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

Abstract

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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References

  1. 1.

    Balachandar S, Eaton JK (2010) Turbulent dispersed multiphase flow. Annu Rev Fluid Mech 42:111–133

    Article  Google Scholar 

  2. 2.

    Eaton JK (2009) Two-way coupled turbulence simulations of gas-particle flows using point-particle tracking. Int J Multiphase Flow 35:792–800

    Article  Google Scholar 

  3. 3.

    Elghobashi S (1994) On predicting particle-laden turbulent flows. Appl Sci Res 52:309–329

    Article  Google Scholar 

  4. 4.

    Elghobashi S (2019) Direct numerical simulation of turbulent flows laden with droplets or bubbles. Annu Rev Fluid Mech 51:217–244

    MathSciNet  Article  Google Scholar 

  5. 5.

    Fullmer WD, Hrenya CM (2017) The clustering instability in rapid granular and gas-solid flows. Annu Rev Fluid Mech 49:485–510

    MathSciNet  Article  Google Scholar 

  6. 6.

    Maxey M (2017) Simulation methods for particulate flows and concentrated suspensions. Annu Rev Fluid Mech 49:171–193

    MathSciNet  Article  Google Scholar 

  7. 7.

    Poustis J-F, Senoner J-M, Zuzio D, Villedieu P (2019) Regularization of the Lagrangian point force approximation for deterministic discrete particle simulations. Int J Multiphase Flow 117:138–152

    MathSciNet  Article  Google Scholar 

  8. 8.

    Hinch EJ (1991) Perturbation methods. Cambridge University Press, Cambridge

    Google Scholar 

  9. 9.

    Proudman I, Pearson JRA (1957) Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J Fluid Mech 2:237–262

    MathSciNet  Article  Google Scholar 

  10. 10.

    van Dyke MD (1975) Perturbation methods in fluid mechanics, annotated edn. Parabolic Press, Stanford

    Google Scholar 

  11. 11.

    Elghobashi S, Truesdell GC (1993) On the two-way interaction between homogeneous turbulence and dispersed solid particles. I: turbulence modification. Phys Fluids A 5:1790–1801

    Article  Google Scholar 

  12. 12.

    Capecelatro J, Desjardins O (2013) An Euler–Lagrange strategy for simulating particle-laden flows. J Comput Phys 238:1–31

    MathSciNet  Article  Google Scholar 

  13. 13.

    Sierakowski AJ, Prosperetti A (2016) Resolved-particle simulation by the Physalis method: enhancements and new capabilities. J Comput Phys 309:164–184

    MathSciNet  Article  Google Scholar 

  14. 14.

    Wang Y, Prosperetti A (2018) Heat transfer from an array of fully-resolved particles in turbulent flow. Phys Rev Fluids 3:084305

    Article  Google Scholar 

  15. 15.

    Wang Y, Sierakowski AJ, Prosperetti A (2017) Fully-resolved simulation of particulate flows with particles-fluid heat transfer. J Comput Phys 350:638–656

    MathSciNet  Article  Google Scholar 

  16. 16.

    Chang ID (1961) Navier–Stokes solutions at large distances from a finite body. J Math Mech 10:811–876

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge

    Google Scholar 

  18. 18.

    Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Butterworth-Heinemann, Oxford

    Google Scholar 

  19. 19.

    Lagerstrom PA (1964) Laminar flow theory. In: Moore FK (ed) Theory of laminar flows. Princeton University Press, Princeton, pp 88–102

    Google Scholar 

  20. 20.

    Prosperetti A (1976) Laminar flow at large distances from an infinite two-dimensional grid. J Mécanique 15:209–235

    MATH  Google Scholar 

  21. 21.

    Prosperetti A (2011) Advanced mathematics for applications. Cambridge University Press, Cambridge

    Google Scholar 

  22. 22.

    Stakgold I (1967) Boundary value problems of mathematical physics. Macmillan, New York

    Google Scholar 

  23. 23.

    Feng ZG, Musong SG (2014) Direct numerical simulation of heat and mass transfer of spheres in a fluidized bed. Powder Technol 262:62–70

    Article  Google Scholar 

  24. 24.

    Sun B, Tenneti S, Subramaniam S, Koch DL (2016) Pseudo-turbulent heat flux and average gas-phase conduction during gas-solid heat transfer: flow past random fixed particle assemblies. J Fluid Mech 798:299–349

    MathSciNet  Article  Google Scholar 

  25. 25.

    Bergman TL, Lavine AS, Icropera FP, DeWitt D (2011) Fundamentals of heat and mass transfer, 7th edn. Wiley, New York

    Google Scholar 

  26. 26.

    Schiller L, Naumann Z (1935) A drag coefficient correlation. Z Ver Deutsch Ing 77:318–320

    Google Scholar 

  27. 27.

    Clift R, Grace JR, Weber ME (1978) Bubbles. Academic Press, reprinted by Dover, Drops and particles, p 2005

  28. 28.

    Gradshteyn IS, Ryzhik IM (2015) Table of integrals, series, and products, 8th edn. Academic Press, New York

    Google Scholar 

Download references

Acknowledgements

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

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:

$$\begin{aligned} \sum _{m=-\infty }^\infty F(m,n) = \sum _{k=-\infty }^\infty \int _{-\infty }^\infty F(\xi ,n) \exp [2\pi \mathrm{{i}} k\xi ]\mathrm{{d}}\xi \, . \end{aligned}$$
(A.1)

Iterating this formula we find

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

In applying this relation to (29) we take

$$\begin{aligned} F(m,n)= \exp \left( - 2\pi |z|\sqrt{\frac{m^2}{d_x^2}+\frac{n^2}{d_y^2}} \right) \exp 2\pi \mathrm{{i}} \left( \frac{m x}{d_x} +\frac{ny}{d_y}\right) \,. \end{aligned}$$
(A.3)

Thus

$$\begin{aligned}&\int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi = \exp \left[ 2\pi \mathrm{{i}} \left( \frac{y}{d_y} +j\right) \eta \right] \nonumber \\&\quad \int _{-\infty }^\infty \exp \left( - 2\pi |z|\sqrt{\frac{\xi ^2}{d_x^2}+\frac{\eta ^2}{d_y^2}} \right) \exp \left[ 2\pi \mathrm{{i}} \left( \frac{x}{d_x} +k\right) \xi \right] \mathrm{{d}}\xi \\&\quad = \, 2\exp \left[ 2\pi \mathrm{{i}} \left( \frac{y}{d_y} +j\right) \eta \right] \int _0^\infty \exp \left( - 2\pi |z|\sqrt{\frac{\xi ^2}{d_x^2}+\frac{\eta ^2}{d_y^2}} \right) \cos \left[ 2\pi \left( \frac{x}{d_x} +k\right) \xi \right] \mathrm{{d}}\xi \nonumber \,. \end{aligned}$$
(A.4)

With the help of the relation [28]

$$\begin{aligned} \int _0^\infty \mathrm{{e}}^{-\beta \sqrt{\gamma ^2+x^2}}\cos bx \mathrm{{d}}x= \frac{\beta \gamma }{\sqrt{\beta ^2+b^2}} K_1\left( \gamma \sqrt{\beta ^2+b^2}\right) \qquad \gamma >0 , \end{aligned}$$
(A.5)

the integral can be evaluated with the result

$$\begin{aligned}&\int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi \nonumber \\&\quad =\, \exp \left[ 2\pi \mathrm{{i}} \left( \frac{y}{d_y} +j\right) \eta \right] \frac{ 2(d_x/d_y)|z||\eta |}{\sqrt{z^2+ (x+ kd_x)^2}}K_1\left( \frac{2\pi }{d_y}|\eta | \sqrt{ z^2+ (x+ kd_x)^2}\right) \,. \end{aligned}$$
(A.6)

The final integral to be evaluated is

$$\begin{aligned}&\int _{-\infty }^\infty \int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi \,\mathrm{{d}}\eta \nonumber \\&\quad =\, \int _{-\infty }^\infty \exp \left[ 2\pi \mathrm{{i}} \left( \frac{y}{d_y} +j\right) \eta \right] \frac{ 2(d_x/d_y)|z||\eta |}{\sqrt{z^2+ (x+ kd_x)^2}}K_1\left( \frac{2\pi }{d_y}|\eta | \sqrt{ z^2+ (x+ kd_x)^2}\right) \mathrm{{d}}\eta . \end{aligned}$$
(A.7)

Here we use the result [28]

$$\begin{aligned} \int _0^\infty K_1 (ax)\sin (bx)dx= \frac{\pi b}{2a\sqrt{a^2+b^2}} , \end{aligned}$$
(A.8)

to find

$$\begin{aligned} \int _{-\infty }^\infty \int _{-\infty }^\infty F(\xi ,\eta ) \exp [2\pi \mathrm{{i}} (k\xi +j\eta )] \mathrm{{d}}\xi \,\mathrm{{d}}\eta =\,\frac{d_xd_y }{\pi }\frac{|z|}{2[ z^2+ (x+ kd_x)^2+(y+jd_y)^2]^{3/2}}. \end{aligned}$$
(A.9)

Upon substitution into (29) we then have

$$\begin{aligned} p= & {} p_i-\frac{f}{2d_xd_y}\left[ 1 + \sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty d_xd_y \frac{z}{2\pi [ z^2+ (x+ kd_x)^2+(y+jd_y)^2]^{3/2}} \right] \nonumber \\= & {} p_i -\frac{f}{2d_xd_y} + f\sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty \frac{\partial }{\partial z} \frac{1}{4\pi |\mathbf{x} -\mathbf{x}_{jk}|} , \end{aligned}$$
(A.10)

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

$$\begin{aligned} p(0,0,z)= & {} p_i-\frac{f}{2d^2}\left[ 1 + \frac{d^2}{2\pi } \sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty \frac{z}{[ z^2+ d^2(j^2+k^2)]^{3/2}} \right] \nonumber \\\simeq & {} p_i-\frac{f}{2d^2}\left[ 1 + d^2 z\int _0^\infty \frac{R}{[ z^2+ d^2R^2]^{3/2}}\mathrm{{d}}R \right] \nonumber \\= & {} p_i-\frac{f}{d^2}\left[ 1 + {\mathrm{sgn}}(z) \right] \,. \end{aligned}$$
(A.11)

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

$$\begin{aligned} X= & {} -\frac{d_x d_yU}{4\pi \nu } \sum _{j,k=-\infty }^\infty \int _{-\infty }^\infty \exp (2\pi \mathrm{{i}} j\eta ) \mathrm{{d}}\eta \nonumber \\&\quad \int _{-\infty }^\infty \frac{\exp \left[ -\frac{U}{2\nu }\sqrt{(x-d_x\xi )^2+(y-d_y\eta )^2+z^2} \right] }{\sqrt{(x-d_x\xi )^2+(y-d_y\eta )^2+z^2}} \exp (2\pi \mathrm{{i}} k\xi )\mathrm{{d}}\xi . \, \, \, \, \, \, \, \, \, \, \, \, \end{aligned}$$
(A.12)

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

$$\begin{aligned} X = -\frac{2}{\pi } \sum _{j,k=-\infty }^\infty e_{kj}(x,y) \int _0^\infty \cos \left( \frac{4\pi \nu j}{Ud_y}{\hat{\eta }}\right) \mathrm{{d}}{\hat{\eta }} \int _0^\infty \frac{\exp \left( -\sqrt{{\hat{\xi }}^2+{\hat{\eta }}^2+{\hat{z}}^2}\right) }{\sqrt{{\hat{\xi }}^2+{\hat{\eta }}^2+{\hat{z}}^2}} \cos \left( \frac{4\pi \nu k}{Ud_x}{\hat{\xi }}\right) \mathrm{{d}}{\hat{\xi }}, \end{aligned}$$
(A.13)

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

$$\begin{aligned}&\int _0^\infty \frac{\exp (-\beta \sqrt{\gamma ^2+x^2})}{\sqrt{\gamma ^2+x^2})} \cos (ax) dx = K_0\left( \gamma \sqrt{a^2+\beta ^2}\right) , \end{aligned}$$
(A.14)
$$\begin{aligned}&\int _0^\infty K_0\left( \alpha \sqrt{x^2+\beta ^2}\right) \cos (\gamma x)dx = \frac{\pi }{2} \frac{\exp \left( -\beta \sqrt{\alpha ^2+\gamma ^2}\right) }{\sqrt{\alpha ^2+\gamma ^2}} , \end{aligned}$$
(A.15)

the integrations can be performed to find

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

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). https://doi.org/10.1007/s10665-020-10052-9

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Keywords

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