begin{array}{l}3\pi \eta d\left(1+\frac{d}{2\delta }\right)u+\\ \frac{3}{4}\pi d^2\sqrt{\frac{2\eta \rho }{w}}\left(1+\frac{d}{9\delta }\right)\frac{du}{dt}\end{array}$	${\left[{\left(\frac{18}{\gamma W{o}^2}+\frac{9}{\gamma Wo}\right)}^2+{\left(\frac{9}{\sqrt{2}\gamma Wo}+\frac{1}{\sqrt{2}\gamma }+1\right)}^2\right]}^{-1/2}$

.

While the described ranges of validity are listed on the right side of Figure 1, they are plotted in the $ϵ$-$Re$ plane on the left side. In case two or three models are valid in the same area, the simplest one has a solid color, while the others have a striped pattern.

The first step is to distinguish between a steady and an unsteady flow. The main criterion for differentiating is whether the boundary layer thickness $\delta =\sqrt{\eta {\rho }_f/\omega }$ is larger or smaller than the particle itself. This phenomenon is expressed by the Womersley number $W{o}^2=d^2/{\delta }^2=Re/ϵ$ [7]. The Womersley number is sometimes also referred to as Stokes number [8] or frequency parameter $M^2$ [9] in literature. While the Womersley number is one bisector of the $ϵ$-$Re$ plane, the other one is the Streaming Reynolds number $R{e}_S=Reϵ$, which is central in the consideration of Steady Streaming [10]. If $W{o}^2\ll 1$, the flow can be considered steady [11]. In case the Reynolds number is also smaller than unity $Re<1$, the viscous forces dominate and creeping flow can be assumed. This assumption enables the application of the simple Stokes drag model [12], which considers pure drag only and leads to closed analytic solutions. If $W{o}^2\ll 1$, while the Reynolds number exceeds the validity limit of the Stokes model $Re>1$, the flow can still be considered steady, but a drag model with a larger range of validity regarding the Reynolds number has to be applied. In this case, the Schiller & Naumann model [13] (SN) was chosen, but for this case, many other drag models can be applied as well [14]. The SN model differs in nature from the Stokes model in two important aspects. On the one hand, it is an empirical model derived via the correlation of experimental data, rather than being derived from the NSE by omitting inertial forces. On the other hand, applying the SN drag model leads to a non-linear motion ODE, which can only be solved numerically. In the opposite case of $W{o}^2\gg 1$, the drag model by Landau & Lifshitz [11] (LL) can be applied, which linearizes the NSE around small $ϵ<1$ to ensure no detachment of the flow from the particle. In case neither $W{o}^2\ll 1$ nor $W{o}^2\gg 1$, but $Re<1$, the drag model by Basset [15] is valid, which is explicitly derived for unsteady flow conditions. It is an extension of the Stokes model that, besides pure drag, also considers added mass and the pressure gradient. Additionally, the “history” of the fluid is factored in by a theoretically derived term. In this work, only the stationary state is considered, and therefore the lower boundary of the history integral $t_0$ is set to $-\infty$. A discussion of the transient case can be found with Coimbra & Rangel [16]. An uncolored area in the $ϵ$-$Re$ plane is recognizable where none of the simplified models are valid. In this area, the NSEs need to be solved numerically to calculate the slip velocity. The introduced drag models are well established in the literature and have been validated via experiments (and DNS) several times. The SN model was derived from experimental data in the first place. Therefore, in this work, the drag models themselves and their minimal ranges of validity are considered valid.

Since the particle oscillates with the same frequency as the fluid, the particle velocity and, in extension, the slip velocity are bound by this condition and change only slightly with respect to the applied drag model. This is in contrast to a steady flow, where the differences between models tend to keep increasing over time and often grow exponentially. This means that the simple Stokes drag model is adequate over a wider parameter range in an oscillating flow than its perceived limit of validity of small Reynolds numbers $Re<1$ in a steady flow. This phenomenon encourages the substitution of more complex models of drag with Stokes. This simplification is addressed in Section 3.

2.2. Slip Velocity Amplitude

The slip velocity amplitude can be calculated with each respected drag model under the assumption of a harmonic oscillating flow that can be described with $u_f\left(t\right)=U_{f}cos\left(\omega t\right)$, the results of which are displayed in Table 1. The derivation of the slip velocity amplitude with Stokes, SN, and the LL model is given in Appendix A, Appendix B and Appendix C, respectively, while the solution for the Basset model is taken from Hjelmfelt & Mockros [6].

Each solution in Table 1 contains only the Womersley number $W{o}^2$ and the density ratio $\gamma$ (In the case of the SN model also the Reynolds number). The density ratio $\gamma ={\rho }_p/{\rho }_f$ is the ratio between the particle density ${\rho }_p$ and the fluid density. This implies that, in cases where relaxation needs to be considered, the above introduced $ϵ$-$Re$ plane can still be utilized with $\gamma$ as a parameter. The normalized slip velocity amplitude is displayed in Figure 3 for $\gamma$ = 25,000, corresponding to iron particles in hot air (left), and $\gamma$ = 8, corresponding to iron particles in water at STP (right). First, neglecting the particle relaxation will be discussed, followed by the comparison of drag models, both by means of Figure 3.

2.3. Particle Relaxation

The $ϵ$-$Re$ plane can be loosely classified into two areas with respect to the slip velocity amplitude. In one part, related to large $W{o}^2$, $U/U_f\approx 1$ holds true. In this area, relaxation can be neglected. The size of this area depends only on $\gamma$, while its border can be represented by a value of the Womersley number $W{o}^2$. This means each value of $W{o}^2$ can be related to a density ratio value set (and vice versa) for which relaxation can be neglected. The lowest value of this set is now defined as the critical density ratio ${\gamma }_{crit}\left(W{o}^2\right)$. Since relaxation is always present with this problem a cut-off criterion needs to be defined for neglecting particle relaxation. Depending on the specific applied case, the criterion can, for example, be set to $U/U_f>0.9$, $U/U_f>0.95$, or $U/U_f>0.99$. The critical density ratio ${\gamma }_{crit}$ with respect to $W{o}^2$ is displayed in Figure 4 for the suggested cut-off criteria. If the density ratio of a specific problem is above the line for the respective Womersley number, relaxation can be neglected, while the choice of the cut-off criteria becomes more important for high Womersley numbers, as can be derived from Figure 4. Graphically, it is even suggested that this finding is also valid for the white area of the $ϵ$-$Re$ plane, where the NSEs need to be solved numerically.

In the other part of the $ϵ$-$Re$ plane, $U/U_f\not\approx$ 1 and relaxation does need to be considered via adequate drag models.

3. Result and Discussion

For large $\gamma$, the quasi-steady approximation is valid for the entire relaxation part of the $ϵ$-$Re$ plane, and the slip velocity amplitude can be calculated by Stokes or SN. This case is displayed on the left side of Figure 3, where $\gamma$ is large enough for Stokes to reach $U/U_f\approx$ 1, within its limit of validity $Re<1$. For small $\gamma$ (right side of Figure 3) this is not the case. The same approach can be applied to the comparison between Stokes and SN. Here it is not $\gamma$, but rather the ratio between $Re$ and $Stk$ that is the determining factor. This relates to the distinction of $U/U_f\approx$ 1 being reached before or after $Re$ becomes unity. Even though the limit $W{o}^2<<1$ was set here to $W{o}^2=0.01$, the slip velocity amplitude calculated with the Basset model still deviates substantially from the Stokes model. (Note here the logarithmic scale exaggerating the deviation between Basset and Stokes). Similar to the comparison between Stokes and Basset, on the right-hand side of Figure 3, the values at the model limits between Stokes and SN and even between Basset and LL do not match well, although this effect is not as pronounced. This is contrary to the high-density ratio case displayed on the left side of Figure 3, where the values at the model limits do match well. This raises the question of under which conditions the models connect to each other well or can even be applied interchangeably. The relation between Stokes and other models can be calculated with the expressions presented in

Table 2. Relations of the drag models listed in Table 1 to the Stokes model, expressed with the oscillation Stokes number $Stk$ and the density ratio $\gamma$. Additionally, the limit is given up to which the deviation stays below 5%, in which case the Stokes model can be applied in the validity range of the respective model. Multiple criteria are set in relation to each other via the logic operator ∧—‘and’.

Name	Relation to Stokes ${\mathit{U}}_{\mathit{i}}/{\mathit{U}}_{\mathbf{Stk}}$		$|{\mathit{U}}_{\mathit{i}}-{\mathit{U}}_{\mathbf{Stk}}|/{\mathit{U}}_{\mathbf{Stk}}<5\%$
Schiller & Naumann	${\left\{\frac{St{k}^2+1}{St{k}^2+S{N}^2}\right\}}^{1/2}$	$SN=1+0.158R{e}^{2/3}$	$\frac{\sqrt{3Re}}{Stk}<1$
Basset	${\left\{\left[1+\frac{1}{St{k}^2}\right]\left[1-{(1+f_1)}^2-f_2^2\right]\right\}}^{1/2}$	$f_1=\frac{2\left(1-\gamma \right)\left(1+2\gamma +3\sqrt{\frac{\gamma }{Stk}}\right)}{{\left(2\frac{\gamma }{Stk}+3\sqrt{\frac{\gamma }{Stk}}\right)}^2+{\left(1+2\gamma +3\sqrt{\frac{\gamma }{Stk}}\right)}^2}$ $f_2=\frac{2\left(1-\gamma \right)\left(2\frac{\gamma }{Stk}+3\sqrt{\frac{\gamma }{Stk}}\right)}{{\left(2\frac{\gamma }{Stk}+3\sqrt{\frac{\gamma }{Stk}}\right)}^2+{\left(1+2\gamma +3\sqrt{\frac{\gamma }{Stk}}\right)}^2}$	$(\gamma St{k}^{6/5}>370)$∧$\left(\gamma >5\right)$
Landau & Lifshitz	${\left\{\frac{St{k}^2+1}{St{k}^2\left[{\left(\frac{3}{\sqrt{2\gamma Stk}}+\frac{1}{Stk}\right)}^2+{\left(\frac{3}{2\sqrt{\gamma Stk}}+\frac{1}{\sqrt{2}\gamma }+1\right)}^2\right]}\right\}}^{1/2}$		$(\gamma >1000)$

, while harmonic particle velocities and slip velocities are assumed. The derivation of the deviation between Stokes and the SN model is given in detail in Appendix B. The expressions in Table 2 are derived from the expressions in Table 1 and also depend only on $\gamma$ and $W{o}^2$. This time, however, the oscillation Stokes number $Stk=\omega {\tau }_p=\gamma W{o}^2/18$ is utilized [17]. It sets the characteristic time of the particle ${\tau }_p={\rho }_p{d}^2/18\eta$ in relation to the characteristic time of the flow ${\tau }_f=1/\omega$. It works as a criterion for how well the particle can adapt to the changing flow conditions. When the normalized slip velocity amplitude in an oscillating flow is calculated with the Stokes model, it is only dependent on the oscillation Stokes number $U/U_f=1/\sqrt{1+1/St{k}^2}$. Therefore, the oscillation Stokes number is utilized in comparison of the Stokes model with the other models. The criterion for an acceptable deviation between the models was defined as 5%, and the parameter areas where the deviation is above and below this criterion are displayed in Figure 5 for the various models. The density ratio $\gamma$ is the decisive parameter for the comparison between Stokes and Basset as well as Stokes and the LL model, while for the comparison between Stokes and the SN model it is the Reynolds number. Additionally, at least one criterion is presented in Table 2 for the deviation between Stokes and the other drag models to stay below 5%, while multiple criteria are set in relation to each other via logical operators. In this case, the Stokes model can be applied in the validity range of the respective model. These criteria are also plotted in Figure 5. Simple criteria were chosen for the sake of convenience, as they can not match the 5% error interface perfectly. This means that the deviation will stay below 5% if the criterion is met, but it can also stay below 5% if the criterion is violated, as can be seen in Figure 5. Stokes can be used in order to calculate the slip velocity amplitude in the entire colored area of the $ϵ$-$Re$ plane when all criteria in Table 2 are met, hence $(\gamma >1000)\wedge (\gamma St{k}^{6/5}>370)\wedge (Stk<\sqrt{3Re})$.

4. Conclusions

The special case of a rigid particle in a one-dimensional oscillating flow was considered. The $ϵ$-$Re$ plane was introduced, in which all flow states of this problem are defined. The conditions under which particle relaxation can be neglected were derived. Subsequently, various drag models from the literature were considered to model particle relaxation as simply as possible in large parts of the $ϵ$-$Re$ plane. It was pointed out under which conditions the simple Stokes model can be applied in the validity range of various other drag models. This was conducted graphically and analytically.

It was shown that the Stokes drag model can be applied validly for the calculation of particle relaxation for large sets of parameter combinations. This can even be true for $Re\gg 1$, as long as other criteria, connected to high-density ratios between fluid and particle and/or high oscillating Stokes numbers, are met. These parameter combinations are found with many gas-solid multi-phase flows, which makes this work especially interesting for these kinds of systems.