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Heat and Mass Transfer

, Volume 54, Issue 8, pp 2177–2186 | Cite as

Parametric analysis of laminar pulsating flow in a rectangular channel

  • Richard BlythmanEmail author
  • Sajad Alimohammadi
  • Tim Persoons
  • Nick Jeffers
  • Darina B. Murray
Original

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.

List of symbols

a, b

channel width, height [m]

Dh

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

Re

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]

Wo

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]

Greek symbols

β

function defined by Eq. 4b

δν

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]

Subscripts & Superscripts

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

Notes

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.

Compliance with Ethical Standards

Conflict of interests

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Manufacturing EngineeringTrinity College DublinDublin 2Ireland
  2. 2.Thermal Management Research Group, Efficient Energy Transfer (ηET) Department, Nokia Bell Labs, Blanchardstown Business & Technology ParkDublin 15Ireland

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