Flow in the Entrance Region of Ducts

Wiginton, Chris; Wendt, Regine

doi:10.1063/1.1692504

Cited by 9 publications

(2 citation statements)

References 1 publication

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“…Numerous studies have described and summarised analytical solutions to the parabolised Navier–Stokes equation for entrance flow (Fargie & Martin 1971; Mohanty & Asthana 1978; Reci, Sederman & Gladden 2018). These solutions can be divided into two categories: category (i) involves linearising the inertia term with axial velocity in the momentum equation, yielding solutions as series functions (Langhaar 1942; Sparrow, Lin & Lundgren 1964; Wiginton & Wendt 1969; Boussinesq 1981). Category (ii) assumes the growth of the boundary layer along the pipe wall.…”

Section: Introductionmentioning

confidence: 99%

Analytical solution for laminar entrance flow in circular pipes

Kim

2024

J. Fluid Mech.

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This study introduces an analytical solution for the laminar entrance flow in circular pipes, aiming to confirm the occurrence of velocity overshoot. Velocity overshoot is characterised by the maximum axial velocity appearing near the pipe wall instead of the central axis. Similar to the previous studies, the analytical solution is derived from the parabolised Navier–Stokes equation; however, the specific approach used in linearising the momentum equation has not been attempted before. The accuracy of this analytical solution has been verified through a comprehensive comparison with various published experimental data. The existence of velocity overshoot at a short distance from the inlet, which is evident in numerous numerical calculations based on the full Navier–Stokes equations and corroborated by recent magnetic resonance (MR) velocimetry experiments, is identified analytically for the first time. The parabolised Navier–Stokes equation has inherent self-similarity with respect to the Reynolds number, implying that $Re$ is incorporated into the dimensionless variables rather than serving as an independent flow parameter. According to both MR velocimetry measurements and the present analytical solution, the self-similarity is not valid immediately following the pipe inlet, and this becomes more evident as $Re$ decreases; hence, the analytical solution derived from the parabolised Navier–Stokes equation cannot accurately predict the evolution of the velocity profile within this region near the pipe inlet.

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Section: Introductionmentioning

confidence: 99%

Analytical solution for laminar entrance flow in circular pipes

Kim

2024

J. Fluid Mech.

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“…Given the aforementioned focus, experimental measurements have generally been made at relatively large distances from the entrance to the pipe and a discrepancy between the predictions of analytical and numerical methods about the development of the velocity profile very close to the entrance to the pipe has not been investigated experimentally. Early approaches considering the development of the velocity profile very close to the entrance of the pipe were based on analytical solutions of approximate Navier-Stokes equations obtained by performing linearization of the inertial terms (Boussinesq 1891;Langhaar 1942;Sparrow et al 1964;Wiginton & Wendt 1969) or using Prandtl's boundary layer assumptions and solving these approximate equations using integral methods (Schiller 1922;Campbell & Slattery 1963;Mohanty & Asthana 1978), series expansions (Collins & Schowalter 1962;Schlichting 1969;van Dyke 1970;Wilson 1971) or numerical finite-difference methods (Bodoia & Osterle 1961;Hornbeck 1964;Christiansen & Lemmon 1965). These approximations were relaxed with the advent of digital computation, which enhanced the capability to solve the full Navier-Stokes equations by numerical methods.…”

Section: Introductionmentioning

confidence: 99%

Experimental evidence of velocity profile inversion in developing laminar flow using magnetic resonance velocimetry

2018

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A discrepancy exists between the predictions of analytical solutions of approximate Navier–Stokes equations and numerical finite-difference solutions of the full Navier–Stokes equations regarding the development of laminar flow at the entrance to cylindrical pipes for Newtonian fluids. Starting from a uniform velocity profile at the entrance to the pipe, analytical solutions of approximate Navier–Stokes equations predict the velocity profile to have a maximum at the centre of the pipe at all times. In contrast, numerical finite-difference solutions of the full Navier–Stokes equations have suggested that the location of the velocity maximum moves from the wall towards the centre of the pipe at a short distance from the entrance, after which it remains at the centre of the pipe. This study presents the first experimental evidence of the moving velocity maximum from the wall towards the centre of the pipe. The initial uniform velocity profile was achieved by flowing the fluid through a monolith composed of narrow parallel channels and the flow development was investigated using magnetic resonance velocimetry. The experimentally observed variation of the position and size of the velocity maximum with the Reynolds number and the distance from the entrance to the pipe is shown to be in good agreement with the predictions of numerical finite-difference solutions of the full Navier–Stokes equations.

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Transient flow in the hydrodynamic entrance regions of long closed ducts

Downing¹

1971

AIChE Journal

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Velocity profiles were calculated for transient fluid flow in the hydrodynamic entrance regions of a long narrow slit and of a long tube. The solutions were obtained by use of a linearized momentum equation. An estimate was also made of the manner in which the resistance to flow of the entrance region develops with time. The theoretical approach used by Sparrow, Lin, and au dU i a at ax rs ar -+ ~( x ) u,-= A(%, t ) + V--Lundgren ( 2 ) in their analysis of steady flow in the entrance regions of closed ducts has been extended to the corresponding transient flow situation.where U, is the average steady velocity and h ( x , t ) includes the pressure gradient and residual inertia terms. The e(x) are the same stretching parameters defined and evaluated by reference 2 and discussed by reference 3; thev will not be further discussed here other than to note VELOCITY PROFILES tha't C ( X ) for a tube ranges from a value of 0.42 at x = 0 responding values for a slit are 0.37 and 1.135. Defining the following dimensionless variables Consider the unsteady laminar incompressible flow of a and (B) a slit formed between two large parallel plates. The flow is described by the dimensional equations Newtonian fluid in the entrance regions of (A) a tube to an asymptotic Of 1*82 at large ' 7 and that ' Or-U T vt v xQ a = -, q = -r = -and X' =-(1) U, R' R2 ' R2 U, where dx = C ( X ) dx', Equation (2) becomes a a -(P u ) + -(P u ) = 0 ax ar au au au 1 ap a" aw i a -+ u -+ u -= -----= AQ(XQ,r) + -at ax a~ P ax aT +ax* qS a7,where s = 0 and 1 for flow in a slit and in a tube, respectively. Taking the axial molecular transport of momentum to be negligible relative to the radial transport, and assuming that the pressure is constant across the section, one can linearize Equation ( 2 ) :Carl G. Downing is now a consulting engineer in Corvallis, Oregon.

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Flow in the Entrance Region of Ducts

Abstract: A technique for the solution of an eigenvalue problem arising in entrance region flow is presented along with a simplification of the nonlinear transformation used with such problems. From this, certain other results are obtained.

Cited by 9 publications

References 1 publication

Analytical solution for laminar entrance flow in circular pipes

Analytical solution for laminar entrance flow in circular pipes

Experimental evidence of velocity profile inversion in developing laminar flow using magnetic resonance velocimetry

Transient flow in the hydrodynamic entrance regions of long closed ducts

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