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Exploring Roughness Effect on Laminar Internal Flow-Are We Ready for
Change?
Satish G. Kandlikar a
a
Mechanical Engineering Department, Rochester Institute of Technology, Rochester, New York, USA
Online Publication Date: 01 January 2008
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Change?',Nanoscale and Microscale Thermophysical Engineering,12:1,61 — 82
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Nanoscale and Microscale Thermophysical Engineering, 12: 61–82, 2008
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ISSN: 1556-7265 print / 1556-7273 online
DOI: 10.1080/15567260701866728
EXPLORING ROUGHNESS EFFECT ON LAMINAR
INTERNAL FLOW–ARE WE READY FOR CHANGE?
Satish G. Kandlikar
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Mechanical Engineering Department, Rochester Institute of Technology,
Rochester, New York, USA
Laminar flow is often encountered in the channels of microdevices as a result of the small
hydraulic diameters. The roughness introduced on the walls of these channels through various
fabrication techniques, such as etching, micromachining, laser drilling, etc., results in a high
value of relative roughness (defined as the wall surface roughness to channel hydraulic
diameter ratio). Laminar flow in rough tubes, therefore, is an important topic in the study
of transport processes in microdevices.
This article begins with a review of the applications where roughness or wall surface features
are present, such as mixing, enhanced mass transfer, and heat transfer. Next, the effect of
roughness on fluid flow is reviewed in terms of eddy generation and transition to turbulence.
The competing effects of destabilization of flow and relaminarization processes are considered. Drawing from the available literature, the criteria for transition to turbulence based on
roughness Reynolds number are evaluated and some of the recent experimental data on
repeated 2-D roughness structures are compared. The present article addresses these issues
and provides a framework for quantifying the roughness effect at microscale.
KEY WORDS: roughness, laminar flow, transition flow, micromixers
INTRODUCTION
Laminar fluid flow of incompressible fluids is amenable to analytical treatment
of frictional effects through viscous dissipation at the wall and the resulting pressure
gradient. For incompressible fluid flow of Newtonian fluids in smooth-walled
channels, the linear relation between the velocity gradient and shear stress at the
wall yields the classical solution of constant Poiseuille number (Po ¼ fRe), its value
being dependent on the channel geometry. The effect of wall roughness on the flow
is, however, less well understood; earlier simplifications presented in the literature
of roughness independence in laminar flow have come under intense scrutiny in
recent years.
Surface roughness on channel walls introduces surface-fluid interactions that are
not well defined and cannot be easily handled through analytical treatment.
Researchers have therefore largely relied on empirical models based on experimental
measurements. Such an approach has resulted in the widely accepted Colebrook
equations and Moody diagram for turbulent fluid flow. The experimental basis for
Received 10 July 2006; accepted 10 October 2007.
Address correspondence to Satish G. Kandlikar, Mechanical Engineering Department, Rochester
Institute of Technology, 76 Lomb Memorial Drive, Rochester, NY 14623, USA. E-mail: [email protected]
61
62
S.G. KANDLIKAR
NOMENCLATURE
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b
bcf
D
Dcf
Dh
Dh,cf
f
fcf
H
L
Po
r
channel height
channel height (constricted flow)
diameter
constricted flow diameter
hydraulic diameter
hydraulic diameter (constricted
flow)
friction factor
constricted flow friction factor
channel width
channel length
Poiseuille number
channel radius
Ra
Re
Recf
Rez
Ret
Ue
Um
w
a*
e
v
r
tw,av
average roughness
Reynold’s number
Reynolds number based Dh,cf
roughness Reynold’s number
shear stress Reynold’s number
velocity at roughness height
average channel velocity
channel width
channel aspect ratio, b/w
roughness height
kinematic viscosity
density
average shear stress (at wall)
these results is provided by well-known experiments [1–6]. The focus of these investigations was largely on turbulent flow, being of more practical importance in fluid flow
applications (hydraulics), including the 5.486-m-diameter Ontario tunnel tested by
Colebrook. Our understanding of pressure drop in rough channels is largely based on
applications involving fluid transport in large diameter tubes (greater than 3-mm
diameter).
The laminar flow studies by the early investigators, particularly Nikuradse, were
conducted on the same experimental setups used in turbulent flow studies [4]. The
lower range of fluid flow and pressure drop in laminar flow yielded significantly higher
values of uncertainties in friction factor calculations. In a recent paper, Kandlikar
critically evaluated Nikuradse’s experiments and showed that the experimental uncertainty for pressure drop measurements in the laminar region was extremely large [7]. In
their experiments, the pressure drop corresponding to a Reynolds number of 600 was
less than 0.1 mm of water head. With pressure taps located 1.5 m apart and the
estimated accuracy of a water manometer of 0.5-mm water head, the uncertainty
greatly overshadows the measurements. Maintaining the pipeline in the same horizontal plane within such small tolerances is also a nearly impossible task for long
pipes. Furthermore, the effect of roughness was studied in the lower range of relative
roughness (highest e/D of only 0.05), where this effect is expected to be quite small.
Systematic experiments were conducted by Kandlikar et al. using saw-tooth
roughness elements in rectangular flow channels [8]. The pitch-to-height ratio of the
roughness elements was set at 6.86 and the highest relative roughness tested was 14%.
They proposed the use of a constricted flow diameter, where Dcf is the constricted flow
diameter, given in Eq. (1).
Dcf ¼ D 2"
ð1Þ
For rectangular channels, the constricted hydraulic diameter is calculated by subtracting the roughness element heights. The results indicated that the use of
a constricted flow diameter instead of the root diameter of the pipe in the friction
factor calculations yielded satisfactory agreement with the established characteristics
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EXPLORING ROUGHNESS EFFECT ON LAMINAR FLOW
63
of fcf ¼ Po/Re. In the turbulent region, the friction factors and Reynolds numbers,
both based on the constricted flow diameter, were employed in replotting the Moody
diagram. The modified Moody diagram showed that the friction factor reached a
plateau corresponding to fcf ¼ 0.42 for 0.03 e/D 0.05. Since the Moody diagram
covers e/D of only up to 0.05, further experimental data are needed for higher e/D
values to validate or extend the modified Moody plot.
The effect of surface roughness on the transport processes at channel walls at the
microscale level is an active area of research. Kandlikar et al. found that heat transfer
was improved over the rough surfaces for small diameter stainless steel tubes [9]. Wu
and Cheng studied silicon microchannels with different surface roughness conditions
and noted a consistent improvement of up to 25% in rough channels (e/Dh in the range
from 3.26 102 to 1.09 102) [10]. Dawson and Trass [11] evaluated mass transfer
rates in rough channels and reported a significant improvement over rough surfaces,
especially in the transition region, during their experiments in square nickel ducts and
a liquid ferro-ferri-cyanide electrolyte; early transition is believed to be the reason.
PRACTICAL APPLICATIONS OF ROUGHNESS OR WALL SURFACE
FEATURES IN MICROFLUIDICS
A number of new channel designs are being proposed in the literature to enhance
the transport processes in microchannels. Silicon fabrication technology makes it
possible to generate complex structures that cannot be easily produced using conventional machining techniques in copper. Some examples of incorporating structured
roughness elements in laminar flow to generate increased interaction between the fluid
layer near the wall and the bulk flow are reviewed in this section.
Grooves and Protruding
Interfacial Area
Roughness
Elements
for
Increasing
Rough channels in Microfluidics applications include specially designed roughness microstructures that address specific process requirements. One such example is
micromixers used in the mixing of miscible liquids and gases. Hessell et al. presented
experimental results for laminar interdigital micromixers and provided a comprehensive summary of different mixing methods [12, 13]. The microstructures currently
employed under the passive techniques essentially rely on providing a larger interfacial
area for efficient diffusive mixing. A good review of the different types of static or
passive mixers of micromixer structures is given by Bayer et al. [14].
Wang et al. [15] studied patterned grooves that were placed at an angle to the
main flow as shown in Figure 1(i). The size of the grooves influenced the flow pattern
considerably. A short aspect ratio (height to width of the groove) of 0.05 resulted in no
significant difference, whereas ratio of 0.3 caused the flow pattern to become more
intermixed but flow did not become chaotic. The recirculation length was related to
the groove aspect ratio and independent of the flow velocity. The grooves caused
rotation and transverse motion that effectively increased the interfacial surface area.
The numerical study was conducted over a Re range of 0.01 to 5.
Sato et al. [16] fabricated microchannels with inclined grooves in the bottom as
well as on the side walls of a rectangular channel as shown in Figure 1(ii). This pattern
caused a stronger swirling motion in the flow as compared to the inclined grooves in
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S.G. KANDLIKAR
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Figure 1. (i) Patterned grooves at an angle on one of the channel walls [15].
Figure 1. (ii) Inclined grooves on three walls [16].
the bottom wall only. The flow experienced an increased interfacial area, which was
mainly responsible for increased mixing. The inclined structure shown in Figure 1(iii)
was placed on the bottom channel wall. It also resulted in a swirling motion of the bulk
flow with increased interfacial area.
Fu et al. [17] placed staggered oriented ridges on the bottom wall as shown in
Figure 1(iv). Different arrangements and patterns of the heights of the cross-ridges
were investigated. The CFD simulation revealed the severe distortion of the interface
as the fluid stretched and folded due to inertial forces. By placing alternate ridges of
different heights on each side, a swirl was generated, causing a more efficient mixing.
The Reynolds number investigated was in the range of 0.12–58.
Figure 1. (iii) 45 Inclined roughness structures protruding in the flow [16].
EXPLORING ROUGHNESS EFFECT ON LAMINAR FLOW
65
Figure 1. (iv) Staggered oriented ridges in the bottom wall [17].
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Enhanced Mixing by Disturbing the Flow—Bending and Tee-Mixers
The introduction of vortices in laminar flow was employed in a tee-mixer used in
microreactors and analytical equipment. Kockmann et al. [18] showed that the vortices in the bends and in the entrance regions of tee-mixers enhance the transport
processes in microchannel devices. Laminar mixing is also important in microchannels
used in single-phase cooling applications for fuel cells [19].
Although the chaotic motion observed in turbulent flows is not present in the
laminar flows, designing passages to generate chaotic advection in laminar flows was
first introduced by Aref [20]. Since then, many researchers have studied this technique.
The chaotic advection refers to a phenomenon in which particles are advected by a
periodic velocity field and result in chaotic trajectories. These trajectories may be
considered to be similar to the turbulent eddies generated at the wall but are a result of
vortices formed by the abrupt changes in the flow direction of the channel. A serpentine microchannel consists of short channel segments that have a 90 bend and is offset
with the previous channel plane as shown in Figure 2(i) for a 2-D case and Figure 2(ii)
for a 3-D case. These serpentine channels were first studied by Liu et al. and Beebe
et al. [21, 22].
Comini et al. [23] studied the effect of channel width H to length L (between two
successive bends) ratio and the bend angle in 2-D serpentine channels and found that
the critical Reynolds number is closely related to these parameters. For a bend angle of
30 and an H/L ratio of 0.45, the critical Re was as low as 260. Above this critical
Reynolds number, the transverse vortices created after the bends started to detach
periodically, leading to an oscillatory flow.
Figure 2. (i) 2-D Serpentine micromixer [21].
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S.G. KANDLIKAR
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Figure 2. (ii) 3-D Serpentine micromixer [22].
The chaotic motion generated by eddies following the bends are effectively
utilized in designing a new class of channels called chaotic channels. Lasbet et al. [19]
proposed three different variants—Figures 3(i)–(iii) show the chaotic C-shape,
V-shape, and W-shape channels. The friction factors for these channels are considerably higher than those for straight channels, but the motion of particles as they flow
through the chaotic sections provides more efficient mixing compared to 2-D or 3-D
serpentine channels.
Wong et al. [23] combined the tee-mixing with the flow interruptions caused by
ridges in the side walls. The mixing occurs largely because of the increase in the
interfacial area at the bends and interruptions rather than through eddy generations.
Figure 4 shows a schematic of the channel geometry employed by them.
Figure 3. (i) Chaotic C-shape channel [19].
Figure 3. (ii) Chaotic V-shape channel [19].
Figure 3. (iii) Chaotic W-shape channel [19].
EXPLORING ROUGHNESS EFFECT ON LAMINAR FLOW
67
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Figure 4. Cross-shaped micromixers with static mixing elements [24].
Eddy Generation from Roughness Structures
The mixing resulting from eddies generated from the microscale roughness
structures provides another means to achieve passive mixing. However, the effect of
such structures on the flow characteristics is not well understood from both theoretical
as well as experimental standpoints. Generation of eddies from these roughness
elements has been examined by a number of investigators.
The chaotic motion was generated by Stroock et al. [25] by using a staggered
herringbone micromixer as shown in Figure 5. The two legs of the herringbone shape
are of unequal lengths. A group of several similarly shaped herringbones are alternated with another group that has the unequal legs shifted to the other side. This
arrangement produces two swirling cells of unequal sizes alternating as they pass over
the two groups of the herringbone structures.
Distributed Roughness and Local Suction and Blowing
The effect of distributed roughness on transition behavior was studied experimentally by a number of investigators including Reshotko and Leventhal [26],
Figure 5. Staggered herringbone micromixer with alternating groups of unequal legs causing chaotic fluid
flow through alternating cells of different size [25].
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68
S.G. KANDLIKAR
Reshotko [27], Corke et al. [28]; a good review of this topic is given by Floryan et al.
[29]. The transition seems to be resulting from the growth of disturbance waves at the
tip of the roughness elements. In both cases the disturbance frequencies are lower than
that at which Tollmein-Schlitching waves become unstable. Tadjfar et al. [30] experimentally investigated flow over arrays of spheres attached over a flat surface. They
measured the velocity distribution between the spheres and concluded that the transition was generated by horseshoe and hairpin vortices. The frequency was dependent
on the spacing between the spheres, as the disturbance waves from one sphere reached
the wake of the following sphere. These observations provide a critical link between
the flow in 2-D grooves, 3-D elements, and distributed roughness [31].
Another method to introduce the advection is to introduce a periodic sourcesink combination that alternately takes fluid out from a sink and reinjects it at the
source locations (same port alternating as source and sink is also considered). Such
flows were investigated by Jones and Aref [32]. A number of subsequent researchers
have studied this configuration in detail. Floryan et al. [29] used this concept to
simulate the periodic roughness structures, although the proof of the similarity
between the two configurations is not clearly established. A good summary of the
development of in-chaotic advection in micromixer application is presented by
Stremler et al. [33].
Floryan et al. [29] simulated the roughness structure with distributed suction and
studied the flow stability. They classified roughness as (i) single isolated 2-D roughness, such as a trip wire; (ii) single isolated 3-D roughness, such as a grain of sand; and
(iii) distributed roughness, as found on machined surfaces. They departed from the
classical approach followed by Schlichting and studied the flow and stability characteristics in the presence of the surface roughness elements [34]. In all cases studied,
the instability was induced by the appearance of stream-wise vortices when the suction
amplitude (similar to the height of the roughness elements) reached a critical value.
Another important observation they made was that the velocity distortion generated
by a constant suction had no effect on instability, contrary to earlier models linking
the flow characteristics of the plain flow and then superimposing the secondary
vortices generated by the simulated roughness elements. The presence of roughness
shifts the low-momentum flow away from the wall and leads to the formation of
highly distorted stream-wise velocity profiles that are subject to very strong secondary
instabilities caused by the vortices.
Eliahou et al. [35] introduced periodic perturbation through the disturbancegenerating ports surrounding a circular pipe. The resulting acoustic waves were used
to generate periodic suction and blowing. Constant blowing of air through the ports
was also investigated. They observed that at low amplitudes, the disturbances decayed
in the direction of the flow. At intermediate amplitudes, the disturbances grew initially
but decayed with increasing distance downstream. This phenomenon was identified as
relaminarization [36] and is discussed in greater detail later. At higher disturbance
amplitudes, the disturbances grew and caused transition to turbulent flow. The
growth was accompanied by the appearance of higher harmonics in the flow. The
injection was accompanied by a shift in the velocity profile. Further experiments were
carried out by introducing jets that constantly introduced air at four alternate slots
along the flow direction. The weak longitudinal vortices generated by these jets caused
transition at lower amplitudes and their role in creating an earlier transition in locally
disturbed flow was also observed.
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EXPLORING ROUGHNESS EFFECT ON LAMINAR FLOW
69
The turbulent eddy structure was observed to be quite different during instabilities caused by mass injection through a porous wall in the study of propellant burning
in a nozzle-less solid rocket motor [37]. This mass injection causes vortices that are
essentially two-dimensional in structure and the transition to turbulence occurs
farther away from the wall compared to a case with no mass transpiration.
The chaotic advection produced by source-sink pairs was studied by Stremler
et al. for mixing in genomic research [33]. The Poincare diagram provided the trajectories of particles that returned to the same location during a suction/blowing cycle
and also indicated the chaotic regions where the particles advected in some periodic
manner. The location of the source-sink and the number of such pairs produce
different effects that can be utilized in what the authors call designing for chaos, a
term originally proposed by Liu et al. [38].
The use of electric fields to generate local velocity fields was utilized effectively to
cause mixing in lab-on-chip and MEMS devices [39]. The electric field governed the
fluid flow path and various source-sink combinations were demonstrated. Although
the authors did not pursue it further, this technique can be implemented in conjunction with time-varying electric field to generate chaotic motion in the fluid to enhance
the mixing and transition to turbulence as well.
Mixing in Slugs
Mixing of microfluidic liquid slugs in an immiscible liquid medium is another
interesting application. Currently this type of mixing relies on the recirculation of the
liquid in the slug [40, 41]. Perturbing the circulation patterns inside the slugs with
structured roughness elements offers a new technique that can be implemented under
passive mixing devices.
Roughness structures are being utilized in microfluidic applications to enhance
transport processes and mixing in different applications, as seen from the above
discussion. However, the effect of roughness on the fluid flow, pressure drop characteristics, and transition to turbulence is not well understood. In the remainder of this
article, some of the issues related to turbulent eddy generation due to roughness
elements in laminar flow and relaminarization of flow past roughness elements are
discussed from a theoretical standpoint. Results from specifically designed experiments utilizing structured roughness elements and computational fluid dynamics
(CFD) simulations are presented. Finally, research needs in this field are outlined.
EFFECT OF ROUGHNESS IN OPEN CHANNEL FLOWS
The effect of roughness in inclined open channel flow of water was experimentally studied by Phelps [42]. The bottom channel wall was coated with different relative
roughnesses obtained with sand and glass particles. Phelps noted the difficulty in
actually locating the channel wall in the presence of roughness. The bottom was
considered to be at a distance one third the diameter of sand grains below the tip
where roughness interacted with the flow. The results showed that the friction factors
in the laminar region followed the linear relationship ( f ¼ Po/Re) on a log-log plot, but
the actual value increases with increasing roughness. The transition to turbulence was
also noted to shift to lower Reynolds numbers. The critical Reynolds numbmer was
around 2000 for smooth channels and decreased to 400 for the highest relative
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S.G. KANDLIKAR
roughness vale tested (around 37%). The presence of roll waves was also noted at the
transition location for higher roughness channels. For smooth channels, the roll waves
occurred prior to transition to turbulence.
The above study is important since it provides a systematic verification of
roughness on the flow characteristics in laminar flow. The roll waves occur prior to
the transition point on smooth surfaces, indicating that the damping effect of the
laminar flow delays the actual transition, whereas the roughness causes the transition
to occur as soon as the waves are set in. The role of instability is thus seen to be closely
linked to the transition.
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INSTABILITY IN GROOVED CHANNELS
Available research on channels with periodic grooves provides valuable insight
into the effect of localized instabilities on the laminar flow, its transition to turbulence,
and the effect on the transport processes at the wall. Transverse grooves are often used
in heat transfer applications to enhance the heat transfer coefficient in low Reynolds
number applications. The research in this area has been focused on obtaining a higher
heat transfer coefficient without significantly increasing friction factors. This is
accomplished by creating localized eddies that enhance the mixing in the fluid without
causing the transition. The oscillatory flows of this nature provide guidance in
projecting the possibilities of roughness elements in microfluidics applications.
Amon and Patera [31] conducted a numerical study to investigate the instability
in grooved channels. At low Reynolds numbers, the recirculation in the grooves is seen
to be similar to that shown in Figure 6. The profile of velocity gradients plotted at the
center plane of the groove shows a strong local inflection point. The instability in this
geometry still exhibits the characteristics of the Tollmein-Schlitching waves in the
plain channel flows. The grooves destabilitze the native T-S waves and introduce
the bifurcation behavior. They concluded that these waves play a central role in the
transition to turbulence, rather than the groove vortex or shear layer. The geometryinduced shear layer destabilization, however, yields a low Reynolds number supercritical instability leading to a narrow band transition near the transition region.
Self-sustained oscillations in the grooves were later investigated by Amon and Mikic
[43]. Above a certain critical Reynolds number, the flow over the grooves becomes
unstable in the presence of small disturbances. However, the disturbances are stabilized
at some fixed amplitude, leading to a periodic secondary self-sustained oscillatory flow.
Figure 6. Flow structure following a rib [60].
EXPLORING ROUGHNESS EFFECT ON LAMINAR FLOW
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These self-sustained oscillations result in well-mixed flows with more efficient thermal
transport processes at the wall prior to the transition to turbulence and require lower
pumping power than in turbulent flows. Additional work on the self-sustained oscillatory flows and their effect on wall transport processes is reported by Amon et al.,
Majumdar and Amon, Mikic et al., Kapat et al., and Mikic et al. [44–48]. As a result of
these investigations, a transition criterion, based on the friction velocity Ret given in
Eq. (2) for a rectangular channel, was proposed. This criterion is similar to that
proposed by Poll for transition induced by an isolated roughness element in the
external flow over a flat plate [49].
Transition criterion, Poll [49] and Mikic et al. [48]:
H=2
Re ¼
rffiffiffiffiffiffiffiffiffi
w;av
¼ 40 60
ð2Þ
EFFECT OF ROUGHNESS ON TRANSPORT MECHANISM AND TRANSITION
TO TURBULENCE — EDDY GENERATION, INSTABILITY,
AND RELAMINARIZATION
External Flows
In the case of external flows, the effect of surface roughness on the transition to
turbulence has been studied by many investigators. For example, Fischer and
Haramura [50] placed short rectangular obstructions in an open channel and studied
the effect of their width and arrangement on the transverse and vertical mixing of the
pollutants in the water stream. The experimental work of Klebanoff and Tidstrom
[51] showed that the mean flow is affected in the region immediately following the
roughness and the basic mechanism of early transition is by destabilizing the flow
within the recovery (or relaminarization) zone. The placement of roughness elements and strips on the swept wings and flat plates has been studied by a number of
investigators [52–55]. Kyriakides et al. [56] studied the effect of cylindrical disturbance on the transition and Funazaki et al. [57] studied the transition in the presence
of periodic waves generated by a sphere creating an isolated turbulence spot.
Reshotko and Tumin [58] proposed a roughness-induced transition that is based
on the transient growth theory. It is clear from these studies that our understanding
of the transition mechanism in this well-studied geometry (external flow over a 2-D
roughness element) is still far from clear in spite of the extensive research efforts in
this field.
Internal Flows The roughness elements cause local flow disturbances, which
are easily seen as the recirculation zones behind 2-D roughness structures, such as
cross ribs. The flow characteristics in these recirculation zones are primarily responsible for the interaction between the roughness elements and the fluid flow.
As a first simplification, the effect of roughness can be considered without taking
into account the local fluid flow fields. Such an approach was taken by Kleinstreuer
and Koo [59], who constructed a parallel flow model of the core flow and the flow
through the roughness layer. The roughness layer was modeled as a porous medium as
shown in Figure 7. This model is useful at very low Reynolds numbers, with very high
values of relative roughness.
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S.G. KANDLIKAR
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Figure 7. Porous medium model of roughness for low Reynolds number flows [59].
A number of investigators have conducted numerical studies of fluid flow
behind a 2-D roughness element considered in the form of a cross rib. These analyses
lead to flow recirculation zones as shown in Figure 6 by Rawool et al. [60] for a single
obstruction. For a series of obstructions, as in a 2-D structured roughness surface,
shown in Figure 8, the steady-state analysis yields solutions that the recirculation
zones are confined between two roughness elements.
To understand the roughness effect in the laminar flow from a theoretical
standpoint, we need to look at the classical developments in the hydrodynamic
stability theory; e.g., see Panton [61]. In internal flow analysis, application of the
laminar flow equations results in any irregularities at the wall being damped out. The
transition to turbulent flow can be analyzed by the application of hydrodynamic
stability theory. Although the roughness structures are 3-D in nature in general, the
2-D case is also important, as it is easier to implement in experimental and numerical
investigations. Further, the study of 2-D becomes even more relevant in light of
Squire’s work, which showed that for any unstable 3-D disturbance there is a corresponding 2-D disturbance that is more unstable [62]. In case of viscous parallel flows,
the instability is induced through the Tollmien-Schlichting waves near the wall.
The local wall velocity fluctuations are introduced by the presence of roughness
elements. The instability at the boundary of the recirculating cells and the main flow
leads to the eddy vortex generation. In case of a single rib, these eddies are introduced
downstream, but they are not sustained beyond a certain distance. At this point, it is
Figure 8. (i) Flow structure in the presence of closely spaced 2-D roughness elements [60].
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EXPLORING ROUGHNESS EFFECT ON LAMINAR FLOW
73
Figure 8. (ii) Flow structure in the presence of 2-D roughness elements at higher pitch [60].
useful to understand the concept of relaminarization. A sound foundation of this
concept was formulated by Sreenivasan [63] and Narasimha and Sreenivasan [36].
Relaminarization of turbulent flow can be accomplished by a number of means:
(a) turbulent energy in the eddies is absorbed or destroyed by body forces such as
buoyancy, (b) turbulent eddies are dissipated through viscous dissipation, or (c) the
turbulent boundary layer undergoes relaminarization under the influence of severe
streamwise acceleration. In the case of a single step shown in Figure 6, the local flow
disturbance induced in the flow is damped out. The flow becomes laminar again a certain
distance away, depending on the Reynolds number as well as the height and shape of the
roughness element. When the roughness elements are closely placed, the individual
recirculation cells are formed between the roughness elements as shown in Figures 8(i)
and 8(ii). The main flow also seems to be affected in terms of deviations from the laminar
flow as seen by some of the transverse velocity components. This net effect of roughness
on flow may thus be characterized in terms of two basic instability mechanisms:
1. Individual roughness elements cause a local disturbance, which decays downstream
due to the relaminarization process. Instabilities are introduced locally beyond the
roughness element and cause departure from laminar flow characteristics.
2. The recirculation cells formed between two successive roughness elements present
another source of instability. This instability is the result of eddies being introduced
from the recirculatory cells in the flow field.
The first type of instability causes the flow to depart more from the fully
developed laminar behavior of roughness elements with shorter pitch lengths,
since the relaminarization process continues in the flow direction. The second type
of instability is somewhat more complex, as it is governed by the oscillations set up in
the recirculatory flow behind the roughness elements. The roughness-induced transition is explosive in nature compared to the natural transition caused by T-L waves
[26, 27].
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74
S.G. KANDLIKAR
The relaminarization process can be defined in terms of wall shear stress values or
centerline velocities. A third method of describing the relaminarization is by plotting the
turbulence intensities. Earlier numerical work by various investigators, discussed above,
indicates that the relaminarization length becomes longer as the Reynolds number
increases. A similar effect is seen with the height of the roughness elements. Gance
et al. [64] show the variation in the local velocity profiles as the Reynolds number is
increased from the laminar to turbulent region with grooved surfaces. Jime
nez [65, 66]
has discussed various numerical techniques that can be applied to laminar flows over
rough surfaces. One such method suggested includes replacing the effect of the roughness layer with an equivalent wall-boundary condition.
The generation and regeneration of turbulence has been studied in the literature
in turbulence flows. For example, Hamilton et al. [67] studied the process numerically
and concluded that the process leading to sustained turbulence consists of three
distinct phases: (i) formation of streaks by stream-wise vortices, (ii) breakdown of
the streaks, and (iii) regeneration of the stream-wise vortices. The instability is induced
by the T-S waves at the wall. Applying these mechanisms to the flow over rough
surfaces, the instabilities caused by the vortices behind the roughness elements are
believed to be responsible for the early transition and occur earlier than the onset
condition for the T-L waves. Reuter and Rempfer [68] conducted a numerical study
and suggested a more complex interplay between the factors responsible for the
transition. The formation of streaks from these vortices and their breakdown at
lower transition Reynolds numbers are not investigated in the literature and present
an opportunity for further research.
One of the early studies on understanding the effects of isolated roughness
during external flow on a flat plate by Sydney [69] provides guidance in understanding
the roughness effects in laminar internal flows. The presence of a roughness element at
Reynolds number below the transition Reynolds number still causes disturbances in
the flow that are carried several hundred roughness diameters downstream. Even
though transition may not occur, the flow characteristic is sufficiently different from
the laminar flow. Considering a set of 2-D roughnesses (such as parallel ribs) or 3-D
roughnesses (such as sand grains), the transition to turbulence represents the ultimate
effect of the roughness elements, but even in the absence of such a transition the flow
structure would be sufficiently affected to depart from the laminar flow behavior.
Looking at the steady-state solution of flow over a series of grooves shown in Figure 8,
the flow in the core is seen to depart from the classical laminar behavior.
Micro-PIV visualization was employed by Zeighami et al. [70] to study the
transition process in microchannels. Changes in flow structure were observed near
the transition and the authors proposed this method as a valuable tool in studying
transition behavior. This technique could be employed in the study of microchannel
flows with rough walls.
ROUGHNESS CHARACTERIZATION
As seen from the above discussion, the main purpose of roughness structures
employed in practical applications is for enhancing mixing or wall-to-fluid transport
processes. A methodology is needed to systematically characterize roughness elements
and their effect on the transport processes. Some of the work available in roughness
characterization is briefly reviewed here.
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EXPLORING ROUGHNESS EFFECT ON LAMINAR FLOW
75
Figure 9. Comparison of experimental data for water and laminar flow friction factor prediction using
constricted flow diameter, water flow, Dh,cf ¼ 684 mm, b ¼ 500 mm, bcf ¼ 354 mm, w ¼ 10.03 mm,
e/Dh,cf ¼ 0.1108 [8].
Although a higher relative roughness in the channel will lead to the formation of
eddies and early transition, this method is not directly implemented for either enhancing mixing or other transport processes. Estimation of roughness effect on fluid flow
rates and pressure drop is of practical interest and theoretical understanding of this
subject is rather preliminary; surface roughness characterization therefore becomes an
important consideration. The current method of defining surface roughness through
simple average or rms roughness is not adequate. In this regard, Kandlikar et al.,
Gloss et al., Taylor et al., and Bahrami et al. have proposed new parameters to
characterize the roughness [8, 71–73]. Kandlikar et al. [8] and Taylor et al. [72] propose
a new parameter, Fp, defined as the height of the mean peak line from the mean floor
line in addition to the use of the constricted flow diameter concept which resulted in
good agreement with the friction factor data as shown in Figure 9.
Bahrami et al. [73] used the local channel radius to evaluate the local friction
factor. A different approach was taken by Kleinstreuer and Koo [59], who divided the
flow as the main flow in the core and flow through the roughness elements modeled as
a porous medium. This model considers the constriction effect and the increased
resistance for flow in the roughness region adjacent to the wall.
The characterization of roughness poses significant challenges in roughness modeling. The average roughness Ra does not take into account the various geometrical
differences among rough surfaces produced by different manufacturing techniques.
Such a difference was noted by Krogstad and Antonia [74] in their experimental study
with two types of rough surfaces: (i) woven stainless steel mesh screen glued on to
aluminum wind tunnel wall, and (ii) lateral rods of 1.2-mm diameter and a spacing of
four times the diameter glued on the wind tunnel walls for a length of 3.2 m. The mean
velocity profiles and the Reynolds stresses were noticeably different, suggesting that a
basic understanding in roughness characterization is lacking in turbulent flows as well.
LAMINAR-TURBULENT TRANSITION CRITERION
A number of investigators have presented various transition criteria in the
presence of roughness elements. The following transition criterion is given by
76
S.G. KANDLIKAR
Kandlikar et al. in Eq. (3) and Eq. (4), based on the experimental results similar to
those shown in Figure 9 previously reported by Kandlikar et al. [8, 71].
For
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For
0 < "Dh;cf 0:08 :
0:08 < "Dh;cf 0:15
Ret;cf ¼ 2300 18; 750 "=Dh;cf
ð3Þ
Ret;cf ¼ 800 3270 "=Dh;cf 0:08
ð4Þ
The laminar-turbulent transition is thus expressed in terms of roughness height
and constricted flow diameter. The viscous effects, or relaminarization effects,
are dominant to dampen out any eddies that are formed behind the roughness
element.
On the other hand, the transition criterion proposed by Mikic et al. provides the
upper limit at which the transition will take place [48]. This criterion is given in Eq. (2),
where Ret is the Reynolds number based on the shear velocity, H is the channel height,
and tw,av is the average wall shear stress.
Morkovin [75] introduced a criterion defining a hydraulically smooth surface,
which was based on the Reynolds number using the roughness height as the characteristic dimension. A surface may be considered as hydraulically smooth if the
roughness Reynolds number is below a certain value as given by Morkovin in Eq.
(5). The velocity Ue is the flow velocity at the tip of the roughness height e, and v is the
kinematic viscosity.
Re" ¼ 25 where Re" ¼
U" "
v
ð5Þ
For a rectangular channel, the maximum velocity in the vicinity of the wall will occur
at the center of the long side of the channel. The velocity Ue at this location is given in
Eq. (6) by Kakac et al. [76]. Because the velocity is calculated at the center of the
channel, the z term in the equation will be zero.
"
#
z m i m þ 1 n þ 1
b" 2 h
U" ¼ Um 1 1
b
a
m
n
8
<2
where m ¼ 1:7 þ 0:51:4 and n ¼
: 2 þ 0:3 1=
3
for
1=3
for
1=3
ð6Þ
The Reynolds number based on the main flow is thus linked to the roughness
Reynolds number criterion given in Eq. (2) by the following expression for a rectangular channel:
Re ¼ 25
U"
"
Dh;cf Um
ð7Þ
Note that the Reynolds number Re is based on the root dimensions of the channel,
whereas Dh,cf is based on the constricted flow diameter. For a circular channel, the
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EXPLORING ROUGHNESS EFFECT ON LAMINAR FLOW
77
Figure 10. Comparison of transition data of Kandlikar et al. [8] with stability and transition criteria
available in literature. Solid line, limit of stability by Morkovin [75]; small dashed line, transition criteria
by Mikic et al. [48]; long dashed line, empirical criterion by Kandlikar [8].
transition criterion expressed in terms of the Reynolds number is obtained by using the
parabolic velocity profile given in Eq. (8) in conjunction with Eq. (3).
U" ¼ 2Um 1 r "2 r
ð8Þ
The above criteria are expected to work well with 2-D roughness elements.
Their applicability to the uniform 3-D roughness structure remains to be
validated.
Figure 10 shows a comparison of the experimental data on transition
Reynolds number obtained by Kandlikar et al. [8] and the criteria by Mikic
et al. [48] and Morkovin [75]. Morkovin’s criterion represents the upper limit
above which the flow is unstable. The theoretical limits represent the upper limit
of transition, but the actual transition may occur sooner in the presence of
roughness elements.
CONCLUDING REMARKS
In order to understand the roughness effect on fluid flow at microscale, we first
need to characterize a surface using unique surface descriptors that are relevant and
lend themselves to fluid flow modeling. The surface representation needs to take into
account:
2-D roughness—roughness profile, spacing, orientation, and spatial and temporal
non-uniformity (in spacing as well as roughness height, from one roughness element
to next, and within each roughness element)
3-D roughness—roughness feature description, its distribution, spatial and temporal non-uniformity.
The channel and flow representation also become important:
78
S.G. KANDLIKAR
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channel shape, flow bends and curves, appropriate definition of hydraulic
diameter
flow Reynolds number, vortex or instability generation
departure from laminar flow behavior and transition to turbulence
Although the complete description of the surface and flow characteristics may
be too difficult to accomplish, interim goals based on specific applications may be set.
For example, in the area of mixing, the effect of surface features on the generation of
eddies and their interaction with the main flow field is of particular interest. In case of
heat transfer from the wall, the increase in the heat transfer in relation to the increase
in the pressure drop is of concern. In addition, the spiral motion in the fluid may also
be generated by the surface features to enhance the mixing as well as wall transport
characteristics.
Roughness studies at microscale are currently in their infancy. It is not yet clear
how roughness itself and its effect on transport processes can be modeled at microscale. Perhaps more intriguing at this stage is this question: what new opportunities
are available to utilize roughness modification in accomplishing specific processes in
yet-unknown roles in microscale applications and devices? The future awaits us in
transforming these mysteries into scientific descriptions.
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