Energy loss and developing length during reciprocating flow in a pipe with a free end

M.K. Hasan,1 R. Manasseh,1 and J.S. Leontini1

Swinburne University of Technology, John St, Hawthorn,

3122 Australia

(Dated: 27 June 2020)

Direct numerical simulations of reciprocating pipe flow in a straight pipe with a free

end are presented. The range of amplitudes and frequencies studied span the laminar

regime, and the beginning of transition towards conditionally turbulent flow. Two

primary results are reported; the measurement of the flow development length, and

the loss of energy, both due to the presence of the free end.

Two regimes of flow are identified with distinct length scales. For low frequencies,

the development length scales with the pipe diameter D. However, for higher fre-

quencies, the development length scales with the Stokes layer thickness δ =√

2ν/ω.

The energy loss is studied by calculating the viscous dissipation function, indicating

where energy is lost, and allowing the energy lost due to the presence of the free end

to be isolated. While strong vortices are formed and convected from the exit, most

of the energy they dissipate is lost within a few pipe diameters of the exit.

It is shown that these trends continue even as the amplitude and frequency are

increased so that the flow begins to transition from a laminar towards a turbulent

state. Two modes of instability are observed in the Stokes layers near the free end,

one short wavelength mode with a wavelength set by the Stokes layer thickness,

and another long wavelength mode with a wavelength set by the amplitude of the

oscillatory flow. These modes are related to those observed in the fully-developed

oscillatory flow.

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I. INTRODUCTION

Wall-bounded oscillatory flows appear in many industrial and biological processes includ-

ing water column oscillation in wave energy converters, wave-height damping and sediment

movement by water waves, water hammering and surging in pipe networks, blood flow in the

cardiovascular system, and air flow in the respiratory system. The focus of this manuscript

is on the subclass of these flows where the mean flux is zero, i.e. there is no net bulk flow

rate and the flow is reciprocating. Arguably the canonical flow in many of these situations

is the reciprocating flow in a straight, infinitely-long circular-cross-section pipe for which

an analytic solution exists for the laminar flow1,2. A number of studies, detailed below,

have investigated various aspects of this setup, including its overall friction coefficients in

the laminar and turbulent regimes, dynamics and mechanisms of transition to turbulence,

and turbulence structure. These studies were performed in the fully-developed region of the

flow, deliberately avoiding the impact of end effects. Clearly, one of the most important

prerequisites for study of the fully-developed flow is to know the flow developing length.

Unlike unidirectional flow, there is no established correlation between the Reynolds number

and the flow developing length in oscillatory flows, and the present study sets out to provide

this for pure reciprocating flow, i.e., those with zero-time-mean. Linked to this developing

length is the dynamics of the flow entering and exiting the pipe, and the associated energy

loss. One primary motivating application is that of the Oscillating Water Column (OWC),

an ocean wave energy converter (WEC) which consists of a large duct with a free end sub-

merged below the ocean surface and in which a reciprocating flow is generated by the passing

waves. In this application, understanding the energy loss is crucial to maximising the power

production.

It must be emphasized that the present study is fundamental in character and not in-

tended to be an engineering scale model of an actual OWC. The present study is essentially

laminar and transitional. Meanwhile, full-scale OWCs are likely to generate flows with that

are conditionally turbulent; as explained below, conditional turbulence is a feature peculiar

to reciprocating flows, in which the flow is laminar for some portions of the cycle and tur-

bulent for other portions. Reynolds numbers based on the boundary layer (defined below)

for full-scale OWCs can be estimated to range from O(102) to O(103), which includes the

conditionally-turbulent range3.

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Engineering scale-model studies of OWCs have been undertaken, for example by Flem-

ing et al. 4 , who stated that it would be difficult to partition the energy losses measured

in experiments into specific processes. However, understanding the losses attributable to

different flow process is required if full-scale systems are to be designed to minimize losses.

An important distinction is the difference between losses in internal boundary layers and

losses due to vortex formation at the entrance, which would require different engineering

interventions to address. While we do not expect the details of the boundary layers to be

the same in an actual OWC, the partitioning between boundary-layer and vortex-formation

losses is, by analogy with other systems in fluid dynamics, likely to be a function of geometric

parameters in both laminar and turbulent flows, and hence the present study of geometric

influences on energy dissipation should be relevant. Furthermore, the process of transition

to turbulence studied in the present paper should help to inform our understanding of the

processes occurring during conditional turbulence in the actual OWC.

Numerous studies on measuring the flow developing length in unidirectional laminar flow

are available in the literature, e.g., McComas 5 , Atkinson et al. 6 , Fargie and Martin 7 , Durst

et al. 8 . Some studies have estimated the developing length in oscillatory or pulsatile flows

with a non-zero mean, e.g., Florio and Mueller 9 , Ohmi et al. 10 , however very few works

are found in the literature that evaluated the developing length in purely reciprocating

flow. This limiting case does not appear to be a smooth transition from a pulsatile flow as

the mean approaches zero. Two notable works on the reciprocating flow are Gerrard and

Hughes 11 , and Yamanaka et al. 12 , where the developing length at different phases of the flow

cycle was measured. These studies showed that the velocity profile in reciprocating flow as

a function of xν/δ2u0, where x is the distance from the entrance, δ =√

2ν/ω is the Stokes

layer thickness and u0 is the cross-sectional mean velocity, is similar to that of the steady

flow when the velocity profile in steady flow is plotted as a function of xν/R2u0, where

R is the radius of the pipe. However, these studies were limited to a Womersley number

α = 12D√ω/ν = 14.4 and velocity parameter A0 = 2πu0max/(ωD) = 2.6 - presenting a slowly

varying flow where a quasi-steady analysis is most likely to be applicable. What happens at

lower α (which results in a larger Stokes layer thickness and therefore increases the potential

for interaction of the flow on one side of the pipe with the flow on the other side) or higher

A0 (which increases the peak velocity for a given α and therefore the potential for nonlinear

inertial effects) are not currently known.

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Like the study of developing length, energy loss due to a sharp entrance or exit in unidirec-

tional flow has been studied extensively. Loss coefficients determined from these studies have

been used in engineering problems for decades, e.g. Streeter 13 , Barbin and Jones 14 , Amer-

ican Society of Heating and Air-Conditioning Engineers Issuing Body 15 . However, when

the flow is reciprocating, the free end of the pipe works as both the entrance and exit in

one complete cycle. Thus, two processes, the generation of streamline convergence and its

evolution during inflow, and the separation of the boundary layer, generating eddies outside

of the pipe during outflow, cause the energy loss due to the free-end. Very few studies have

been conducted to investigate the free-end losses in reciprocating pipe flow; notably, Knott

and Mackley 16 and Knott and Flower 17 conducted experimental studies on submerged tubes

to measure the energy dissipation in reciprocating flow owing to the shear stress and the

free end. These experiments were conducted for high α and low A0.

While the present study focuses on the laminar regime, it is important to understand when

transitions to turbulence occur in reciprocating flow. Similarly to the questions of devel-

oping length and energy loss, transitions to turbulence in reciprocating flow have received

much less attention than in unidirectional flow. Experimental studies in fully-developed

reciprocating pipe flows from Hino, Sawamoto, and Takasu 18 and Akhavan, Kamm, and

Shapiro 3,19 clearly show that these flows are conditionally turbulent. For a fixed frequency,

there exists a critical amplitude of oscillation over which turbulent bursts occur over small

intervals of time near the beginning of the deceleration phase of the oscillation. However,

as the deceleration decreases and the flow reverses direction, the flow relaminarizes. The

length of these turbulent intervals increases with increasing amplitude of oscillation, and for

relatively high frequencies (which result in a Stokes layer thickness δ much less than the

pipe diameter), the turbulent bursts emanate from the Stokes layers. The result of this is

a flow which is turbulent for some times and laminar for others. Also for high frequencies,

the critical amplitude for the onset of conditional turbulence is effectively independent of

the frequency.

In the fully-developed flow in a straight pipe, the appearance of this conditional turbu-

lence appears to be via either bypass transition or a type of structural instability. Early

linear stability analysis on reciprocating flow in pipes and channels predicted that the flow

is unconditionally stable20. More recent analysis from Blennerhassett and Bassom 21 and

Thomas, Bassom, and Blennerhasset 22 has shown the presence of a linear instability acting

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Dimensionless numbers used Other dimensionless numbers

as input parameter known in the literature

α = (D/2)√ω/ν ReD = u0maxD/ν = 4A0α

2

A0 = u0max/ωD Reδ = u0maxδ/ν = 2√

2A0α

TABLE I. Dimensionless groups in reciprocating pipe flow

on the Stokes layers, however the predicted critical amplitude is much higher than that ob-

served experimentally. Luo and Wu 23 showed that the linear modes could be prematurely

destabilized by introducing a slight waviness at the wall, which supported the direct nu-

merical simulations of Verzicco and Vittori 24 which showed that the introduction of a very

small perturbation at the walls could destabilize a reciprocating channel flow. Further work

from Thomas et al. 25 on the reciprocating channel flow has also shown other possible modes

of transition, demonstrating that disturbance packets introduced at a particular place and

time can grow and spawn further packets, with the overall disturbance growing along the

channel and over time.

There are numerous ways to scale the reciprocating flow in a pipe or a channel, and

previous results have typically been presented in terms of two types of oscillatory Reynolds

number; ReD and Reδ. Conversion between the parameters used in the present study (α and

A0) and the oscillatory Reynolds numbers (ReD and Reδ) are presented in Table I. Table

II shows the critical dimensionless numbers at the onset of conditional turbulence, observed

from different experimental studies.

Critical numbers determined by these studies do not differ much except the one found

in Merkli and Thomann 26 as shown in table II. An explanation of this large variation

is given in Ohmi et al. 27 . According to Hino, Sawamoto, and Takasu 18 , there are two

stages of transition from laminar to turbulent regimes in reciprocating flow; the first is from

laminar to distorted laminar flow (where “distorted laminar” refers to a flow that is spatio-

temporally complex and potentially chaotic, but not fully turbulent) and the other second is

from weakly turbulent to conditionally turbulent. Ohmi et al. 27 presumed that the critical

Reynolds number measured in Merkli and Thomann 26 is for the transition from laminar to

distorted laminar whereas others measured the critical Reynolds number for the transition

from weakly turbulent to conditionally turbulent flow.

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Reference RecritD Recritδ (A0α)crit Range of α

Sergeev 28 700α 495 175 4 ≤ α ≤ 40

Merkli and Thomann 26 400α 283 100 42 ≤ α ≤ 71

Hino, Sawamoto, and Takasu 18 780α 550 195 1.91 ≤ α ≤ 8.75

Ohmi et al. 27 780α 550 195 2.6 ≤ α ≤ 41

Eckmann and Grotberg 29 707α 500 177 8.9 ≤ α ≤ 32.2

TABLE II. Critical dimensionless numbers at transition in a fully-developed pipe flow without end

effects.

FIG. 1. Schematic diagram of the geometry. Lengths are presented in terms of the pipe diameter

D.

The present work uses a Direct Numerical Simulation (DNS) code to study the impact

of low α and high A0 on the developing length and energy loss in reciprocating flow. All the

studies are conducted for the parameter range of α (2.24 ≤ α ≤ 10) and A0 (1 ≤ A0 ≤ 9).

Hence the range of A0α studied is from 2.24 to 90, where the maximum value is less then

the lowest critical value mentioned in Table II for the fully-developed reciprocating flow,

hence the deviation from the laminar state reported here is due to the presence of the free

end, and it is shown in section VI that the free end excites flow features reminiscent of the

instability modes summarized above.

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II. METHODOLOGY

A. Computational setup

The present study investigates the flow field in a pipe with free-ends terminating in

reservoirs as shown in figure 1. All the dimensions are normalized by the pipe diameter.

The flow rate at the reservoir inlets is driven sinusoidally with time, and equal and opposite

at each end. The DNS code used in the present study uses a nodal-based spectral-element

method to solve the incompressible Navier-Stokes equations. It is assumed that the flow is

axisymmetric throughout the flow domain, resulting in a two-dimensional, three-component

simulation on a two-dimensional domain bounded by the pipe centreline, the inlet and outlet,

and the pipe and reservoir walls. On this domain the final equations solved are

∂u

∂t= −

(v∂u

∂r+ u

∂u

∂x

)− 1

ρ

∂p

∂x+ ν

(∇2u

)(1)

∂v

∂t= −

(v∂v

∂r+ u

∂v

∂x

)+w2

r− 1

ρ

∂p

∂r+ ν

(∇2v − v

r2

)(2)

∂w

∂t= −

(v∂w

∂r+ u

∂w

∂x

)− vw

r+ ν

(∇2w − w

r2

)(3)

for momentum, and

1

r

∂(rv)

∂r+∂u

∂x= 0 (4)

for the conservation of mass, where x, r, and θ are the coordinates in the streamwise, radial

and azimuthal directions respectively, u, v and w are the velocity components in these

directions, t is time, ρ is the fluid density, ν is the kinematic viscosity, and the operator ∇2

is defined as

∇2 =∂2

∂r2+

1

r

∂

∂r+

∂2

∂x2(5)

This domain was discretized into 4720 quadrilateral elements, concentrated near the pipe

walls, the entries/exits of the pipe and the areas along the centreline in the reservoirs where

large flow gradients were expected to occur. Figure 2 shows the mesh structure first for one

half of the domain (the mesh structure is reflected about the midpoint of the pipe), as well

as a close-up image of the mesh in the vicinity of the free end of the pipe. These elements

were further discretized using 7th-order tensor-product Lagrange polynomials as shape func-

tions (these are the basis functions from which the solution on each element is constructed),

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’

FIG. 2. The spectral element mesh shown from the midpoint of the pipe to the end of the reservoir

(left) and a close-up view of the same mesh in the vicinity of the free end of the pipe (right). The

yellow line shows the domain boundary where axisymmetry is imposed, the cyan line shows the

pipe wall.

associated with Gauss-Legendre-Lobatto quadrature points. These same Lagrange polyno-

mials were used as weighting functions in the variational formulation employed, resulting in

a Galerkin scheme.

For time integration, a three-way time-splitting scheme was used. This results in separate

“sub-step” equations for the advection, pressure, and diffusion terms of the Navier-Stokes

equations. For the advection equation, a second-order Adams-Bashforth scheme was em-

ployed. A second-order scheme was also employed for the pressure which, when combined

with enforcing the divergence-free condition from the incompressible continuity equation,

results in a Poisson equation for the pressure correction (or the change in pressure from the

previous to the current timestep). The diffusion equation was solved using a second-order

Crank-Nicolson scheme. Further details of the spectral-element method in general can be

found in Canuto et al. 30 , Karniadakis and Sherwin 31 , and details of the implementation used

here, including in an axisymmetric context can be found in Griffith et al. 32 . The boundary

conditions on the velocity field were a sinusoidally-time-varying parabolic velocity profile at

the inlet/outlet of the reservoirs,

u(r) = U0

(1−

( rR

)2)cos(ωt) (6)

where U0 is the maximum velocity at r = 0 on the reservoir boundary and R is the radius of

the reservoir. This imposed a volumetric flow rate at the reservoir boundary, and therefore

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in the pipe, of the form

Q = 2πU0

∫ R

0

(1−

( rR

)2)r.dr cosωt =

π

2U0R

2 cosωt = Q0 cosωt. (7)

This could then be related to the maximum cross-sectional-mean velocity in the pipe via

u0max =Q0

πD2/4=

4Q0

πD2. (8)

For implementation, the flow rate was set such that was always u0max = 1 for all simulations,

and the dimensionless groups were varied by varying ω and ν.

A no-slip boundary condition (u = v = w = 0) was applied at the pipe and reservoir

walls, and a reflection condition on the axis of axisymmetry.

For the pressure, a Neumann condition was used to solve the Poisson equation for the

pressure correction resulting from the three-way time splitting, imposing the gradient of the

pressure in the normal direction, with the value of the gradient derived from the Navier-

Stokes equations i.e, ∂p/∂n = ν∇2(u ·n)− (∂(u ·n)/∂t+u.∇(u ·n))33. Note that at no-slip

walls this reduces to the commonly used ∂p/∂n = 0. All simulations were started from rest.

B. Decomposition of power dissipation to assess the impact of the pipe free

end

While calculating the energy dissipation rate, or dissipated power, in the control volume

the dimensionless viscous dissipation equation for axisymmetric flow is integrated over the

entire domain,

P =

∫V ∗

1

A0α′

[2

(∂v∗

∂r∗

)2

+ 2

(v∗

r∗

)2

+ 2

(∂u∗

∂x∗

)2

+

(r∗

∂

∂r∗

(w∗

r∗

))2

+

(∂w∗

∂x∗

)2

+

(∂v∗

∂x∗+∂u∗

∂r∗

)2]dV ∗, (9)

where V ∗ is the dimensionless volume of the control volume (i.e, volume normalized by

D3, and α′ = 4α2, the starred quantities (u∗, v∗ and w∗) represent velocities normalized

by the maximum cross-sectional-average velocity u0max , and the starred quantities (x∗, r∗)

represent lengths normalized by the pipe diameter D. The exact energy dissipation rate

can be calculated by multiplying the dimensionless dissipation P by ρu30maxD2. The overall

dissipation in the domain, P , can be written as the sum of the free-end loss, Pe and the

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shear stress loss that would have occurred only inside the pipe if the flow was assumed to

be fully developed throughout the pipe, Ps,

P = Pe + Ps. (10)

For all the simulations conducted in this study, it was confirmed that the flow returned to the

fully-developed analytical profile inside the pipe far from the free end. Therefore, the shear

stress loss Ps can be calculated analytically from the laminar solution for the reciprocating

flow in a straight pipe2, applied over the entire finite length of the pipe used. Any other

losses - those inside the pipe generated by the deviation from the flow that would have been

generated if the free end were not present, and any losses generated outside the pipe - can

be attributed to the presence of the free end. Hence, the loss attributable to the presence of

the free end is simply calculated by subtracting Ps from P , where P is calculated over the

entire computational domain according to equation (9).

Of course, equation (10) is only a decomposition, and is not unique. The total loss could

be decomposed in any number of ways (such as the loss inside and outside of the pipe, loss

during inflow and outflow, etc.) However the decomposition outlined in equation (10) does

give an indication of the loss induced by the vortices outside the pipe, and the deviation

of the flow inside the pipe from the fully developed solution that is independent of the

developing length.

III. VALIDATION

To test the validity of the setup - driving fluid through the pipe connected to two reservoirs

- axial velocity profiles at the midpoint of the pipe, where the flow is expected to be fully

developed, are compared with the exact solution for the reciprocating flow in an infinitely

long pipe2. The comparison is shown in figure 3 for α′ = 50 and A0 = 3, and for α′ = 400

and A0 = 3. In both cases the match between the analytic solution and the simulation is

excellent.

A mesh resolution and domain size study was also completed. Since a focus of the study

is the energy loss induced by the free end of the pipe, table III compares this energy loss for

a series of resolutions and domain sizes for the case α′ = 400 and A0 = 3. The resolution was

changed by simply changing the order of the shape functions used on each element, which

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-0.5

0

0.5

-2 -1 0 1 2 -1.5 -1 -0.5 0 0.5 1 1.5

5π

3

2π

3

4π

3π

π

32π

5π

3

2π

3

4π

3π

π

32π

u u

r/D

(a) (b)

FIG. 3. Comparison between theoretical results (—) and simulation results (•) of the velocity

profiles in fully-developed flow for (a) α′ = 4α2 = 50, A0 = 3; (b) α′ = 400, A0 = 3 at six phases

of the oscillation cycle.

changes the number of internal points on each element. The mesh structure and domain

extent for the resolution test were the same as those outlined in figures 1 and 2. Both the

length and the width of the reservoirs was also varied for the domain study, while the total

number of elements and internal points was held constant. The mesh was only deformed in

the far field, so that the mesh structure and resolution in the pipe, and near the free end,

remained constant. The data show that the changes in resolution and domain size have little

effect on the energy dissipation, providing some confidence that the mesh employed resolves

all of the relevant flow features accurately.

IV. MEASURING THE DEVELOPING LENGTH

Figure 4 demonstrates the flow development along the pipe at different phases of the cycle

for α′ = 400 and A0 = 3. It is apparent that the velocity profile at and near the free end

(x/D = 0, 1, 2) differs significantly from the profile in the fully-developed region (x/D = 50).

The present study focuses on measuring the length required for the reciprocating flow to

become fully-developed. The transverse velocity gradient at the wall, ∂u/∂r|wall, is used

to measure the developing length. The flow is considered fully-developed when the local

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Pemin Pemax P e

Resolution

p = 6 0.06980 0.13563 0.09993

p = 7 0.07176 0.13071 0.09877

p = 8 0.07197 0.13212 0.09989

Domain length

50 0.07720 0.13217 0.09997

75 0.07200 0.13215 0.09993

100 0.07197 0.13212 0.09989

125 0.07187 0.13204 0.09979

150 0.07181 0.13196 0.09970

Domain width

15 0.07195 0.13215 0.09988

20 0.07197 0.13212 0.09989

25 0.07198 0.13209 0.09988

TABLE III. Resolution and domain test results for α′ = 400 and A0 = 3. Here p represents the

order of the shape functions on each element, the domain length is the length of each reservoir

and the domain width is the radius of the reservoir. Pemin , Pemax and P e represent the minimum,

maximum and cycle-averaged viscous dissipation rates.

∂u/∂r|wall becomes equivalent to 99% of (∂u/∂r|wall)∞, where (∂u/∂r|wall)∞ is the value of

∂u/∂r|wall in the fully-developed region which was calculated from the analytic solution of

Uchida 2 . Figure 5 shows the developing lengths measured by this method superimposed

on contours of ∂u/∂r|wall measured along the wall over time for α′ = 100, 200, 300 and

400, and for A0 = 3. It shows that the developing length does not follow a sinusoidal

or co-sinusoidal pattern with time though the flow is purely periodic. During the inflow

(φ = 0 to π/2 and 3π/2 to 2π) the developing length increases with time following a

parabolic profile, which is expected. However, during the outflow (φ = π/2 to 3π/2) all the

perturbations introduced by the free-end (at the time of inflow) are pushed back towards

the pipe-end. Hence, it is possible that during the outflow the developing length can become

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0

0.5

0 0.4 0.8 0 0.4 0.8

0

0.5

0 0.4 0.8 0 0.4 0.8

u u

u ur/D

r/D

x/D

=0 1 2 50

x/D

=0 1 2 50

x/D

=0 1 2 50

x/D

=0 1 2 50

φ = π/6

φ = 10π/6φ = 7π/6

φ = 4π/6

FIG. 4. Evolution of velocity profile along the pipe, at different phases of the cycle for α′ = 400

and A0 = 3.

very short (effectively zero), and remain so for a portion of the cycle. This phenomenon is

shown in the data in figure 5(a). Importantly, this shows that the variation of the flow from

the inflow to the outflow caused by flow separation and the formation of vortices, introduces

a phase modulation - the flow is not spatio-temporally symmetric, and it is for this reason

that the instantaneous development length deviates from a pure sinusoid. Additionally,

it can be seen that during the change of the direction of ∂u/∂r|wall the developing length

deviates from the trend.

The maximum developing length is found when ∂u/∂r|wall changes its sign. At these

points in time, the developing length can vary rapidly as the flow near the pipe wall reverses

its direction and develops in the flow direction. The streamwise development can also be

complex, as vortex structures developed from instabilities triggered by the presence of the

free end in the previous half-cycle are convected. Further description of this complicated

flow structure is presented later in section VI. In general, the developing length variation

is complicated yet periodic, with a maximum developing length occurring near φ = π/2 or

maximum inflow.

Figure 6(a) shows how the normalized maximum developing length, l0/D varies with α′,

for a fixed oscillation amplitude (i.e. A0 = 3). The developing length l0/D increases linearly

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0 2 4 6 8 10 12 14

0

π/2

π

3π/2

2π

-100 -50 0 50 100

0 2 4 6 8 10 12 14

0

π/2

π

3π/2

2π

-100 -50 0 50 100

0 2 4 6 8 10 12 14

0

π/2

π

3π/2

2π

-100 -50 0 50 100

0 2 4 6 8 10 12 14

0

π/2

π

3π/2

2π

-100 -50 0 50 100

x x

x x

∂u/∂r|wall ∂u/∂r|wall

∂u/∂r|wall ∂u/∂r|wall

φ φ

φ φ

(a) (b)

(c) (d)

FIG. 5. Contours of ∂u/∂r|wall for A0 = 3 and (a) α′ = 100, (b) α′ = 200, (c) α′ = 300 and (d)

α′ = 400, as a function of the distance from the free-end (x) and the phase variation φ; the symbols

(•) shows the location where ∂u/∂r|wall = 99%(∂u/∂r|wall)∞.

with α′ up to α′ = 60. In the range 60 ≤ α′ ≤ 400, l0/D oscillates between 7 and 10. This

indicates that, at least for this relatively low amplitude, there are possibly two flow regimes

present as a function of α′. Figure 6(b) shows how the developing length varies with the

oscillation amplitude A0 for a fixed α′ (i.e. α′ = 400). It can be seen that l0/D increases

linearly with the oscillation amplitude, indicating the high α′ regime is reasonably consistent

across a wide range of amplitudes.

The linear increase in the developing length with A0 is not surprising, as A0 can be

interpreted as the amplitude of the oscillation of the flow. It would therefore be expected

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4

6

8

10

0 100 200 300 400

0

5

10

15

20

25

30

0 2 4 6 8 10

l 0/D

l 0/D

α′ A0

(a) (b)

FIG. 6. Maximum developing length of the cycle measured at ∂u/∂r|wall = 99%(∂u/∂r|wall)∞ (a)

as a function of α′, for A0 = 3, (b) as a function of A0, for α′ = 400.

0

5

10

15

20

25

30

0 1000 2000 3000 4000

0

100

200

300

400

0 100 200 300

l 0/D

l 0/δ

ReD Reδ

(a) (b)

α′ > 50 ◦α′ ≤ 50 •

α′ > 50 ◦α′ ≤ 50 •

FIG. 7. (a) Maximum developing length to diameter ratio as a function of ReD. The straight

line represents the correlation l0/D = 0.07ReD, (b) maximum developing length to Stokes-layer

thickness ratio as a function of Reδ. The straight line represents the correlation l0/δ = 1.43Reδ.

that as this amplitude increases, the advection distance of a given fluid packet also increases,

and so the influence of the free end, and therefore the developing length, should increase

in step with the amplitude. However, this straightforward explanation does not explain the

increase in the developing length with α′ when α′ is small.

The presence of these multiple flow regimes is further elucidated in figure 7. Figure 7(a)

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shows the maximum developing length to diameter ratio (l0/D) plotted as a function of

ReD = A0α′ = u0maxD/ν. The data are grouped into low and high α′ classes. Guided by

the result of developing length as a function of α′ shown in figure 6, a threshold value of

α′ = 50 is used. Figure 7 shows that for α′ ≤ 50, l0/D collapses to a linear trend when the

length scale is taken as the pipe diameter D. Hence the correlation between l0/D and ReD

for α′ ≤ 50 is

l0/D = 0.07ReD (11)

which is similar to the correlation that is found in steady laminar flow34.

The same data can be plotted using an alternative scaling. Figure 7(b) plots the develop-

ing length normalized by the Stokes layer thickness δ =√

2ν/ω instead of the pipe diameter

D. This is plotted against Reδ = A0

√2α′ = u0maxδ/ν (i.e. the Reynolds number using δ as

the length scale instead of D). Again, the data are separated into low and high α′ classes.

Here, all the data for α′ > 50 collapse towards a single linear trend as shown in figure 7(b),

while the data for α′ ≤ 50 sit below this trend. Hence, a linear correlation between l0/δ and

Reδ for α′ > 50 is

l0/δ = 1.43Reδ. (12)

There is some fluctuation about this linear trend, especially for the higher Reδ cases. It is

hypothesized that this is due to an increase in flow complexity due to instability in these

cases, and these instabilities are investigated in more detail in section VI.

The change in in relevant length scale can be somewhat explained by considering the

change in flow structure with α′. For high values of α′ where the relevant scale is δ, the

Stokes layer formed in the pipe is thin and well-defined. For low values of α′ where the

relevant scale is D, the Stokes layer is much thicker, approaching the radius of the pipe such

that the layer on one “side” of the pipe interacts with the layer on the other side. Further

decreases in α′ would result in thicker Stokes layers - which of course is not possible due to

the constraint of the pipe. At this point, the length scale saturates to the diameter of the

pipe. This argument is also supported by the flow visualisation provided below in section

VI.

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V. ENERGY LOSS DUE TO THE FREE END

Different mechanisms cause energy loss due to the free end in reciprocating flow. During

the inflow, it is the generation of the streamline convergence and its evolution. During the

outflow, it is the separation of the boundary layer, causing the generation of vortices outside

of the pipe. Knott and Mackley 16 and Knott and Flower 17 studied the vortex formation

and the consequent energy loss at the mouth of partially submerged vertical tubes subjected

to reciprocating flow. These studies were conducted for high α′ and for low A0. The present

work focuses on a more detailed study of the generation of vortices due to the free-end, their

evolution throughout the period and the consequent energy loss for a lower range of α′ ,

however, for a much wider range and greater value of A0 than these earlier studies.

To demonstrate how the free end contributes to the energy loss in the reciprocating flow

system, the generation and evolution of vortices near the free end and the consequent energy

dissipation on a sectional plane are presented in figure 8, for an example case at α′ = 400

and A0 = 3. It shows the contours of vorticity on the left and contours of energy dissipation

rate Φ on the right. Since the contours of vorticity are presented on a sectional plane, the

vortices near the upper wall are opposite in sign to the vortices near the lower wall - although

of course they are simply the component of azimuthal vorticity normal to the page.

Focusing first on the development of vorticity, during the inflow (φ = π/3), owing to

the sharp entrance, high vorticity is noticeable within the first few diameters of pipe length

concentrated in the boundary layers. However, all the vorticity generated in the pipe during

the inflow is pushed towards the free end when the flow reverses and the outflow starts

(φ = 2π/3). During the outflow, the stationary fluid in the reservoir detaches the outer

layer of the fast moving fluid that comes out from the pipe, resulting a vortex ring to form

just outside the pipe (φ = π). Note the initially complex structure of this vortex, as it

wraps to layers of opposite-signed vorticity around each other, generating a a region of high

shear between these layers. As the outflow progresses and more fluid is ejected from the

pipe (φ = 4π/3), this vortex ring continues to grow in size. The layer of vorticity of opposite

sign from that being fed from the boundary layer in the pipe (which was being wrapped

around the outside of the vortex) is concentrated into a second weaker vortex outside the

primary vortex. As the flow again reverses in direction and the flow begins to re-enter the

pipe (φ = 5π/3) the primary vortex ring propagates away from the free-end, however the

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smaller secondary vortex is left effectively stationary. With further progression of the inflow

(φ = 2π), the intensity of vorticity once again increases in the boundary layers at the pipe

entrance, while the bigger ring continues self-advect away from the free-end, and the smaller

secondary vortex begins to diffuse.

Changing focus to the dissipation fields (Figure 8(right)) it can be seen that during the

inflow (φ = π/3), the flow separates from the wall due to the sharp entrance, and the highest

dissipation takes place in the separated shear layer. As the flow crosses the vena-contracta,

it gets attached to the wall and the dissipation takes place due to the shear stress in the

boundary layer now attached to the wall. As the flow begins to reverse (2π/3) it is clear

that there is high dissipation in both the separated shear layer created from the boundary

layer during the inflow and the attached boundary layer being formed at the wall. As the

outflow progresses and the two-layer vortex is formed at the free end (φ = π) most of the

dissipation inside the pipe takes place near the wall due to the boundary layer, however

there is also significant dissipation outside the pipe in the high-shear region in the layer of

vorticity being wrapped around the outside of the vortex. As more fluid is ejected from the

pipe (φ = 4π/3) the boundary layers in the pipe (and their continuation as free shear layers

outside the pipe) continue to have high levels of dissipation, as does the high-shear braid

region that connects the primary vortex and the well-formed weaker secondary vortex. As

the flow reverses and the flow begins to re-enter the pipe (φ = 5π/3), the dissipation is

almost completely concentrated in the forming boundary layers inside the pipe; even though

there are strong vortices present in a reasonably complex configuration, the shear in these

vortices is low, meaning they dissipate little energy. This continues as the inflow continues

to increase (φ = 2π). An important point to note from this is that the strong vortices ejected

from the pipe dissipate little energy; in fact within a few diameters downstream from the

free-end, the dissipation caused by the vortices becomes insignificant.

A primary interest of this study is to establish the energy loss due to the presence of the

free end of the pipe. This can be done by considering how energy is removed from the flow

domain. The boundary conditions at the end of the reservoir are equal and opposite, and

so over one cycle the net energy input via kinetic energy flux is zero. This effectively leaves

energy input at the boundaries due to pressure (assuming an isothermal fluid). After any

initial transient, there is no energy accumulation in the domain over a period of oscillation,

and so the energy input must be balanced by the loss of energy via viscous dissipation. The

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φ = π/3

2π/3

π

4π/3

5π/3

2π

FIG. 8. Contours of vorticity (left) and viscous dissipation rate Φ (right) at the free end, for

α′ = 400 and A0 = 3, at various oscillation phases φ. Red and blue colours on the vorticity contour

plot represent positive and negative values of the dot-product of vorticity and the unit normal to

the page, respectively. Note that the vorticity and dissipation structure outside the pipe that was

generated in previous cycles has been erased to only show the structure generated during one cycle

of oscillation.19

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total average power loss over a cycle P can be therefore calculated by integrating the viscous

dissipation function over the cycle and the entire flow domain, as shown in equation (9).

This total power loss can then be used to find the power loss due to the presence of the

free end. The parameter range studied here produces a laminar flow when the flow is fully

developed. Hence, the average rate of loss over a cycle of oscillation in the entire pipe - if

it is assumed that the flow is fully developed - can be calculated from the analytic laminar

solution. Here, this quantity is referred to as P s, or the power loss due to the shear stress

in the pipe. Any other loss above this is due to the presence of the free end, either via the

production of vortices at the exit or deformation of the flow profile inside the pipe, and is

here referred to as P e. So, this power loss due to the free end can be found by subtracting

the loss due to the hypothetical fully-developed flow from the total power loss,

P e = P − P s. (13)

Figure 9 shows the dimensionless cycle-average free-end loss P e, as function of α′ and

ReD for different A0. The value of P e decreases with α′ and converges for higher A0. A

better convergence of P e at higher A0 can be seen when the same data are plotted against

ReD. With respect to both α′ and ReD, P e decreases exponentially for lower values, however

becomes nearly constant for higher α′ and ReD. The solid line in figure 9(b) provides an

approximate upper bound for the power loss due to the end as

P e =10

Re3/4D

+ 0.07. (14)

There is some slight deviation or noise in the data for the higher values of A0 around the

general exponential decay described by equation (14). This is particularly noticeable for the

A0 = 7 case in figure 9(b) This is due to the fact that as ReD increases, the flow begins to

deviate from a laminar state to something more spatio-temporally complex. This point is

expanded upon in section VI.

To understand the significance of the free-end loss in the overall loss, the cycle-average

shear stress dissipation per unit length of the pipe in the fully-developed region, P s/l is

plotted as function of α′ and ReD for different A0 in figure 10. Like P e, P s/l shows a better

convergence when plotted against ReD. P s/l decreases exponentially with both α′ and ReD

at the beginning, however, P s/l becomes constant at much lower values than that for the

P e. Comparison between P e and P s/l within the range of ReD where both the parameters

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0.05

0.1

0.15

0.2

0.25

0.3

0 100 200 300 400

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1000 2000 3000 4000

α′ ReD

Pe

Pe

(a) (b)

A0 = 1 +

A0 = 3 ∗A0 = 5 ×

A0 = 7 ◦A0 = 9 •

A0 = 1 +

A0 = 3 ∗A0 = 5 ×

A0 = 7 ◦A0 = 9 •

FIG. 9. Cycle-average energy dissipation due to the free-end, P e for different A0; (a) as a function

of α′, (b) as a function of ReD.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 100 200 300 400

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1000 2000 3000 4000

α′ ReD

Ps/l

Ps/l

(a) (b)

A0 = 1 +

A0 = 3 ∗A0 = 5 ×

A0 = 7 ◦A0 = 9 •

A0 = 1 +

A0 = 3 ∗A0 = 5 ×

A0 = 7 ◦A0 = 9 •

FIG. 10. Cycle-average shear stress dissipation per unit length in the fully-developed region of the

pipe, P s/l, for different A0; (a) as a function of α′, (b) as a function of ReD.

are nearly constant (e.g. between 2000 and 3600) shows that P s would be equal to the P e

for a pipe which is approximately 10 times its diameter long, i.e., l ≈ 10D. Hence, P e is

significant within this range of ReD. This finding is quite contrary to one of the assumptions

of Knott and Flower 17 , which states that the dissipation between ReD = 0 and 7000 is only

from the shear stress loss in the pipe. Knott and Flower 17 used pipes of approximately

2.3D, 4D and 7D long which are shorter than 10D. In fact, the present study shows that

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the effect of the free end remains significant up to almost 28D inside the pipe at ReD = 3600

(figure 7(a)).

VI. THE DEVELOPMENT OF TRANSITORY EFFECTS

Figure 11 shows contours of vorticity and viscous dissipation rate at the end of the pipe

at the point in the cycle where the net mass flux is zero, coinciding with the point where

the net deceleration is maximum. Examples are shown for both increasing A0 and α′.

The first feature of note is the dependence of the viscous dissipation on the scale of the

Stokes layers in the pipe. The low A0 and/or low α′ cases (which coincide with large values

of δ) show shear layers that effectively meet at the centre of the pipe, and there are high

levels of viscous dissipation over almost the entire interior of the pipe. Increasing either

parameter (effectively increasing δ) sees the thickness of the shear layers diminish, and the

areas of the flow with significant viscous dissipation diminish accordingly.

The second feature of note is the development of instability in the shear layers inside the

pipe, for the highest A0 and/or α′ cases. Both the vorticity and the viscous dissipation show

a pronounced streamwise dependence with a wave-like structure that is far more pronounced

than the self-similar development of the boundary layers forming from the end of the pipe.

This is most pronounced for the highest A0/α′ case shown A0 = 9, α′ = 400. However,

evidence of this instability is present for all of the values of A0 shown at α′ = 400 in the last

column of figure 11.

Concentrating on the α′ = 400 case shown in this last column shows that there are two

characteristic wavelengths of this instability. The first of these - a long wavelength - is clear

in the case at A0 = 5 which shows long shear layers that roll up into a distinct vortex at

a distance (at the instant shown) of around 4D. We propose that this long wavelength

instability is related to the convection of a disturbance introduced by the presence of the

free end of the pipe by the bulk motion of the fluid. Its wavelength, therefore, should scale

on the amplitude of the reciprocating motion A0.

The second wavelength - a short wavelength - is more obvious in the case at A0 = 9,

which shows that the almost constant-thickness shear layers of the lower amplitude cases

are completely lost due to a break-up into a complicated vortex pattern. We propose that

this break-up is due to the triggering of the inherent Stokes layer instability modes by the

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α′=

20

α′=

240

α′=

400

A0=1 A0=5 A0=9

FIG

.11.

Con

tou

rsof

vort

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yan

den

ergy

dis

sip

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free

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dat

the

osci

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ph

aseφ

=π/2

(i.e

.,th

efl

owis

chan

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irec

tion

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ou

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),fo

rva

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sα′

andA

0.

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xx

|v||v|

φφ

(b)(a)

FIG. 12. Contours of absolute radial velocity component, |v| for α′ = 400 and (a) A0 = 5, (b)

A0 = 9 as a function of the distance from the free-end, x and the phase variation, φ.

disturbance introduced by the presence of the free end.

To investigate this in further detail, space-time diagrams of the two cases at α′ = 400

and A0 = 5 and A0 = 9 are presented in figure 12. These diagrams have been produced

by sampling the radial velocity along a line at a distance 0.1D from the wall, over one

period of oscillation. Time is presented in terms of the phase of the oscillation φ, with

φ = 0 corresponding to the time when the instantaneous flow rate is maximum into the

pipe (to the right in the images shown in figure 11) or equivalently to the time at the end

of the acceleration of the flow into the pipe and the beginning of the deceleration. The

radial velocity is chosen, because its deviation from zero clearly shows the development of

secondary structures on top of the almost-parallel flow inside the pipe that is expected if

there is no instability. Both images in figure 12 show the same basic structure. There are

large dark bands present over the time period from the beginning of the acceleration phase

into the pipe (φ = 3π/2) to just beyond the beginning of the deceleration (which occurs at

φ = π/2). There are clearly two length scales present in these diagrams. The longer one

is shown by the spacing between the groups of finer lines, which at φ = π/2 are spaced by

around 2D in the A0 = 5 case, and by around 4D in the A0 = 9 case. In both cases, this

spacing is slightly less than A0/2, indicating that there is a wavelength of approximately

A0 present (the dark regions represent high magnitude of the radial velocity and therefore

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local minima and maxima). This is reminiscent of the “family-tree-structure” of disturbance

growth reported by Thomas et al. 25 when a localized linear disturbance is introduced to the

reciprocating flow in a channel. That study also reported on localized disturbances in space

that would convect a distance down the channel that scaled with the amplitude of oscillation,

with most of the growth of the introduced wave packets occurring over the portion of the

cycle where the deceleration is increasing (φ = 0 to φ = π/2 in figure 12). It should be

noted that while the scaling of the wavelength with amplitude can be inferred from the

results of Thomas et al. 25 , the connection between the convection length and the amplitude

of oscillation was not explicitly stated. We hypothesize that the pipe free end introduces a

local disturbance which is convected into the pipe as flow enters the pipe, with the largest

growth of this disturbance occurring over the deceleration phase, in a similar fashion to that

outlined in the perturbed streamwise homogeneous flow from Thomas et al. 25 .

Effectively superimposed on these dark bands is a much finer-scale pattern with a much

shorter length scale or wavelength. This shorter scale seems almost independent of A0, with

a value somewhere around 0.6D. Such a short wavelength mode is predicted to grow on

the oscillatory flow in an infinite pipe. Thomas, Bassom, and Blennerhasset 22 used Floquet

analysis to show that for flows with a large diameter to Stokes layer ratio (as occurs for

the higher A0 cases here) the first mode to become unstable has a spanwise wavenumber

normalized by the Stokes layer thickness αc = 0.38. Luo and Wu 23 also report that flows

that are not linearly unstable can be triggered via a detuned resonance by introducing a

small-amplitude waviness with a wavenumber around this same value of αc = 0.38. The

ratio of the Stokes layer thickness to the diameter of the infinite pipe can be calculated as

δ

D=

√2

α′, (15)

which can be used to convert the wavenumber αc to a wavelength. Doing so gives a critical

wavelength of

λcD

=2π√

2/α′

αc(16)

so that for αc = 0.38, λc ' 1.17D - around twice the observed wavelength evident in figure

12. However, this number is calculated based on a fully-developed Stokes layer, whereas the

layers in the pipe entry are spatially developing, are significantly thinner near the pipe entry

which may explain the reduction in the wavelength of this instability.

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Therefore, while there is not an exact quantitative match, the development of instability

in the end of the truncated pipe seems to be due to similar mechanisms to fully-developed

Stokes layers - a long wavelength mode with a wavelength that scales on the amplitude

of the oscillation develops due to the local disturbance caused by the free end, and a short

wavelength mode develops due to the inherent modes of the Stokes layers that can be excited

by the perturbation introduced by the free end.

VII. THE VALIDITY OF THE AXISYMMETRIC INSTABILITY

MECHANISM

The observations made in the previous sections are based on axisymmetric simulations,

and therefore may not be observed if the flow is, in fact, susceptible to instability due to non-

axisymmetric perturbations. To assess this validity, here the instabilities of two canonical

flows are used as a guide to the stability of this flow. First, the flow outside of the pipe is

assessed against a series of vortex rings, or a zero-net-mass-flux jet, which is effectively the

flow generated outside of the pipe. Second, the flow inside of the pipe is assessed against

the stability of Stokes layers and the flows in infinitely long pipes.

Glezer 35 , and more recently from Shuster and Smith 36 , studied the stability of trains of

vortices generated from a circular orifice. The study from Glezer 35 investigated the stability

of vortices generated using experiments consisting of a constant-speed piston that pushed

fluid from an orifice, with the piston travel stopping with the piston face flush with the

wall containing the orifice. It was found that the critical Reynolds number (defined as

Re0(crit) = PA20α′/π, where P = 1 for the constant-speed piston, and A0 and α′ are as

defined in this paper) increased with increasing A0 - for A0 ' 1, Re0(crit) ' 6000, with

Re0(crit) asymptoting to a value around Re0(crit) = 3 × 104 for A0 > 4. Here, the highest

Re0 = PA20α′/π = 1×92×400/π ' 104, suggesting all the cases of this paper would produce

stable vortices.

However, the later study from Shuster and Smith 36 , investigating trains of vortices using

a similar experimental setup but with a sinusoidal piston motion profile, found critical

Reynolds numbers almost a factor of five smaller. This dramatic difference was attributed

to the impact of vortices from a previous cycle. If this lower transition Reynolds number

scenario is used, then cases in this paper for α′ = 400 and A0 > 4 could produce vortices

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that are unstable and therefore the axisymmetric assumption for a small number of the

simulations of this paper are not physically realisable.

Taken together, the experimental evidence suggests that at least the initial vortex for-

mation process is likely to be laminar and therefore axisymmetric. A breakdown of these

vortices as they travel away from the pipe may break the axisymmetry, however this is un-

likely to have a large impact on the overall loss, as figures 8 and 11 show the majority of

the viscous loss outside of the pipe occurs in the vortex as it is forming.

Perhaps more importantly is a consideration of the flow inside the pipe, in particular the

validity of the short wavelength mode developing on the Stokes layers. The linear stability

analysis of Thomas, Bassom, and Blennerhasset 22 investigated this problem in detail for

the infinite pipe, showing that first mode to become unstable on the Stokes layer was an

axisymmetric mode, except for very low values of α′ (where a separate Stokes “layer” are is

not well defined as significant gradients exist across the entire pipe). In section VI it was

argued that the appearance of the short wavelength mode is due to the excitation provided

to this Stokes layer mode by the free end, and so it seems likely that again the most unstable

mode is axisymmetric.

To definitively decide on the applicability of these axisymmetric results, a fully three-

dimensional simulation, or at least a Floquet stability analysis, is required, and these anal-

yses will be the topic of future work.

VIII. CONCLUSIONS

The flow developing length, energy dissipation due to the free end in reciprocating flow,

and the loss of stability of the laminar flow has been studied via DNS. The present study

focuses on the flow dynamics and the energy dissipation in reciprocating flow.

It is found that the maximum developing length, l0 for α′ ≤ 50 follows a linear trend when

normalized by the diameter, D and plotted against ReD. Hence a correlation between l0/D

and ReD for α′ ≤ 50 is established as, l0/D = 0.07ReD, which is close to the correlation that

is found in steady laminar flow. However, for α′ > 50, l0 follows another linear trend when

it is normalized by the Stokes-layer thickness, δ and plotted against Reδ. Hence another

correlation, which is between l0/δ and Reδ for α′ > 50 is established as, l0/δ = 1.43Reδ.

A detailed study of the generation of vortices due to the free end, their evolution through-

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out the period and the consequent energy dissipation field is presented. The energy dissipa-

tion field shows that the loss due to the free end is intense at the free end and diminishes

with distance from the free end. In fact, the effect on the energy dissipation of the free

end outside the pipe is significant only within the first few diameters. The cycle-average

free-end loss, P e and the shear stress loss considering fully-developed flow throughout the

pipe, P s decrease with both α′ and ReD for all the A0 values. A comparison between P e

and P s shows that the contribution of the free end loss to the overall loss is not negligible.

On the contrary, the free-end loss is so significant that it is equivalent to P s of a 10D long

pipe within the range of 2000 ≤ ReD ≤ 3600.

It has also been shown that the deviation from laminar flow is brought on by the free

end. Two characteristic wavelengths occur in the instability that grows in the pipe near the

free end, one on the scale of the amplitude of oscillation and one on the scale of the Stokes

layers. It is conjectured that these are related to instability modes of similar scale that are

present in the fully-developed reciprocating pipe flow that are excited by the disturbance

introduced by the free end.

IX. ACKNOWLEDGEMENT

A portion of this work was conducted with the support of the National Computational

Infrastructure, which is supported by the Australian government. A portion of this work

was performed on the OzSTAR national facility at Swinburne University of Technology.

OzSTAR is funded by Swinburne University of Technology and the National Collaborative

Research Infrastructure Strategy (NCRIS).

X. DECLARATION OF INTERESTS

The authors report no conflict of interest.

XI. DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding

author upon reasonable request.

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’

-0.5

0

0.5

-2 -1 0 1 2 -1.5 -1 -0.5 0 0.5 1 1.5

5π

3

2π

3

4π

3π

π

32π

5π

3

2π

3

4π

3π

π

32π

u u

r/D

(a) (b)

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0

0.5

0 0.4 0.8 0 0.4 0.8

0

0.5

0 0.4 0.8 0 0.4 0.8u u

u u

r/D

r/D

x/D

=0 1 2 50

x/D

=0 1 2 50

x/D

=0 1 2 50

x/D

=0 1 2 50

φ = π/6

φ = 10π/6φ = 7π/6

φ = 4π/6

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0 2 4 6 8 10 12 14

0

π/2

π

3π/2

2π

-100 -50 0 50 100

0 2 4 6 8 10 12 14

0

π/2

π

3π/2

2π

-100 -50 0 50 100

0 2 4 6 8 10 12 14

0

π/2

π

3π/2

2π

-100 -50 0 50 100

0 2 4 6 8 10 12 14

0

π/2

π

3π/2

2π

-100 -50 0 50 100

x x

x x

∂u/∂r|wall ∂u/∂r|wall

∂u/∂r|wall ∂u/∂r|wall

φ φ

φ φ

(a) (b)

(c) (d)

4

6

8

10

0 100 200 300 400

0

5

10

15

20

25

30

0 2 4 6 8 10

l 0/D

l 0/D

α′ A0

(a) (b)

0

5

10

15

20

25

30

0 1000 2000 3000 4000

0

100

200

300

400

0 100 200 300

l 0/D

l 0/δ

ReD Reδ

(a) (b)

α′ > 50 ◦α′ ≤ 50 •

α′ > 50 ◦α′ ≤ 50 •

φ = π/3

2π/3

π

4π/3

5π/3

2π

0.05

0.1

0.15

0.2

0.25

0.3

0 100 200 300 400

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1000 2000 3000 4000

α′ ReD

Pe

Pe

(a) (b)

A0 = 1 +

A0 = 3 ∗A0 = 5 ×

A0 = 7 ◦A0 = 9 •

A0 = 1 +

A0 = 3 ∗A0 = 5 ×

A0 = 7 ◦A0 = 9 •

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 100 200 300 400

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1000 2000 3000 4000

α′ ReD

Ps/l

Ps/l

(a) (b)

A0 = 1 +

A0 = 3 ∗A0 = 5 ×

A0 = 7 ◦A0 = 9 •

A0 = 1 +

A0 = 3 ∗A0 = 5 ×

A0 = 7 ◦A0 = 9 •

α′=

20α′=

240

α′=

400

A0=1 A0=5 A0=9

xx

|v||v|

φφ(b)(a)