12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

A COMPARISON OF SPATIAL AND TEMPORAL ACCELERATION IN
TURBULENT CHANNEL FLOWS

Matthew A. Falcone
Department of Mechanical Engineering
University of Sheffield
Sir Frederick Mappin Building, Mappin Street, Sheffield, S1 3JD, UK
mfalcone1@sheffield.ac.uk

Shuisheng He
Department of Mechanical Engineering
University of Sheffield
Sir Frederick Mappin Building, Mappin Street, Sheffield, S1 3JD, UK
s.he@sheffield.ac.uk

ABSTRACTtures. Eventually, the flow undergoes a spontaneous transition,
in a process resembling bypass transition. Recently, we havecarried out a study of spatially accelerating flow incorporating moving walls to investigate the effect of flow acceleration
without the streamline contraction intrinsic to a more conventional spatial acceleration (Falcone & He, 2022). This movingwall flow has been found to exhibit very similar phenomena tothat observed in temporal acceleration. It was found that thisflow could similarly be understood in terms of the theory developed in He & Seddighi (2013). In this paper, we directlycompare the temporal and moving wall accelerations to understand the similarities and differences in the transition processesof each flow.

A direct comparison of a temporal and an idealised spatial
acceleration incorporating moving walls has been performed
using Direct Numerical Simulation by equating the bulk flow
inertia of the two cases, which resulted in both flows having thesame bulk velocity when a convective distance was defined. It
was found that the key features of the two flows were similar and conformed to the transition theory proposed by He &Seddighi (2013) for temporal acceleration. However, a number of differences were found. These include the response of
the skin friction coefficient, the initial increase of which wassignificantly reduced in the moving-wall acceleration, and thelocation of transition, which occurred significantly earlier inthe moving wall case. A novel pseudo-body force approachhas been developed to understand the differences in the flow
evolution in the moving wall and temporal accelerations.

METHODOLOGY
Direct Numerical Simulations of turbulent channel flow
have been performed using the solver CHAPSim (He & Seddighi, 2013; Seddighi-Moormani, 2011). The spatial acceleration has been implemented by applying a streamwise decreasing velocity on the channel walls. In this case, a linear
acceleration profile has been used such that the wall velocity,
Uw = −Cx, where x is the distance from the onset of the acceleration and C is a positive constant. This results in a relative
acceleration between the fluid and the wall with the relative
bulk velocity defined as Ub = U0 − Uw = U0 + Cx. This is
depicted in figure 1. The temporal acceleration has been imposed by applying a uniform body force. In order to compare
the temporal and spatial accelerations, the bulk temporal inertia, dUb /dt (for the temporal acceleration) and the spatial
inertia of the bulk flow, Ub dUb /dx (for the moving-wall acceleration) were equated. Given the linear relative bulk velocity
profile for the moving-wall acceleration, the bulk velocity inthe temporal acceleration can be obtained from the following
differential equation.

INTRODUCTION
Spatially and temporally accelerating flows are present ina range of engineering applications. This results in a need for
an in-depth understanding of these flows, which can include
complex and not well-understood phenomena, including laminarisation. Temporally developing flows have long been usedin the study of spatially developing flows, including in transition (Wray & Hussaini, 1984), developing turbulent boundary
layers (Kozul et al., 2016), and three-dimensional boundary
layers (Coleman et al., 1996). Spatial and temporal accelerations contain a number of overarching similarities, such as theinitial freezing of the Reynolds stresses. In this context, we
can consider the extent to which temporally accelerating flows
can be used to understand their spatial counterparts.
Significant developments have been made over the past
few years in the understanding of temporal acceleration (He& Seddighi, 2013; Sundstrom & Cervantes, 2018; Guerrero
et al., 2021), which have found that the flow development
can be characterised by the development of a new boundary
layer superimposed on the pre-existing flow. Initially, the newboundary layer does not significantly alter the flow characteristics, although it modulates the pre-existing turbulence struc-dU
dUb (t)= Ub b = CUb
dt
dx

1

(1)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1.

Case setup for the moving-wall and temporal accelerations

Re0

Re1

∆xconv

Domain (X ×Y × Z)

Mesh (X ×Y × Z)

+
∆xmax

∆y+
w,max

∆y+
c,max

∆z+
max

Case M

2800

4237

15

30 × 2 × 41296 × 256 × 270

5.9

0.3

3.3

3.8

Case T

2800

7840

∼ 50

18 × 2 × 51000 × 324 × 480

7.7

0.4

4.3

4.5

(a)

2.5

1.4

Ub

2δ

Ub

U0

1.2

(b)

2.0
1.5

U

1.0

Relative velocity, Ub

1.0
0

1

20

Absolute velocity, U0

x

0

0

20

x
2.5

Ub∗

Wall velocity, Uw

40

t

(c)

Case M
Case T

2.0
1.5

Figure 1. Flow acceleration caused using the moving-wall
approach. Top: The channel and its streamwise boundary condition are shown with the filled black arrows representing thewall velocity. The shaded yellow region is the region where
the acceleration is applied. Bottom: A plot showing the variation of the absolute velocity (dashed), wall velocity (dotted),
and relative velocity (solid).

1.0
0

20

40

60

80

xconv
Figure 2. (a) Relative bulk velocity for case M. (b) Bulk velocity for case T. (c) Comparison of case M and T with case T
plotted against convective distance.

The resulting bulk velocity for the temporal acceleration isUb (t) = Ub0 exp(Ct).

the bulk velocity profile in cases M and T when plotted against
convective distance. The statistics for case M have been averaged in the spanwise direction, and at many times. Case T has
been averaged in the spanwise and streamwise directions, andensemble-averaged using six independent simulations. Unlessotherwise stated, the velocity is normalised by the initial bulk
velocity, Ub0 , the distance by the half-channel height, δ , andtime by δ /Ub0 .

(2)

To compare the two flows, a convective distance can be
defined, xconv which considers the distance travelled by a fluid
particle moving at the bulk velocity.

Z t

xconv (t) =0

Ub (τ)dτ(3)

RESULTS AND DISCUSSION
For equation 2, the convective distance is given by xconv =Ub (t)/C, which shows that the bulk velocity in the temporal case varies linearly with its convective distance and henceshows that the two flows are equivalent in the bulk sense. It canalso be shown that the values of acceleration parameter, K are
the same with respect to xconv 1 . Each case was accelerated until its transition was completed. This resulted in the temporal
acceleration having a higher final Reynolds number to accommodate the later transition in this flow (see figure 2c). In thisstudy, an initial Reynolds number, Reb = Ub δ /ν of 2800 isused, where δ is the half-channel height. The details of thecases are shown in table 1. The bulk velocity profiles for themoving wall (case M) and temporal (case T) accelerations areshown in figure 2. Figure 2c confirms the equivalence between

1 For the moving-wall acceleration, K =Figure 3 presents the instantaneous streamwise and wallnormal velocity fluctuations for case M. Figure 3a shows thatthe near-wall streaks mildly strengthen from the onset of theacceleration (x = 0) while v0 (figure 3b) does not initially respond. The strengthening of the streaks is linked to the stretching of the near-wall turbulent structures by the mean shear associated with the development of the new boundary layer that
forms with the onset of the acceleration. At x ≈ 8, coincident
spots can be observed in u0 and v0 that are initially localisedin space but grow as they are convected downstream until theentire surface of the wall is covered in new turbulence structures. Figure 4 presents u0 and v0 for case T at a time aroundthe onset of transition showing the strengthened and elongatedstreaks coexisting with the localised turbulent spots. These results are consistent with previous studies of the moving-walland temporal acceleration, which have split the flow development into three regions: Pre-transition (0 < x ≤ 8), the initial
phase characterised by the strengthening of near-wall streaks;
transition (8 < x ≤ 14), the phase characterised by the forma-

1 dUb. For the tempoRe·Ub2 dx

1 dUb
ral acceleration, K = Re·U
3 dt
b

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3.

Contours of fluctuating velocity for case M at y+0 = 5.2. (a) u0 . (b) v0
observed u0 v0 are related to increases in mean shear. This isconsistent with previous studies of accelerating channel flows(Falcone & He, 2022; He & Seddighi, 2013).

Skin friction coefficient and FIK Identity
While figures 3 to 5 show that qualitatively the flows
develop in a remarkably similar fashion, differences emerge
when the accelerations are compared against convective distance. Figure 6 shows the skin friction coefficient, C f and its
FIK identity (Fukagata et al., 2002), which is defined as1
6−6 (1 − y)(u0 v0 )dy
Reb
0
{z
}
|{z} |

Z

Cf =

CLf

Figure 4. Contours of fluctuating velocity for case T at a timenear the onset of transition (t ∗ = 20) showing the presence of
turbulent spots. (a) u0 . (b) v0 .

CTf


00 
Z 1
∂p
− 3 (1 − y)2 I +
dy .
∂x
0
|
{z
}

(4)

CIf

tion and growth of turbulent spots; and fully turbulent (x > 14),
the region after the merging of the near-wall turbulent spots
which is largely characterised by the wall-normal diffusion of
the newly generated turbulence.
The flow characteristics can also be understood throughthe second-order statistics, which are shown in figure 5 withlocations before the onset of transition coloured blue and thoseafter coloured red. The strengthening of the near-wall streaks
during pre-transition in both cases M and T is indicated by
the increases in u0 u0 (figures 5(a, e)). The initial lack of response from the other turbulent components can be seen in
figure 5(b, f ), which shows v0 v0 . These figures also show thesudden change in the behaviour of v0 v0 with the onset of transition with growth occurring over a large wall-normal region,
particularly for the case T, which grows by nearly an order ofmagnitude. The growth of the −u0 v0 (figures 5(c, g)) in bothcases is also similar, with increases during pre-transition being limited to the near-wall region with broad increases during
the transition phase. The eddy viscosity, which can be considered an indicator of turbulence activities, is presented in figures 5(d, h). During pre-transition, there is no change to thisquantity, indicating that the fundamental turbulence characteristics of flow remain largely unchanged and that the increases

The behaviour of C f is generally similar in both cases,
with an initial increase before decreasing to a minimum andthereafter increasing. For accelerating flows, the minimumin the skin friction coefficient is considered an approximate
marker for the onset of transition (He & Seddighi, 2013), andit can be seen that the convective distance where the minimum occurs is substantially delayed in case T compared withcase M. It should also be noted that the initial increases in C f
are very weak in case M. The FIK decomposition reveals thatthese differences can be associated with inertial contribution,
CIf , which exhibits significant increases in case T but remains
limited in case M. The differences in the development of theinertia terms between the temporal and moving-wall acceleration are explored below.

Momentum balance and pseudo body force approach
In order to better understand the differences between theaccelerations, we can analyse the streamwise mean momentum
equations which are given below for the temporal (equation 5)
and moving-wall acceleration (equation 6) with only the dominant right-hand side terms shown.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

(b)

×10−2

x=0
x=3
x=6
x = 10x = 14x = 18

2
0

(f)

×10−1

0.5
0.0

0.0

0

(g)

×10−2

2.0

(h)

×10−2

1.0

0.0

15

1.5
1.0

10
5

0.5

0.5
1.0

0.5

0

1.5

v0 v0

u0 u0

1.0

10

2

2.0

t =0
t = 12
t = 18t = 22t = 27t = 35

1.5

15

5

0

(e)

4

νt /ν

4

νt /ν

2

20

−u0 v0

4

(d)

×10−3

6

−u0 v0

u0 u0

6

(c)

×10−3

6

v0 v0

(a)

0.0

0.5

y

0.0

1.0

0.0

y

1.0

0.5

0
0.0

y

1.0

0.5

y

Figure 5. Second-order statistics. (a, e) u0 u0 , (b, f ) v0 v0 , (c, g) u0 v0 , (d, h) νt . Case M is shown in (a, b, c, d) and case T is shown in(e, f , g, h). Locations or times before the onset of transition are shown in blue and those after in red for clarity. The streamwise locations
for case M are located in (a) and the times for case T are located in (e).

(a)
0.010

Cf

CIf

CLf

CTf

Total

∂ ū
∂ p̄
∂
=−
+
∂t
∂x ∂y
|{z}



∂ ū
∂ ū
∂ p̄
∂
ū
+ v̄
=−
+
∂x
∂y
∂x ∂y



1 ∂ ū
− u0 v0
Re ∂ y


(5)

− fT

0.008

(6)

Cf

0.006


1 ∂ ū
− u0 v0 .
Re ∂ y0.004

The development of the moving wall acceleration can be better
understood as a relative acceleration using the substitution ū =ūrel − Uw in equation 6, which results in modified convection
terms:

0.002

0.000

0

10

5

20

15

x

∂ ū
∂Uw
∂ ū
∂Uw
∂ ū
ūrel rel + ūrel+Uw rel +Uw+ v̄ rel = RHS.
∂
x
∂
x
∂
x
∂
x
∂y
| {z } |{z} | {z } | {z
}

(b)

− fA

Cf

0.006

0.004

0.002

0

20

40

− fC

− fD

(7)
Equation 1 was determined to equate the bulk temporal
inertia with the bulk relative acceleration of the moving wall
acceleration. As a result, term fA can be considered broadly
comparable with fT in equation 5. The substitution also resultsin the additional terms fB and fC , as well as term fD , whichis unchanged by the substitution. The effect of these terms on
the flow response needs to be determined, particularly in the
pre-transition region.
The theory that has been developed for the temporal andmoving-wall accelerations considers the flow development tobe characterised by the formation and development of a new
boundary layer superimposed on the pre-existing turbulent
flow. As a result, the flow development can be considered withthe following decomposition:

0.008

0.000

− fB

60

xconv
Figure 6. Skin friction coefficient and FIK identity for cases
M (a) and T (b). Vertical line indicates the minimum in C f foreach case.

ū = ū0 + ū∧ ,4(8)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
where ū0 is the pre-existing flow and ū∧ is a perturbation flow,which results from the imposition of the acceleration and incorporates the new boundary layer. We can consider the termson the left-hand side of equations 7 and 5 as pseudo-bodyforces and moved to the right-hand side of the equation withthe resulting equations for the pre-existing flow and the flow
after the onset of the acceleration given as

0=−

∂ p̄0
∂
+
∂x
∂y

∂
∂ p̄
+
∂x ∂y






1 ∂ ū0− u0 v0 0 ,Re ∂ y
1 ∂ ū
− u0 v0 + f .
Re ∂ yfAN + fBN + fDN
fAN
fBN
fDN
fT N

Gain

0.04

0.02

0.00

(9)

Loss

0=−

x = xconv = 5

−0.02

−0.04

(10)
0.0

0.2

0.4

For the case T, f = fT and for case M, f = fA + fB + fC + fD .
An equation for the perturbation flow, ū∧ can be developed bysubtracting equation 9 from equation 10.

0=−

∂ p̄∧
∂x

+

∂
∂y



1 ∂ ū∧Re ∂ y− u0 v0

∧

+ f,

(11)
into equation 13 results in an equation for the perturbation flow
associated with acceleration in terms of the non-uniform inertia pseudo-body forces:

where u v = u0 v0 − u0 v0 0 . During the pre-transition region,
figure 5(d, h) shows that the eddy viscosity, νt remains unchanged from the pre-existing flow hence νt = νt0 . Also notethat νt0 is the same for case T and case M as their pre-existingflows are the same. Invoking the Boussinesq hypothesis andconsidering the dominant strain rate only,

∧

νt0
Re



∂ ū ∂ ū0
−
∂y
∂y


=

νt0 ∂ ū∧.
Re ∂ y

0=−

(12)

0=

Substituting equation 12 into equation 10 results in

0=−

∂ p̄∧
∂
+
∂x
∂y



1 + νt0 ∂ ū∧
Re ∂ y


+ f.

∂
∂y



∂
∂y



1 + νt0 ∂ ū∧
Re ∂ y1 + νt0 ∂ ū∧
Re ∂ y+ fT N .

(15)+ fAN + fBN + fDN ,

(16)



with fCN = 0. The profiles of these body forces at xconv = 5
for cases T and M are given in figure 7, which shows that fAN
has a similar overall profile to fT N , although its magnitude issignificantly larger. Body forces fBN and fDN are significant
and oppose fAN , which results in the total non-uniform bodyforce being a significantly different shape to fT N particularly
close to the wall. This different near-wall behaviour of the
body forces helps to explain why there is a much more limited increase in skin friction coefficient for case M. Further
understanding of the effect of these differences on the transition process will be an aspect of future work with an emphasis on comprehensively understanding the substantially earliertransition in the moving-wall acceleration.

(13)

∧

The perturbation pressure gradient, ∂∂p̄x can be eliminatedby considering the following assumptions (these can be confirmed by analysing momentum equation balance for cases M
and T):
1. The pressure gradient is constant across the wall-normal
extent of the channel.
2. The flow development during pre-transition is limited to∧
the near-wall region and hence ∂∂ūy = 0 around the channel centreline.

CONCLUSION• A comparison of a temporal and an idealised spatial acceleration using moving walls has been conducted suchthat the bulk velocities are the same with respect to a convective distance. This was achieved by equating the bulk
temporal inertia with the relative bulk inertia of the moving wall acceleration• The behaviours of the two flows are qualitatively similar to each other, as shown through the Reynolds stresses
and instantaneous results. These results were also foundto conform to the transition theory developed from He &Seddighi (2013).

Hence, taking the force balance at the channel centreline
results in∂ p̄∧= fU ,∂x

1.0

Figure 7. Non-uniform body forces for cases T and case M
at xconv = 5



0 0∧

u0 v0 =

0.8

0.6

y

(14)

where fU is the body force at the centreline, which is hereafterreferred to as the uniform body force with the non-uniform
component defined as fN = f − fU . Substituting equation 14

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Falcone, M. & He, S. 2022 A spatially accelerating turbulent
flow with longitudinally contracting walls. Journal of Fluid
Mechanics (submitted) .
Fukagata, Koji, Iwamoto, Kaoru & Kasagi, Nobuhide 2002
Contribution of reynolds stress distribution to the skin friction in wall-bounded flows. Physics of Fluids 14, L73–L76.
Guerrero, Byron, Lambert, Martin F. & Chin, Rey C. 2021
Transient dynamics of accelerating turbulent pipe flow.
Journal of Fluid Mechanics 917, A43.
He, J., Tian, R., Jiang, P.X. & He, S. 2021 Turbulence in aheated pipe at supercritical pressure. Journal of Fluid Mechanics 920, 45.
He, S. & Seddighi, M. 2013 Turbulence in transient channel
flow. Journal of Fluid Mechanics 715, 60–102.
Kozul, M., Chung, D. & Monty, J. P. 2016 Direct numerical simulation of the incompressible temporally developing
turbulent boundary layer. Journal of Fluid Mechanics 796,
437–472.
Seddighi-Moormani, Mehdi 2011 Study of turbulence andwall shear stress in unsteady flow over smooth and roughwall surfaces.
Sundstrom, L. R. Joel & Cervantes, Michel J. 2018 The selfsimilarity of wall-bounded temporally accelerating turbulent flows. Journal of Turbulence 19, 49–60.
Wray, A. & Hussaini, M. Y. 1984 Numerical experiments inboundary-layer stability. Proceedings of the Royal Societyof London. A. Mathematical and Physical Sciences 392,
373–389.

• Despite the similarities, the detailed flow development
contains a number of significant differences, particularlyin the response of the skin friction coefficient in the earlystages of the acceleration, where the increases in C f inmoving-wall acceleration are much more limited. It wasalso found that the point of transition occurred substantially earlier in the moving wall case.
• The convection terms of the momentum equation for themoving wall acceleration were decomposed to consider
the relative flow acceleration. This resulted in additionalterms which were analysed using a pseudo-body force approach similar to He et al. (2021). This showed that theconvection in the moving-wall acceleration behaves in asignificantly different manner to the temporal acceleration, which may be used to explain the quantitative differences in the turbulence response in the two flows.

Acknowledgements
This work makes use of the DNS code CHAPSim, which
is currently maintained by CCP-NTH (EP/T026685/1). Highperformance computing resources on ARCHER2 were provided by the UK Turbulence Consortium (EP/R029326/1).

REFERENCES
Coleman, Gary N, Kim, John & Le, Anh-Tuan 1996 A numerical study of three-dimensional wall-bounded flows. International Journal of Heat and Fluid Flow 16, 333–342.

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