Theory of Flow Access With Apparent Obstacles: Cascades, Jumps, Roll Waves, and Turbulence

In this article I question of whether a flow system that has freedom to morph also exhibits the natural tendency to configure itself to provide greater access to its currents. In other words, I test the validity of the constructal law. There is considerable empirical evidence in support of the natural tendency toward greater flow access. Examples are reviewed briefly in the next section.

The reason for questioning this evolutionary tendency is that there are flow architectures that seem to contradict the tendency. Three examples are mounted together in Fig. 1, and they are highly familiar.

First, the flow of rain water on an inclined street covered with gravel does not carve a smooth channel with uniform depth and width. Instead, it carves a straight path with cataracts, which are periodic “dams” interspaced with pools followed by “cascades.” The pools are most evident after the rain (Fig. 1(a), right).

Second, a sheet flow that is smooth and rectilinear terminates itself into a “wall” posed by a thicker layer of fluid that flows more slowly. This is known as a “hydraulic jump,” Fig. 1(b). It is considered classic and understood.

Third, the sheet of rain water that slides on smooth pavement exhibits periodic water falls. The flow that precedes the fall is inclined (it is not a pool), Fig. 1(c). The phenomenon is known by several names (roll waves, ripples, ridges).

In sum, are these abrupt morphological changes hindering or liberating the flow? I do not know, but I want to know. By questioning the directionality of these changes we have an excellent opportunity to check the validity of the constructal law of evolutionary design toward greater access. For now, the three examples of Fig. 1 give the impression that the falls and the jumps are obstructing the flows, like the fences and the water trap in the 3000 m steeplechase run in track and field.

The straight line is the shortest path between two points, and the straight duct with round cross section poses minimum resistance to flow. These truths are known scientifically since antiquity [1,2], and arguably are much older. Today, we know them intuitively even when we encounter them in homework problems of fluid dynamics.

Why should the intuitive truths be questioned? Here are three reasons:

The first is a general trend in science: what is intuitively correct is taken for granted (i.e., understood), and so it escapes scrutiny.

The second is that the straight path and the round cross section are about flow architecture. The science of configuration, or design, is a relatively new domain that hovers over many of the sciences (geophysics, physiology, engineering, urbanism). With the advent of fire, boating, wind power, steam power, and aviation, the science of flow architecture covers fluid dynamics as well [3–5].

Third, in scientific thought the straight line and the round cross section emerge along deductive sequences of imagined changes in configuration. The changes are oriented with purpose (with directionality) toward providing easier access from point to point. Observable changes with discernible direction in time are the phenomenon of evolution, or evolutionary design. Evolution is the phenomenon that unites everything that is animate and inanimate.

The three reasons escape scrutiny because they are known, invoked, and accepted intuitively. They should not be. Evolutionary design is the mother of all engineering science, innovation, and evolution. With Fig. 2, for example, we showed that the number of possible flow architectures that connect two points is infinite [6]. With direction in the changes that we contemplated, we eliminated one class of architecture at a time, in favor of better classes.

Specifically, the physics phenomenon of “economies of scale” [7] is why two ducts in parallel are inferior to a single duct with the same volume as the two ducts combined. Next, the many curved ducts between the two points are inferior (as a class) to the straight duct. The many cross sections that have sharp corners are inferior (as a class) to those that do not have sharp corners.

The numerous polygonal cross sections are inferior to the few that are shaped as regular polygons. This last stage of evolution by elimination is shown in Fig. 3 [7]. The flow resistance between the two points is indicated qualitatively on the abscissa (for details see Ref. [6]). On the ordinate is plotted the number of sides (n) of the regular polygon, such that a larger n value accounts for greater freedom in the mind and hand of the person who contemplates and draws the design.

“Flow access” is defined by the arrow on the abscissa of Fig. 3. The discernible direction of change is indicated with two arrows, to the left for greater flow access, and downward with greater freedom. The world of point-to-point flow designs is divided into two worlds, the possible designs versus the impossible designs. The apparent frontier between the two is populated by the regular polygonal cross sections. These are the most valuable, and on that frontier is located the key milestone, the round cross section (n = ∞).

The evolving flow architecture is illustrated on the horizontal line at n = 6. Hexagonal shapes become better as they migrate to the left, toward the regular hexagon. The world of possible designs is to the right of the frontier. The effect of the increasing freedom in evolutionary change is to extend the domain of the possible into the domain of the impossible, if possible.

The evidence that the design evolution phenomenon illustrated in Figs. 2 and 3 is universal in nature is massive. Our group is one of several that have been reviewing the evidence, bio, and nonbio [8–21]. If design in nature is this way, then why is it that not all the point-to-point flow designs have evolved to be straight ducts with round cross sections? We questioned and answered this several times, class by class [6]:

• River cross sections are lemon-slice shaped, not round. In the simplest, a river cross section has a free surface and two scales, the width (w) of the free surface, and the maximum depth (d). Empirically, it is known that large rivers are wide and deep, and tiny channels are narrow and shallow. Measurements of river channels of all sizes revealed that w/d is essentially a constant in the range of 2–10 [22]. In other words, measurements show that the river cross section has one scale (w or d). We showed that the proportionality between width and depth emerges in the direction of increasing flow access through the cross section (of size w × d) by varying w (or d) subject to fixed cross-sectional area. Four designs are aligned with direction in Fig. 4. Their relative flow resistance is indicated under each drawing (relative to the semicircle, which is represented by 1).

• River channels are not straight. They exhibit meanders, which are quasi-sinusoidal shapes with wavelengths (λ) that scale with the stream width (w). The empirical data on the scaling λ ∼ w were reviewed in Ref. [23]. The average ratio λ/w ranges from 2 to 3 in laboratory studies to 6.5–11 in natural channels with long history. Two explanations for the deviation from the straight path have been proposed, and both account for the fact that the channel flow is relatively inviscid (high Re), and the medium in which it evolves is deformable, erodible. One is the theory of incipient hydrodynamic instability [24–39]. The other is the incipient buckling of the river column in apparent “longitudinal compression” due to the impulse of the stream entering the column and the reaction of the same stream as it exits the column [40–46]. Meanders occur naturally in many other deformable channels (e.g., blood vessels, hoses) and are covered by the same theory [44].

• River basins, deltas, lungs, snowflakes, and all vascular systems are dendritic (tree shaped), not straight channels from one point to another point. The reason is that every dendritic flow architecture connects one point with an infinity of points (area or volume), not with another point. Our group has shown on multiple occasions how to predict tree shaped flows (bio, nonbio) by invoking the evolution of the architecture toward greater access point-area and point-volume. This voluminous work is reviewed in books and review articles [13–22].

Figures 1(a)–1(c) are just three classes of flow architectures from nature. They are not human made. Many more examples of flow architectures from nature owe their evolving design to the same principle: to facilitate flow access by morphing with freedom. For example, our group has shown how to predict temporal design (rhythm) in animal locomotion (swimming, running, flying) [47], and respiration and blood circulation (heart beat) [48]. We showed that even greater opportunities for predicting and fast-forwarding flow design evolution exist in technology: the convergent evolution of the shapes of airplanes [49], helicopters [50], boats with sails [51], fire [52], and wheels [53].

On this diverse background, what is that remains to be elucidated? Three puzzles are identified in the next sections. They seem to go against the tendency toward facilitating flow access via evolutionary design. Cataracts, hydraulic jumps, and ripples seem to “snuf” the flow and contradict the smooth (straight) design that is expected from the textbook.

Cataracts and hydraulic jumps are two of the most common phenomena in nature, Figs. 1(a) and 1(b). They are different, obviously: the abrupt fall of the stream cannot be confused with the abrupt jump of another stream. Examined together, cataracts and hydraulic jumps have in common the pull of gravity. Cataracts go along gravity, and jumps go against gravity. They also share the impression given to the observer, which is that the abrupt changes “interfere” with the easy flowing of the straight channel expected by the scientist.

Are these differences and impressions correct? Two recent and unconnected publications challenged me to answer this question:

In 2019, Scheingross et al. [54] conducted laboratory experiments with water and debris in open channel flow and showed that cataracts happen naturally. They exhibit pattern, periodicity, and reproducibility. They are not due to pre-existing forms of relief (rocks, seismic faults).

In 2018, Bhagat et al. [55] showed that in a thin liquid film the hydraulic jump does not depend on the orientation of the film with respect to gravity. They showed that at the jump the viscous forces and surface tension balance the momentum change in the flowing film.

Can cataracts and their periodicity be predicted? The most recent study of this phenomenon [54] demonstrated through extensive laboratory visualization and measurements that cataracts form by themselves, and that the mechanism is the removal and deposition of debris [56]. This is in accord with the movement of gravel (Fig. 1(a)). The evolutionary design literature reviewed in the preceding section includes the erosion (smoothing) of rolling stones into spheres [57] and the phenomenon of self-lubrication [5], as a more general term for erosion and the tendency of channels to shape themselves to provide greater access to what flows in them. Self-lubrication owes its name to human made systems, such as a rod sliding longitudinally or rotating in its housing, with a film of oil between rod and housing. The rod is free to adjust its position (its center), and this is synonymous to the freedom of the oil film to morph its thickness and shape.

To address the question of flow access in the case of cataracts, consider the water flow in a straight river channel that forms the angle β with the horizontal line (Fig. 5). Most rivers flow this way, and β is small, only a few degrees. The geographical distance traveled is L, the river bed length is $Lβ≅L$⁠, and the mean water velocity in the inclined channel is V_β.

Figure 5 shows a stepped alternative to the β-inclined path. The long stretch of the river bed is close to the horizontal, i.e., close to being a pool. This is followed by a drop of height z along which the water is in free fall. Along the stepped path ACB the water starts from A and ends at B, just like along the incline.

Which alternative offers greater access to the water flow? In the following analysis I compare the flow time on the path ACB with the straight path AB. This comparison requires assumptions about the Reynolds number regime for the channel flow AB, and the condition of the riverbed. The question about greater flow access should lead to a predictable choice between the two paths, which should shed light on the periodicity of steps when steps provide greater access.

Figure 6 shows another common observation of natural fluid flow behavior. When the water jet from the faucet impinges on the sink bottom, it first spreads smoothly as a wall jet. Suddenly, the wall jet forms a thicker layer that continues to flow radially, with rolls and bulges that indicate a three-dimensional flow (turbulent flow), not a laminar radial flow. This phenomenon is known as a “jump” based on the established view [58] that at the transition from radial flow to three-dimensional flow the water jumps against gravity and acquires an amount of gravitational potential energy that the original radial jet did not have.

The following analysis goes against the established view: we analyze the radial (r) jet flow in the absence of gravity. The mass flowrate and velocity of the faucet jet $(m˙,V)$ are fixed. The analysis proceeds in three steps based on scale analysis:

• Consider the initial (smooth, radial) wall jet and determine the scales of its thickness (δ) and radial velocity (u). Model the flow as inertial, i.e., frictionless.

• Model the radial wall jet as viscid, with viscous diffusion having penetrated the δ thickness (as in laminar boundary layer flow), and then determine the corresponding δ and u scales of this flow regime.

• Intersect the two asymptotes determined earlier and deduce the radial distance where the inertial jet ceases to be inviscid.

In what direction should the change occur, from (a) to (b), or from (b) to (a)? According to the tendency toward greater flow access (constructal law), the change should be toward greater access for momentum vertically, into the plate, from the high momentum (movement) of the arriving jet to the slow movement of the liquid that comes to rest on the plate. In other words, the flow that changes configuration toward greater access is the momentum transfer across the water film, not the radial flow of the liquid (which, by the way, does not change configuration because it is radial in both configurations, (a) and (b)).

The answer then is that the change should be from (a) to (b), because in this direction the liquid film is consistently the thickest. In this direction of change, more of the liquid is slowed down by the wall.

Guided by Fig. 1(b), if the impinging jet flow is water with $m˙∼0.01kg/s$⁠, $V∼1m/s$⁠, ρ = 1000 kg/m³, and μ = 0.01 g/(s · cm), then Eq. (27) yields r ∼ 6 cm, which is comparable with the radial distance to the hydraulic jump observed in Fig. 1(b).

The constructal law dictates the direction in which the transition must take place, that is the natural selection between two changes that are available, AC → CB and CB → AC. The natural direction must be from t_AC to t_CB, because along this piecewise continuous curve the flow access down the incline requires consistently a shorter time.

Finally, another conclusion is that the distance L_TR marks the distance between consecutive ripples. Furthermore, t_TR is the time scale of water flow along the distance L_TR on the inclined surface, and the ridge at the end of L_TR represents the edge (B) shown in Fig. 7.

It may seem contradictory to propose to examine “flow access” in the same sentence with “obstacles,” as I did in the title of this article. The image of obstacles suggests lack of access, not access. As we saw in Secs. 3–6, there was no contradiction. More than that, we saw an illustration of what “theory” means [7,60]. It is a purely mental viewing that (if present in the mind already) empowers the mind to imagine other phenomena that impress the observer with “obstacles” when in fact the features that look like obstacles facilitate flow access.

Theory, the mental viewing, means idea (from the Greek idein, which means seeing with the eyes of the mind). I published this mental image in 1975 [61], Fig. 9, where I showed how to use a cascade of thermal contacts to facilitate the capture of the heat current that leaks along a mechanical support at low temperatures. Lim et al. published a similar image in 1992 [62], Fig. 10, and showed the benefits from apparent “obstacles” in the quest for more power from energy stored by melting a phase-change material. We discovered that the most power than can be produced with a hot stream $(m˙,T∞)$ is accessible in “cascades” of phase-change materials with descending melting points (T_m,1, T_m,2, …). The stream temperature should fall in a particular sequence of steps dictated by the recommended selection of melting materials, each expressed in dimensionless terms [τ₁ = T_m,1/(T_∞T₀)^1/2, τ₂ = T_m,2/(T_∞T₀)^1/2, etc.] that were determined numerically so that the total power (⁠$W˙(2)$ in Fig. 10) is maximum. It is useful to predict the cascade (τ₁, τ₂, …) when the need is to increase the access of the flow of power from the hot stream to the environment that includes you, the user of the installation.

The image of the cascade, today and earlier, is important for two reasons. First, it shows the background on which I felt eligible to question the apparent negatives associated with flow obstacles (Secs. 3–6). Second is a trend that threatens the integrity of all sectors of science [63]: new work is being published without citing the published precedents of the same idea. The most recent example is the 2021 paper by Xu et al. [64], which is about the cascade pictured in Fig. 10 and published by Lim et al. [62] in 1992. Numerous infractions of this kind were signaled in the published papers listed in Ref. [7], and reviewed pictorially in Ref. [65].

By questioning flow access in the context of three distinct fluid flow phenomena (Figs. 1(a)–1(c)), we came to the conclusion that natural flow architectures that look like obstacles owe their occurrence to the tendency toward greater access for what flows: water down an incline (Figs. 1(a) and 1(c)), and momentum perpendicular to the radial sheet flow (Fig. 1(b)), which causes the laminar-turbulent transition.

Agreements between classical notions of turbulence and constructal predictions have happened earlier. In the incipient rolling of slender flow regions (shear layers, jets, and plumes) the natural selection of configuration (called “transition”) between laminar flow and turbulent flow is always in favor of the configuration that offers faster access to the transversal transfer of longitudinal momentum to the neighboring layers. Predictions of the onset of turbulence cover a wide range of previously disconnected classical observations and correlations. For their derivation, terminology, and application, the reader is directed to my convective heat transfer textbook [18] as follows:

• All the “critical” numbers of transition to turbulence (forced convection, natural convection, boundary layers, and duct flow).

• The universal local Reynolds number that is of order 10², for example, Eq. (28).

• The frequency of undulating (buckling) turbulent jets, wakes, and plumes (the universal Strouhal number).

• The constant-angle shapes (wedge, cone) of turbulent flow regions with large scale structure.

• The pulsating frequency of billows rising from pool fires.

• The smallest eddy that universally has a Reynolds number of order 10².

• The thicknesses of the viscous and conduction sublayers in fully turbulent flows, Fig. 11.

• The analogy between fluid friction and heat transfer.

• The analogy between rolling eddies and rolling stones.

Many other stop & go (or in & out) phenomena with apparent obstacles, such as respiration [48], heat exchanger structure [66], ice making and surface cleaning [21], can be conceptualized, analyzed, and unified as demonstrated in this article.

If you know the principle you can predict not only consequences that are observable but you can also make unintended predictions. I made unexpected predictions throughout my career, even before I called them “constructal law” in 1996. In retrospect, the pre-1996 ideas constitute a body of constructal prefigurations, such as the predictions of turbulence listed earlier.

If you know the principles, you know where the observed configuration came from. You know its cause. That is where biomimetics and reverse engineering are successful: in the minds of those who know the science of design, the physics principle. Reverse engineering is the mental process where one uses reasoning in order to deduce how an observed artifact was made (weapon, software, airplane). Artifacts are objects made by humans, and they empower the human and machine species (recall that the word “machine” means artifact, contrivance).

Many more objects happen by themselves, naturally (animals, rivers, winds, valuable minerals, celestial bodies). The many are not artifacts. To claim that a person can “reverse engineer nature” is incorrect [65]. Objects that are not made by people are not contrivances. Nature was not made by a person. If you doubt that, consider the fact that nature on earth and elsewhere is much older than the biosphere of the earth. Rivers and winds like ours are visible on planets without biospheres.

One can examine, describe, and catalog objects that happened by themselves, and that is empiricism. On the other hand, if one questions why many of such objects look similar, or why they behave similarly, the idea that pops up in the mind of the curious is that the coincidences are due to a common cause. That idea is theory, the new physics that empowers future observers not to waste time searching for the new idea.

Theory must not be confused with empiricism. Empiricism is “the facts” established after observation and experiment (measurements) intended to validate or invalidate a theory. For example, the theory presented here for cataracts, jumps, and roll waves can be compared subsequently against data measured in Refs. [54,55,59], respectively. Theory and empiricism are eminently different. For this very reason, they are useful when practiced together.

The broad conclusion from this article is that the constructal law governs the evolution of configuration in flows with apparent obstacles. By analyzing the three phenomena of Fig. 1, we subjected the constructal law successfully to Ellis and Silk’s test [67] of a scientific theory, namely “the issue boils down to clarifying one question: what potential observational or experimental evidence is there that would persuade you that the theory is wrong and lead you to abandoning it? If there is none, it is not a scientific theory.”

I thank the four reviewers for their unanimous and highly supportive evaluation of my manuscript.

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable.

No data, models, or code were generated or used for this paper.

d =

depth, m

g =

gravitational acceleration, m s^–2

$k˙$ =

kinetic energy rate, W

$m˙$ =

mass flowrate, kg s^–1

$m˙′$ =

mass flowrate per unit length, kg s^–1 m^–1

p =

perimeter, m

r =

radial position, m

t =

time, s

u =

radial velocity, m s^–1

w =

width, m

z =

vertical drop, m

A =

cross-sectional area, m²

D =

sheet thickness, m

F =

friction force, N

R =

time ratio, Eq. (5)

V =

jet velocity, m s^–1

C_f =

skin friction coefficient, Eq. (1)

C₁ =

factor, Eq. (14)

C₂ =

factor, Eq. (24)

C₁₀ =

factor of order 10

F_R =

reaction force, N

L, L_β =

flow lengths, m

β =

angle

δ =

film thickness

θ =

angular position

λ =

wavelength, m

μ =

viscosity, kg s^–1 m^–1

()_BL =

boundary layer

()_TR =

transition