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Parametric analysis of laminar pulsating flow in a rectangular channel

Abstract

Pulsating flow has potential for enhanced cooling of future electronics and photonics systems. To better understand the mechanisms underlying any heat transfer enhancement, it is necessary to decouple the mechanical and thermal problems. The current work performs a parametric analysis of the flow hydrodynamics using particle image velocimetry (PIV) measurements, CFD simulations and analytical solutions, reorganised in terms of amplitude and phase values using complex notation. To the best of the authors’ knowledge, the frequency-dependent behaviour of amplitude and phase of wall shear stress has not been studied in a two-dimensional channel. For laminar flow, the amplitudes are directly proportional to pressure. The amplitudes of various local and mean wall shear stress measures are augmented with frequency compared to steady flow, especially near the short walls and corners. The phases of wall shear stress differ at each wall at moderate frequencies – with the bulk-mean values at the short wall leading those at the long wall – and tend to π/4 in the limit of high frequency. The amplitudes of pressure gradient increase more significantly than wall shear stress magnitudes due to accelerative forces. The boundaries to the quasi-steady, intermediate and inertia-dominated regimes are estimated at Womersley number W o = 1.6 and 27.6 in a rectangular channel, based on the contribution of viscous and inertial terms.

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Abbreviations

a, b :

channel width, height [m]

D h :

hydraulic diameter [m]

f :

oscillation frequency [ H z]

\(\mathfrak {I}\) :

imaginary part of complex number

i :

imaginary unit (=\(\sqrt {-1}\))

L :

channel length [m]

m, n :

summation indices

p :

pressure [ P a]

Q :

flow rate [ m 3/s]

\(\mathfrak {R}\) :

real part of complex number

R e :

Reynolds number (= 〈uD h /ν)

\(Re_{\delta _{\nu }}\) :

R e based on Stokes layer thickness (= 〈uδ ν /ν)

t :

time [s]

u :

velocity in the axial direction [m/s]

u〉:

space-averaged velocity [ m/s]

W o :

Womersley number \(\left (=\frac {1}{2}D_{h}\sqrt {\omega /\nu }\right )\)

x :

axial flow coordinate [m]

y, z :

coordinates normal to flow direction [m]

β :

function defined by Eq. 4b

δ ν :

Stokes layer thickness [m]

μ :

viscosity [ k g/(ms)]

ν :

kinematic viscosity [ m 2/s]

ρ :

density [ k g/m 3]

τ :

wall shear stress [ P a]

τ〉:

space-averaged wall shear stress [ P a]

Φ:

function defined by Eq. 4a

ψ :

complex functions defined by e.g. Eq. 3a

ω :

angular oscillation frequency [ r a d/s]

0:

steady flow component

A :

oscillating flow amplitude

yx :

x component with normal y

zx :

x component with normal z

:

phase relative to \(\nabla p^{\prime } = \mathfrak {R}[\nabla p_{A} e^{i \omega t}]\)

:

phase relative to \(Q^{\prime \prime } = \mathfrak {R}[\psi _{Q}e^{i (\omega t - \phi _{Q} - \pi /2)}]\)

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Acknowledgements

The authors would like to acknowledge the financial support of the Irish Research Council (IRC) under grant numbers EPSPG/2013/618 and GOIPD/2016/216.

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Correspondence to Richard Blythman.

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Blythman, R., Alimohammadi, S., Persoons, T. et al. Parametric analysis of laminar pulsating flow in a rectangular channel. Heat Mass Transfer 54, 2177–2186 (2018). https://doi.org/10.1007/s00231-017-2196-z

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  • DOI: https://doi.org/10.1007/s00231-017-2196-z