Laminar flow past an infinite planar array of fixed particles: point-particle approximation, Oseen equations and resolved simulations


In the point-particle model of disperse multiphase flow, the particles, assumed to be very small compared with all the scales of the flow, are represented by singular forces acting on the fluid. The hydrodynamic forces are found from standard correlations by interpolating the velocity field from the grid nodes to the particle positions, with the implicit assumption that the computational cells are much larger than the particles. It is argued here that this model has similarities with the Oseen linearization of the Navier–Stokes equation, the most important one being that, in the Oseen context, the particles are also treated, to leading order, as singularities. For this and other reasons addressed in the paper, the Oseen equations can be used as proxies for the point-particle model and the comparison of their solutions with particle-resolved simulations, both of which are presented in this paper, can shed light on the strengths and weaknesses of the point-particle model. The specific situation considered is the laminar, steady, uniform flow normal to a plane of periodically arranged particles exchanging momentum and heat with the fluid.

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This study was supported partly by the University of Houston. The numerical computations were carried out on the Sabine cluster of the University of Houston Research Computing Data Core.


Funding was provided by National Science Foundation (Grant No. CBET 1335965).

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Correspondence to Andrea Prosperetti.

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Appendix: Reconciliation of the solutions obtained by different methods

Appendix: Reconciliation of the solutions obtained by different methods

We show here that the solutions for p and \(\chi \) obtained by means of the Fourier series and by superposition of singularities do in fact agree.

Reconciliation of the two solutions for p

To establish the identity of (29) and (31) we use the Poisson summation formula:

$$\begin{aligned} \sum _{m=-\infty }^\infty F(m,n) = \sum _{k=-\infty }^\infty \int _{-\infty }^\infty F(\xi ,n) \exp [2\pi \mathrm{{i}} k\xi ]\mathrm{{d}}\xi \, . \end{aligned}$$

Iterating this formula we find

$$\begin{aligned} \sum _{n=-\infty }^\infty \sum _{m=-\infty }^\infty F(m,n)= & {} \sum _{k=-\infty }^\infty \int _{-\infty }^\infty \left[ \sum _{n=-\infty }^\infty F(\xi ,n)\right] \exp [2\pi \mathrm{{i}} k\xi ]\mathrm{{d}}\xi \nonumber \\= & {} \sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi \,\mathrm{{d}}\eta . \end{aligned}$$

In applying this relation to (29) we take

$$\begin{aligned} F(m,n)= \exp \left( - 2\pi |z|\sqrt{\frac{m^2}{d_x^2}+\frac{n^2}{d_y^2}} \right) \exp 2\pi \mathrm{{i}} \left( \frac{m x}{d_x} +\frac{ny}{d_y}\right) \,. \end{aligned}$$


$$\begin{aligned}&\int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi = \exp \left[ 2\pi \mathrm{{i}} \left( \frac{y}{d_y} +j\right) \eta \right] \nonumber \\&\quad \int _{-\infty }^\infty \exp \left( - 2\pi |z|\sqrt{\frac{\xi ^2}{d_x^2}+\frac{\eta ^2}{d_y^2}} \right) \exp \left[ 2\pi \mathrm{{i}} \left( \frac{x}{d_x} +k\right) \xi \right] \mathrm{{d}}\xi \\&\quad = \, 2\exp \left[ 2\pi \mathrm{{i}} \left( \frac{y}{d_y} +j\right) \eta \right] \int _0^\infty \exp \left( - 2\pi |z|\sqrt{\frac{\xi ^2}{d_x^2}+\frac{\eta ^2}{d_y^2}} \right) \cos \left[ 2\pi \left( \frac{x}{d_x} +k\right) \xi \right] \mathrm{{d}}\xi \nonumber \,. \end{aligned}$$

With the help of the relation [28]

$$\begin{aligned} \int _0^\infty \mathrm{{e}}^{-\beta \sqrt{\gamma ^2+x^2}}\cos bx \mathrm{{d}}x= \frac{\beta \gamma }{\sqrt{\beta ^2+b^2}} K_1\left( \gamma \sqrt{\beta ^2+b^2}\right) \qquad \gamma >0 , \end{aligned}$$

the integral can be evaluated with the result

$$\begin{aligned}&\int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi \nonumber \\&\quad =\, \exp \left[ 2\pi \mathrm{{i}} \left( \frac{y}{d_y} +j\right) \eta \right] \frac{ 2(d_x/d_y)|z||\eta |}{\sqrt{z^2+ (x+ kd_x)^2}}K_1\left( \frac{2\pi }{d_y}|\eta | \sqrt{ z^2+ (x+ kd_x)^2}\right) \,. \end{aligned}$$

The final integral to be evaluated is

$$\begin{aligned}&\int _{-\infty }^\infty \int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi \,\mathrm{{d}}\eta \nonumber \\&\quad =\, \int _{-\infty }^\infty \exp \left[ 2\pi \mathrm{{i}} \left( \frac{y}{d_y} +j\right) \eta \right] \frac{ 2(d_x/d_y)|z||\eta |}{\sqrt{z^2+ (x+ kd_x)^2}}K_1\left( \frac{2\pi }{d_y}|\eta | \sqrt{ z^2+ (x+ kd_x)^2}\right) \mathrm{{d}}\eta . \end{aligned}$$

Here we use the result [28]

$$\begin{aligned} \int _0^\infty K_1 (ax)\sin (bx)dx= \frac{\pi b}{2a\sqrt{a^2+b^2}} , \end{aligned}$$

to find

$$\begin{aligned} \int _{-\infty }^\infty \int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi \,\mathrm{{d}}\eta =\,\frac{d_xd_y }{\pi }\frac{|z|}{2[ z^2+ (x+ kd_x)^2+(y+jd_y)^2]^{3/2}}. \end{aligned}$$

Upon substitution into (29) we then have

$$\begin{aligned} p= & {} p_i-\frac{f}{2d_xd_y}\left[ 1 + \sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty d_xd_y \frac{z}{2\pi [ z^2+ (x+ kd_x)^2+(y+jd_y)^2]^{3/2}} \right] \nonumber \\= & {} p_i -\frac{f}{2d_xd_y} + f\sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty \frac{\partial }{\partial z} \frac{1}{4\pi |\mathbf{x} -\mathbf{x}_{jk}|} , \end{aligned}$$

which coincides with (31).

Since this relation has been found from (29), which satisfies the boundary conditions, one may expect that the result in this form also does. Without giving a complete proof, let us consider the special case \(d_x=d_y=d\) and take \(x=y=0\) so that

$$\begin{aligned} p(0,0,z)= & {} p_i-\frac{f}{2d^2}\left[ 1 + \frac{d^2}{2\pi } \sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty \frac{z}{[ z^2+ d^2(j^2+k^2)]^{3/2}} \right] \nonumber \\\simeq & {} p_i-\frac{f}{2d^2}\left[ 1 + d^2 z\int _0^\infty \frac{R}{[ z^2+ d^2R^2]^{3/2}}\mathrm{{d}}R \right] \nonumber \\= & {} p_i-\frac{f}{d^2}\left[ 1 + {\mathrm{sgn}}(z) \right] \,. \end{aligned}$$

It is evident that this expression reproduces the boundary conditions \(p\rightarrow p_i\) and \(p\rightarrow p_i-f/d^2\) at \(-\infty \) and \(+\infty \), respectively. Furthermore, it constitutes a direct proof of (32) for this special case \(d_x=d_y\).

Reconciliation of the two solutions for \(\chi \)

The reconciliation of the two solutions can proceed as before applying the double Poisson summation formula (A.2) to the Fourier series solution (44). It is however of some interest to proceed in the opposite direction starting from the singularity solution (48). To this end, according to (A.2), we need to calculate

$$\begin{aligned} X= & {} -\frac{d_x d_yU}{4\pi \nu } \sum _{j,k=-\infty }^\infty \int _{-\infty }^\infty \exp (2\pi \mathrm{{i}} j\eta ) \mathrm{{d}}\eta \nonumber \\&\quad \int _{-\infty }^\infty \frac{\exp \left[ -\frac{U}{2\nu }\sqrt{(x-d_x\xi )^2+(y-d_y\eta )^2+z^2} \right] }{\sqrt{(x-d_x\xi )^2+(y-d_y\eta )^2+z^2}} \exp (2\pi \mathrm{{i}} k\xi )\mathrm{{d}}\xi . \, \, \, \, \, \, \, \, \, \, \, \, \end{aligned}$$

We set \({\hat{\xi }}= (d_x\xi -x)U/(2\nu )\), \({\hat{\eta }}= (d_y\eta -y)U/(2\nu )\), \({\hat{z}}=U|z|/(2\nu )\) so that the double integral becomes

$$\begin{aligned} X = -\frac{2}{\pi } \sum _{j,k=-\infty }^\infty e_{kj}(x,y) \int _0^\infty \cos \left( \frac{4\pi \nu j}{Ud_y}{\hat{\eta }}\right) \mathrm{{d}}{\hat{\eta }} \int _0^\infty \frac{\exp \left( -\sqrt{{\hat{\xi }}^2+{\hat{\eta }}^2+{\hat{z}}^2}\right) }{\sqrt{{\hat{\xi }}^2+{\hat{\eta }}^2+{\hat{z}}^2}} \cos \left( \frac{4\pi \nu k}{Ud_x}{\hat{\xi }}\right) \mathrm{{d}}{\hat{\xi }}, \end{aligned}$$

in which \(e_{kj}(x,y)\) are the Fourier basis functions defined in (20). With the aid of the relations

$$\begin{aligned}&\int _0^\infty \frac{\exp (-\beta \sqrt{\gamma ^2+x^2})}{\sqrt{\gamma ^2+x^2})} \cos (ax) dx = K_0\left( \gamma \sqrt{a^2+\beta ^2}\right) , \end{aligned}$$
$$\begin{aligned}&\int _0^\infty K_0\left( \alpha \sqrt{x^2+\beta ^2}\right) \cos (\gamma x)dx = \frac{\pi }{2} \frac{\exp \left( -\beta \sqrt{\alpha ^2+\gamma ^2}\right) }{\sqrt{\alpha ^2+\gamma ^2}} , \end{aligned}$$

the integrations can be performed to find

$$\begin{aligned} X= & {} - \sum _{j,k=-\infty }^\infty \frac{\exp \left[ -\sqrt{1+ 16\pi ^2 \nu ^2k^2 /d_x^2+ 16\pi ^2 \nu ^2j^2/d_y^2} (|z|U/2\nu )\right] }{\sqrt{1+ 16\pi ^2 \nu ^2k^2 /d_x^2+ 16\pi ^2 \nu ^2j^2/d_y^2}} \, e_{kj}(x,y)\,, \,\,\,\,\,\, \end{aligned}$$

which reproduces the Fourier series (43) for X.

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Zhang, Y., Wang, Y. & Prosperetti, A. Laminar flow past an infinite planar array of fixed particles: point-particle approximation, Oseen equations and resolved simulations. J Eng Math 122, 139–157 (2020).

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  • Disperse multiphase flow
  • Oseen flow
  • Point-particle model