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

DIRECT NUMERICAL SIMULATIONS OF OSCILLATORY BOUNDARYLAYERS OVER ROUGH WALLS

Umberto Ciri
Dept. of Mechanical Engineering
University of Puerto Rico at Mayagüez
PO Box 9000, Mayagüez, PR 00680, US
umberto.ciri@upr.edu

Sylvia Rodrı́guez-Abudo
Dept. of Engineering Sciences and Materials
University of Puerto Rico at Mayagüez
PO Box 9000, Mayagüez, PR 00680, US
rodriguez.abudo@upr.edu

Stefano Leonardi
Dept. of Mechanical Engineering
The University of Texas at Dallas
800 W Campbell Rd, Richardson, TX 75252 US
stefano.leonardi@utdallas.edu

ABSTRACTof the problem are the amplitude of the platepvelocity oscillations U0 , and the characteristic length δ = 2ν /ω , where νis the fluid viscosity, and ω the frequency of the oscillations.
The length δ is a measure of the distance from the plate up
to which the effect of the oscillation is felt in the fluid domain (thus, from a reference frame on the oscillating plate, it
represents the thickness of the boundary layer). Additionally,
the laminar solution shows that the wall shear stress (also harmonic) has a phase lead of π /4 over the velocity oscillations.
More recently, various authors have investigated the oscillatory boundary layer over smooth walls, either numerically (Spalart and Baldwin, 1989; Akhavan et al., 1991b; Salonet al., 2007) or experimentally (Hino et al., 1983; Jensen et al.,
1989; Akhavan et al., 1991a; Sarpkaya, 1993; Eckmann and
Grotberg, 1991), for a wide range of flow conditions. Although
in the experimental apparatus or numerical frameworks the oscillations are mostly imposed through an harmonic pressure
gradient, the flow regime appears to still scale with the parameters identified by the shear-driven classic analysis, δ and U0 ,and, in particular, with the Reynolds number Reδ = U0 δ /ν .
Up to Reδ ≃ 100, the oscillatory boundary layer is laminarand follows the Stokes solution. As the Reynolds number is
progressively increased, perturbations in the flow field appear
and grow for some phases of the oscillation cycle. The boundary layer develops into two intermediate flow regimes (the‘disturbed laminar’ and the ‘intermittently turbulent’ regime;
Sarpkaya, 1993; Blondeaux & Vittori, 1994), before a fully developed turbulent regime is established. In the fully developed
regime, Reδ & 750 − 1000 (Hino et al., 1983; Jensen et al.,1989), turbulent fluctuations are present essentially throughout the whole period. The intensity of the fluctuations is larger
during the deceleration interval (where perturbations starts toappear before extending to earlier and earlier phases while Reδ
is increased). The phase lead between the peak shear stress (nolonger harmonic) and the maximum velocity is reduced compared to the laminar solution.
While the smooth wall provides a useful theoretical abstraction, in practice the Stokes layer will be bounded by arough wall, in particular in coastal environments. Various studies have emphasized the importance for transition to turbu-

Turbulent oscillatory flow over a rough bed is investigatedwith direct numerical simulations. Flow motion is induced byharmonic oscillation of the bed and the fluid is otherwise atrest, similarly to the classical Stokes boundary layer problem
for the laminar flow over an oscillating flat plate. In the present
case, the bed is rough and consists of two layers of fixed, identical spherical particles. In the turbulent flow, the oscillatory
motion is propagated more rapidly from the bed to the bulk of
the flow. At each distance from the bed, the time delay is reduced compared to the laminar linear trend and follows a curve
which seems to depend primarily on the Reynolds number,rather than the particular bed configuration (rough or smooth).
During the decelerating phase of the cycle, a log layer is observed in the velocity profiles, which is later suppressed in theacceleration phase. The Reynolds stresses resemble the canonical uni-directional boundary layer distribution in the deceleration phase and change sign as the flow reverses. Nevertheless,the production of turbulence kinetic energy does not achieve
large negative values and remains positive. The phase of maximum turbulent kinetic energy production is anticipated compared to smooth beds, and depends on the rough bed morphology.

INTRODUCTION
Turbulent oscillatory boundary layers are a common hydrodynamic feature at the seabed of littoral zones. The oscillatory motion is generated by sea surface waves throughgravity, and it affects the whole bed dynamics and morphology, from sediment erosion to transport and deposition patterns. Thus, the understanding of the oscillatory dynamics ofa wall-bounded flow has both intellectual and practical consequences.
In the laminar regime, the oscillatory motion close to asolid wall is fairly well-known and can be described by the
Stokes boundary layer. The Stokes boundary layer is a sheardriven flow generated by the harmonic oscillations of a flat
plate respect to a fluid at rest in the far field. According to theclassical solution (Batchelor, 1967), the governing parameters112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
lence of even very small imperfections in otherwise “mirrorshine” smooth walls (Blondeaux & Vittori, 1994; Vittori and
Verzicco, 1998; Costamagna et al., 2003; Verzicco and Vittori,
1996; Tuzi and Blondeaux, 2008). Other works have focusedon large scale modulations of the bedform, ripples, which areoften observed in various patterns as a result of sediment transport and deposition process by the oscillatory boundary layer(Grigoriadis et al., 2012; Penko et al., 2013; Rodrı́guez-Abudo
and Foster, 2014; Scandura et al., 2000; Chang and Scotti,
2004; Önder and Yuan, 2019). On the other hand, Sleath(1987) and Jensen et al. (1989) have considered rough flatbeds packed with sand, gravel or pebbles of various size, analogous to the classic configurations extensively analyzed in thecontext of steady boundary layers (Nikuradse, 1933). Sleath(1987) emphasized the role of vorticity ‘bursts’ or ‘jets’ associated with the wakes of the individual roughness elements,
which separate and propagate from the wall to the outer layer
when the flow reverses direction. The experiments of Jensen
et al. (1989) at larger Reδ suggested a similar scenario tosteady rough-wall boundary layers, with the flow approaching a ‘fully-rough’ regime, where the main scaling parameteris the roughness size. In general, a complete understanding ofrough-wall oscillating boundary layers is still lacking. In thiswork, we discuss results from direct numerical simulations ofshear-driven oscillatory flow over a bed of spherical particles.

(1536 × 768 × 256) to have sufficient resolution to model theparticles. The resolution is consistent with the study of ChanBraun et al. (2011).
As boundary conditions, periodicity is applied in thestreamwise and spanwise directions. Free-slip is applied at
the top of the domain. We consider the flow motion under an oscillatory shear-driven forcing, applied as a boundary condition with the immersed boundary method on the bedof spherical particles. The layers of sphere oscillate following the harmonic function in figure 1b, U0 sin (ω t). Fourcases have been run, corresponding to a Reynolds numberReδ = U0 δ /ν = 100, 190,
p765 and 1200, where δ is the depthof the Stokes layer (δ = 2ν /ω ). These last values are above(or at least approaching) the smooth-wall critical value for the
fully-developed regime (Hino et al., 1983; Jensen et al., 1989).

RESULTS
Figure 2a summarizes the various cases analyzed in a
‘Moody-diagram’-like chart, with the friction coefficient as afunction of Reδ . The bed drag varies in time throughout thecycle, and the friction coefficient C f is defined with the maximum drag as C f = 2τmax /ρ U02 , where τmax is the maximum
bed friction over the oscillation period and ρ the fluid density. The simulation results are compared with the analyticalresults for smooth-wall oscillatory flows in the laminar (Batchelor, 1967) and turbulent (Fredsøe, 1984) regimes.
It is apparent from figure 2a that the flow regime overthe spherical bed with H/D = 5.5 scales with Reδ similarly
to the smooth wall case. At low Reynolds numbers, when
the regime is laminar, the friction coefficient has an inverse
dependence on Reδ . The highest value of Reynolds number
tested herein (Reδ = 1200) is approaching the fully turbulent
regime. Simulations for the bed with small spherical particles(H/D = 10) have been performed for the two highest values
only of Reδ = 765, 1200 to limit the computational cost. The
trend in the transitional and turbulent regime is consistent withthe case H/D = 5.5.
The bed friction as a function of time is shown in figure 2b, for the four Reynolds number cases. For the tests at
low Reδ , the shear waveform τ /τmax resembles the laminarsmooth wall case, cos (ω t − π /4), but the maximum value islarger (fig. 2a) because the bed is rough. As the Reynolds number is increased, transition to the turbulent regime occurs andthe friction coefficient increases compared to the laminar case.
The shear stress further departs from the laminar behavior, as
the waveform is no longer harmonic and the phase shift in thezero-crossing increases.
Turbulence over the rough bed induces further modifications in the temporal evolution of the flow, compared to thelaminar case. Figure 3 shows the phase shift in the flow velocity with respect to the oscillatory forcing (U sin ω t), as a function of the distance from the bed. According to the Stokes’
solution, the phase shift increases linearly (in magnitude) withthe distance from the bed as the velocity is proportional tosin (ω t − z/δ ). For the present simulations, the phase shift has
been determined by first computing the time correlation between the sinusoidal forcing and the velocity signal at different
distances from the bed. Then, the time shift τ corresponding to
the maximum correlation has been taken to calculate the phasedelay as ϕ = ωτ .
Figure 3 presents results for both the rough-bed configurations and a corresponding smooth wall simulation at thesame Reδ . Compared to the classical solution, the phase delay
is reduced when the flow becomes turbulent. The decrease is

NUMERICAL METHODOLOGYThe non-dimensional continuity and Navier-Stokes equations are taken as the governing equations:

∂ Ui
= 0;
∂ xi

∂ Ui ∂ UiU j∂P
1 ∂ Ui
+
=−
+
∂t
∂xj
∂ xi Reδ ∂ x2j(1)

where Ui is the velocity component in direction xi and P is
the pressure. The numerical method for solving the governing
equations is described in Orlandi (2000). The substrate particles are treated with the immersed boundary method presented
in detail in Orlandi and Leonardi (2006). Simulations are performed for an open channel with a bed of spherical particles
on the lower wall (figure 1). The spheres are identical and regularly distributed into two superposed layers, with the bottom
layer staggered by a radius. The lower boundary of the computational domain is tangent to the spheres of the upper layer,
so that only the upper cap of the bottom spheres is inside thecomputational domain (see the inset in figure 1a).
This rough wall reproduces one of the configurations analyzed in Chan-Braun et al. (2011) and Mazzuoli and Uhlmann(2017) for a uni-directional pressure forcing at a Reynolds
number based on the bulk velocity equal to U0 H/ν = 2870,
where H is the height of the computational domain. This
would correspond to a smooth-wall channel at Reτ = uτ H/ν =180, where uτ is the friction velocity. In the present work, wehave considered two values for the sphere diameter D, suchthat H ≈ 5.5D and H ≈ 10D. The larger-particle case (H =5.5D) reproduces the geometry in Chan-Braun et al. (2011),
while the second case, for the small spheres (H ≈ 10D), is obtained by halving the diameter of the spheres.
In the directions parallel to the bed, the computational domain is 6H (streamwise) and 3H (spanwise). For the caseH = 5.5D, the mesh consists of 768 × 384 × 256 uniformly
distributed points in the streamwise, spanwise and bed-normal
direction respectively. For the case H = 10D, the points
have been doubled in the streamwise and spanwise direction

2

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

U/U0

1

0

-1
0

0.2

0.4

0.6

0.8

1

t/T

(b)

(a)

Figure 1. (a) Computational domain. (b) Oscillatory forcing and phase identifiers.
1Reδ = 1200
0turbulentτ /τmax

0.1

Cf

Reδ = 7650Reδ = 190
00.01

Reδ = 900laminar

0.001
1

2

10

3

10

4

10

0

10

Reδ

(a)

0.1

0.2

0.3

0.4

0.5

t/T

(b)

Figure 2. a) Friction coefficient C f as a function of the Reynolds number. Symbols, present simulations: ◦ spherical-particle bed,
H/D = 5.5; • H/D = 10;  smooth bed validation (Stokes boundary layer). Lines, theoretical trends for the laminar (Batchelor,1967) and turbulent (Fredsøe, 1984) regimes (smooth bed). b) Wall friction variation over half-cycle for different Reδ and H/D = 5.5(lines are shifted vertically for the sake of clarity). Solid lines: present simulations for rough bed. Dashed line: Stokes solution,
τ /τmax = cos (ω t − π /4).

15

15

10

10

z/δ

20

z/δ

20

5

5

0

0

-5

-5
-1

(a)

-0.8

-0.6

-0.4

-0.2

0

-1

ϕ /2π

(b)

-0.8

-0.6

-0.4

-0.2

0

ϕ /2π

Figure 3. Velocity phase delay as a function of the distance from the bed for Reδ = 765 (a) and Reδ = 1200 (b). Symbols indicate
present simulation over a rough bed: ◦ H/D = 5.5; • H/D = 10. The solid line indicates present results for oscillating flow over asmooth bed. The dashed line shows the (laminar) theoretical trend from the Stokes’ solution. The origin for z in the cases with thespherical-particle bed is taken at 0.2D below the crest plane.

significant already within a few Stokes depths δ from the wall(z/δ . 5). This effect can be expected on the basis of a classic
eddy-viscosity argument, with the macroscopic turbulent mixing enhancing the diffusion by molecular viscosity and propagating more rapidly the driving force from the wall to the bulkof the fluid. Consistently, the phase-shift reduction increases
with the Reynolds number (as the flow approaches the fully

turbulent regime) and it is observed at a shorter distance fromthe bed (figure 3a at Reδ = 765 vs. figure 3b at Reδ = 1200).
It seems that this effect is controlled by the turbulent structures in the bulk of the flow, with little dependence on the
shape or morphology of the bed. Both the cases with spherical particles (H/D = 5.5 and H/D = 10) and the smooth wall
approximately collapse on a single curve with a slope which3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
depends on the Reynolds number. Possibly, at large values ofReδ , in the fully-developed regime, a universal curve for thetime delay may exist, where results collapse regardless of the
bed morphology and Reynolds number.

phases, when, consistently, a logarithmic layer is also established in the velocity profile. For the case with the large particles (H/D = 5.5), the peak in the production appears shiftedat larger distance from the wall. However, this may be due tosome uncertainty in the determination of the virtual origin forz, which herein is taken at 0.2D below the crest of the spherefollowing Chan-Braun et al. (2011). Negative values of production P are observed when the velocity changes sign, although the magnitude is very small. The production depends
on both the Reynolds stress uw and the shear dU/dz. As theforcing changes sign, the Reynolds stresses are large (and with
opposite sign) in the outer layer where the velocity gradient issmall. This is also the case for oscillatory flows over smoothbeds (Salon et al., 2007). However, it is not straightforward togeneralize this situation for other flow regimes or rough beds,
since the sign of the production will ultimately depend on theinterplay between the oscillatory forcing and time-scales associated with the rough bed. For example, ejections of coherent
structures or shedding from the bed crests as the flow reverses
direction may affect and change the sign of TKE production.
Figure 7 shows the evolution in time, throughout half thecycle, of the peak of TKE production Pmax . The peak isnormalized both with the maximum friction velocity (panel
a; normalization as in figure 6) and with the instantaneous
frictionpvelocity (panel b), which depends on the local shear,ũτ = τ (t) /ρ . Figure 7a shows that the maximum value ofPmax across the cycle is not attained at the same phase forthe rough beds and the smooth wall. Instead, a large phaselead compared to the smooth wall reference is observed forthe bed made of large spherical particles (D/H = 5.5). The
phase lead is still present although somewhat mitigated for thecase (D/H = 10). The phase shift in P (which can also be
noted in figure 6) should be attributed to time evolution of theshear, rather than the Reynolds shear stresses uw. From figure 5, it appears that uw are relatively well synchronized forboth values D. The shear dU/dz seems to be more sensitive toparticular configuration of the bed and causes the phase shift.
In figure 7b, the peak of TKE production is shown with
the normalization by the instantaneous friction velocity. Since
the shear vanishes during the cycle (figure 2b), the curve issingular. The zero-crossing of the wall shear stress is anticipated compared to the laminar case (which has a phase lead of
π /4, corresponding to a zero crossing at t = 3/8T = 0.375T ).
Figure 7b confirms the increase in the phase lead for the roughbed cases, as the curve blows-up shortly after t/T = 0.3. Thisbehavior is opposite to that of the turbulent oscillatory flow
over a smooth wall, where the phase lead is reduced comparedto the laminar solution, as it can be observed from the dashedline in figure 7 and literature results (Jensen et al., 1989).

Close to the bed, within approximately z/δ . 1, thephase delay curve has the same slope as the laminar case (i.e.
ϕ /2π ≈ −z/δ ) regardless of the bed details and the Reynolds
number. This region is the analogous of the viscous sub-layer
in canonical uni-directional boundary layers. The flow remains
laminar and the results collapse with the classical Stokes’ theory.
Literature studies of smooth-wall oscillatory boundary
layers report that intense turbulent activity is found in the deceleration phases, while fluctuations are damped in the acceleration phases (Jensen et al., 1989; Salon et al., 2007).
+
Velocity profiles in wall units (U = U/uτ ) seem to confirm this scenario (figure 4). The friction velocity to scale
the profiles
corresponds the maximum friction over the cycle:puτ = τmax /ρ . The overline indicates an ensemble averagein phase and in the streamwise and spanwise directions. The
velocity profiles in figure 4 are evaluated in a reference frame
moving with the oscillatory bed for the cases at Reδ = 1200 for
half a cycle (the other half is symmetric in the present case).
A logarithmic layer develops during the deceleration (purple
lines), while it is not observed in the early phase of the cycle (acceleration, green lines). The logarithmic layer is observed also at Reδ = 765 (not shown here) in the deceleratingphases. As expected, it occupies a thinner region of boundary
layer, but this does suggest that turbulence starts developing
already at Reδ = 765. The size of the spheres does not affecttoo much the time evolution of the profiles, as corresponding
phases show similar features in terms of appearance of the logarithmic layer. The sphere diameter does change the downward shift of the profiles compared to the canonical law of thewall. The downward shift (i.e., the roughness function, ∆U)
is related to the wall drag, and, consistently with figure 2a, it
decreases for the case H/D = 10 (small spheres, and smallerdrag).
Turbulence generation near the wall in the early deceleration phases results in large values of the Reynolds shear
stress uw. Figure 5 shows Reynolds stress profiles for the large
sphere case (H/D = 5.5, panel a) and the small sphere case
(H/D = 10, b). During the deceleration, the profiles resemble
the typical uni-directional boundary layer case, with a (negative) near wall peak and an approximately linear decay in theouter layer, which depends on the total shear. The value of thenear wall peak is approximately the same for both values of theparticle diameter D and is approached at about the same phaseof the cycle. Coherent structures generated during the deceleration phase propagate from the wall to the outer layer in thenext phases of the cycle. Even if turbulence is damped in theacceleration phase, the Reynolds stress remains relatively large
and changes sign as the flow reverses, which may potentiallylead to negative turbulent kinetic energy (TKE) production.
The turbulent kinetic energy production (P = −uw dU/dz) is
plotted in figure 6 for both values of D. The TKE production
for a canonical unidirectional flow is also shown as reference.
The peak value of TKE production appears reduced comparedto the uni-directional flow because results in figure 6 are nor+4
malized withp the maximum friction velocity (P = P ν /uτ ,with uτ = τmax /ρ ) which does not occur at the same phaseas the maximum in the production. The maximum in friction(i.e., maximum in shear) occurs earlier in the cycle, towards
the end of the acceleration phase (see figure 2b). Instead, turbulence production is large and peaks during the deceleration

CONCLUSIONS
Numerical simulations of turbulent oscillatory flow over a
bed made of spherical particles have been performed. We haveconsidered a shear-driven flow, where the bed moves according to an harmonic function and the fluid is otherwise at rest.
The bed consists of two superposed layers of fixed, identicalspherical particles. We have considered two different values ofthe particle diameter to investigate the dependence of the flowon the sphere size. Simulations have been performed for arange of Reynolds number, spanning from the laminar regime
and approaching fully-developed turbulence.
Results show that the temporal evolution of the flow over
the rough bed is modified compared to classical Stokes’ solution for laminar boundary layers. In particular, the phase412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
20

15

15

10

10

U

U

+

+

20

5

5

0

0

-5

-5
1

10

z+

(a)

100

1000

1

10

z+

(b)

100

1000

1

1

0.8

0.8

0.6

0.6

z/H

z/H

Figure 4. Mean velocity profile in wall units at different phases and law of the wall (solid line): (1/κ ) log y+ + B, with κ = 0.41 andB = 5.5; (a) H/D = 5.5 large sphere case; (b) H/D = 10 small sphere case. Following Chan-Braun et al. (2011), the virtual origin of
the profile is taken 0.2D below the crest of the spheres.

0.4

0.4

0.2

0.2

0
-0.004 -0.003 -0.002 -0.001

(a)

0

0.001

uw/U02

0.002

0.003

0
-0.004 -0.003 -0.002 -0.001

0.004

(b)

0

0.001

uw/U02

0.002

0.003

0.004

0.25

0.25

0.2

0.2

0.15

0.15

P+

P+

Figure 5. Reynolds shear stress uw profile at different phases of the oscillation: (a) H/D = 5.5 large sphere case; (b) H/D = 10 small
sphere case. The horizontal dashed line indicates the crest plane. Colors indicate the phase as in figure 1b.

0.1
0.05

0.05

0

0

-0.05

-0.05
0

(a)

0.1

20

40

z+

60

80

100

0

(b)

20

40

z+

60

80

100

Figure 6. Turbulent kinetic energy production P: (a) H/D = 5.5, large sphere case; (b) H/D = 10, small sphere case. The solid blackline shows the production for a uni-directional turbulent boundary layer. Colors indicate the phase as in figure 1b for the oscillatory
flow cases.
shift in the velocity at various distances from the bed is reduced compared to the laminar case, which predicts a linear
increase (in magnitude) with the distance from the bed. The
phase shift reduction seems predominantly dependent on the
Reynolds number rather than the bed morphology, which suggests that a unique curve may exists for large enough valuesof the Reynolds number Reδ . The bed drag also has a different phase shift compared to the laminar solution. In thiscase, as the flow becomes turbulent, the drag wave form is nolonger harmonic and the zero-crossing occurs earlier in the cycle for the rough beds. An opposite behavior, with a delay inthe zero crossing, is observed for turbulent oscillatory flows
over smooth beds.
Strong turbulence activity is observed during the deceleration phases of the cycle, with the presence of a logarithmic
layer in the velocity profile. During the acceleration, the logarithmic region is suppressed and the value of the Reynolds
shear stress also decreases. However, the Reynolds stress re-

5

0.6

0.6

0.5

0.5

0.4

0.4

+
Pmax

+
Pmax

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

0.3

0.3

0.2

0.2

0.1

0.1

0

0
0

(a)

0.1

0.2

0.3

0.4

0.5

0

t/T

(b)

0.1

0.2

0.3

0.4

0.5

t/T

Figure 7. Evolution in time of the maximum of turbulent kinetic energy production Pmax normalized in wall units with the maximum
friction velocity (a) and the instantaneous friction velocity (b): ◦ spherical-particle bed, H/D = 5.5; • H/D = 10;
smooth wall.
The horizontal line indicates the value for a unidirectional boundary layer over a smooth wall.

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Mazzuoli, M. and Uhlmann, M. 2017 Direct numerical simulation of open-channel flow over a fully rough wall at moderate relative submergence. J. Fluid Mech. 824, 722–765.
Nikuradse, J. 1933 Stromungsgesetze in rauhen Rohren.
Forschungshelft 361 (Engl. transl. NACA TM 1292).
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oscillatory flow over a wavy bottom. J. Fluid Mech. 858,
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7, N73
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mains different than zero throughout the cycle and changes
sign (compared to the canonical case in wall-bounded turbulence) as the flow reverses direction. Nevertheless, the turbulent kinetic energy production appears to remain positive (orjust slightly negative) because the velocity gradient is verysmall during these phases. While this happens similarly forboth rough bed configurations, the time evolution of the turbulent kinetic energy is not the same in the two cases. This isbecause production depends on both the time-scales of the oscillatory forcing and the overlying turbulent flow. Therefore, it
may be possible to observe enhanced regions of negative TKE
production for different bed configurations or flow regimes.

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