Boolean logic by convective obstacle flows

Abstract

We present a potential new mode of natural computing in which simple, heat-driven fluid flows perform Boolean logic operations. The system comprises a two-dimensional single-phase fluid that is heated from below and cooled from above, with two obstacles placed on the horizontal mid-plane. The obstacles remove all vertical momentum that flows into them. The horizontal momentum extraction of the obstacles is controlled in a binary fashion, and constitutes the 2-bit input. The output of the system is a thresholded measure of the energy extracted by the obstacles. Due to the existence of multiple attractors in the phase space of this system, the input–output relationships are equivalent to those of the OR, XOR or NAND gates, depending on the threshold and obstacle separation. The ability to reproduce these logical operations suggests that convective flows might have the potential to perform more general computations, despite the fact that they do not involve electronics, chemistry or multiple fluid phases.

1. Introduction

This work highlights the computational abilities that arise in simple convective obstacle flows, highlighting a possible new realm of natural computing, and illustrating the information processing that even simple physical systems perform.

Thermodynamics is the science of energy; its inter- conversions, and its evolution. The simplest physical systems have a unitary relationship between physical state and control parameters such as pressure and temperature. For these systems, history is irrelevant, due to the entropic erasure of information from past states. The system explores its accessible phase space and destroys any correlations that might serve as a memory. Yet other physical systems make very effective memories (e.g. condensed matter systems undergoing very slow relaxation to equilibrium).

The world exhibits a vast spectrum of phenomena and systems that are neither perfectly frozen (memoryful), nor completely disordered (memoryless). There is in fact a myriad of physical mechanisms that generate, retain, manipulate and destroy information. Simple Brownian or quantum fluctuations are a prime source of information, and retention can occur by some form of annealing process, for example. Entropic forces that seek a maximum in accessible states cause information erasure. And between the production and destruction of information, yet other physical processes transform the information that they contain.

Since Turing outlined the universal model for computation, we have produced faster and more powerful electronic computers. However, the natural world is replete with examples of computation. In fact, the perspective of information and the processing thereof can serve as a powerful alternative to the energy-based interpretation of physical systems (see e.g. [1–7] and references therein).

The field of unconventional or natural computing has explored a plethora of non-electronic systems that process information. These include gene regulatory networks [8–11], cellular automata [3,12–15], chemical reaction networks [16–21], interference of physical waves or concentration profiles [16,22–24], nonlinear dynamical systems [25–27] and myriad other exotic methods (e.g. [28,29]).

The use of fluid dynamical systems as a means of computing initially emerged in the 1960s [30–41], but received decreasing attention due to the irrepressible advances of electronics and integrated circuits. However, in recent decades unconventional computing and microfluidics has seen renewed interest. This is due to the many applications that require compact, non-electronic, precisely controllable environments, in particular for manipulating microscale reaction compartments and carrying out high-throughput screening protocols. Note however that the use of thermal flows for logical operations has not yet been explored.

Some notable instances of fluid computing include the use of the parallel nature of a three-dimensional microfluidic system to solve an NP hard problem [42]. A microfluidic analogue of an integrated circuit was introduced by Thorsen et al. [43], with memory functionality akin to random-access memory. This was soon followed by further progress in microfluidic memory and control [44], with potential applications in the medical and chemical industries. Using differential flow resistances, researchers demonstrated all of the primary 2-bit Boolean logic operations in a microfluidic system [45]. Several logic operators were later exhibited by a two-phase droplet system [46]. In this case, the relative hydrodynamic influences of the droplets afforded the system's computational abilities. In the same year, a comprehensive study revealed a range of functionalities in a two-phase bubble system including several 2-bit logic gates (sufficient for universal computation), a toggle flip-flop, a ripple counter, timing restoration, a ring oscillator and an electro-bubble modulator [47]. Contemporary research continues to explore the properties of two-phase fluid computation [48–51], as demand grows for microscale, self-organizing, intelligent systems.

In this paper, we present a novel example of fluidic logic, which uses buoyancy-driven thermal flows. The abilities of the system arise from the non-trivial attractor landscape that arises when obstacles are placed in such systems. This was demonstrated in our recent work, which presented bistability, hysteresis and memory in a single-obstacle convective system [52].

In the present work, we wish to suggest convective logic as a new avenue for natural computing, and use this example to motivate the computational view of nature (in contrast to the purely energetic view). We do so by presenting a simple thermohydrodynamic system that can carry out three Boolean logic operations: OR, XOR and NAND.

2. System

Single phase fluid convection is a prime example of pattern formation in a non-equilibrium system [53–59]. When the thermal driving force (the dimensionless Rayleigh number) is raised above a critical value, the diffusive, static state becomes unstable to perturbations. Fluid parcels that are warmer than average rise and are displaced by colder parcels, which descend. After a transient phase, the system settles into an organized configuration of convection rolls.

In this work we analyse a two-dimensional fluid enclosed by horizontal no-slip plates, shown in figure 1. The aspect ratio is 2 : 1, the Rayleigh number is fixed at Ra = 10⁴ and the Prandtl number at Pr = 1. Periodic boundary conditions are used in the horizontal direction. Two obstacles are placed in the flow at the system's horizontal mid-plane. They are square and only approximately 0.8% of the system height, separated by a distance of Δ_o. Two systems are considered, one with Δ_o = H and one with Δ_o = H/2, where H is the system height.

Figure 1.

System schematic showing the boundary temperatures T_h and T_c that drive convective heat flow from the lower to the top boundary. The two obstacles are shown as white crosses (their actual size is only approximately 0.8% of the system height H). The colouring shows the dimensionless temperature with red corresponding to T_h and blue to T_c. (Online version in colour.)

The fluid dynamics and heat transport are simulated using a thermal lattice Boltzmann model, which has been rigorously analysed and tested across a broad range of thermal flow problems over several decades [60–66]. Momentum extraction is implemented using a simple body force method. Both obstacles absorb all vertical momentum that flows into them. Extraction of horizontal momentum is controlled by the binary input parameter m_ui, where i is the obstacle index. We assume only complete or null extraction, for example, m_u1 = 1 means that the left obstacle is extracting all horizontal momentum that flows into it (in addition to vertical momentum, which is extracted in all cases).

3. Results

For a full discussion of the fluid dynamics of convective obstacle flows, we refer the reader to our recent work [52]. In general, obstacles that extract both momentum components tend to attract convection plumes (maxima of vertical velocity), due to the Coandă effect [67,68]. We denote this the plume attractor.

When only vertical momentum is extracted, the most stable attractor occurs when the centre of either convection cell is co-located with the obstacle (we denote this the vortex attractor). The plume attractor is also observed in the vertical case, but it occurs less frequently. In the present work, these effects are combined in different configurations through the presence of two obstacles (only single obstacles were used in [52]).

(a). System-height obstacle separation

In the first case considered, the two obstacles are separated by Δ_o = H. The first of the four 2-bit inputs consists of neither obstacle extracting horizontal momentum: [m_u1 m_u2] = [0 0]. The resulting steady state is shown in figure 2a. For vertical momentum sink systems, the vortex attractor is the most stable attractor because induced shearing forces from vertical velocity gradients act to push vortex centres towards obstacles (see [52] for full details). In the double obstacle system shown in figure 2a, this effect is approximately doubled, and the steady state shown becomes strongly attractive (no other steady states were observed).

Figure 2.

Typical flow structure of a convective 2-obstacle flow in which the obstacles are separated by Δ_o = H, with different inputs: (a) [m_u1 m_u2] = [0 0], (see animation of this simulation (https://www.youtube.com/watch?v=IKBfdaL9Zr8)), (b) [m_u1 m_u2] = [0 1], (see animation of this simulation (https://www.youtube.com/watch?v=6D2PtErYHHY)). Obstacles are shown as white crosses (actual size is only approx. 0.8% of the system height). Colour shows the normalized temperature of the fluid. (Online version in colour.)

The next 2-bit input is: [m_u1 m_u2] = [0 1], for which the steady state is shown in figure 2b. Because the second obstacle now also extracts horizontal momentum, there is an effective attractive force between the nearest convection plume and the obstacle. As a result, this input causes co-alignment between the two convection plumes of the system and the two obstacles. There is an equivalent effect for the input [m_u1 m_u2] = [1 0], shown in figure 3a, which given the periodic boundary conditions of the system, is physically equivalent to the previous case. The final input, [m_u1 m_u2] = [1 1], also causes co-alignment of obstacles and convection plumes, since both obstacles are attractive to the convection plumes.

Figure 3.

Typical flow structure of a convective 2-obstacle flow in which the obstacles are separated by Δ_o = H, with different inputs: (a) [m_u1 m_u2] = [1 0], see animation of this simulation (https://www.youtube.com/watch?v=vj_tqCfPK-8), (b) [m_u1 m_u2] = [1 1], see animation of this simulation (https://www.youtube.com/watch?v=IHZrO9r0ff4). Obstacles are shown as white crosses (actual size is only approx. 0.8% of the system height). Colour shows the normalized temperature of the fluid. (Online version in colour.)

The output of these systems can be considered to be the total kinetic energy extracted by the two obstacles, W_o. The steady states of the four Δ_o = H systems are summarized in table 1. The first input causes co-alignment between vortex centres and obstacles. Since the vortex centres are locations of vanishing fluid velocity, the obstacles do not extract any momentum. For the other three 2-bit inputs, there is co-alignment of convection plumes and obstacles, producing a high work output. The [1 1] input produces a slightly higher output than the second and third inputs, because it is extracting horizontal momentum from both obstacles, whereas the second and third inputs only extract horizontal momentum from one obstacle.

Table 1.

Truth table for double obstacle convection flows with separation Δ_o = H. Steady-state kinetic energy extraction W_o is given in arbitrary lattice units.

Δo	mu1	mu2	Wo/10−7	Wo > 4.5 × 10−7
H	0	0	0	0
H	0	1	9	1
H	1	0	9	1
H	1	1	10	1

If we consider the output bit to be simply whether the extracted work is greater than a threshold value: W_o > 4.5 × 10⁻⁷ lattice energy units, then this system implements a Boolean OR gate, since the presence of either input bit produces a high output. Indeed, the presence or absence of work output could also be used as the binary output, and the system would again implement an OR gate.

(b). Half system-height obstacle separation

We now consider a smaller obstacle separation of Δ_o = H/2. With the input of [m_u1 m_u2] = [0 0], there is co-alignment of the obstacles with a vortex centre and plume, or vice versa (both are equally stable). For vertical momentum extraction, co-alignment with the vortex centre or plume are both attractors, albeit with different frequencies of occurrence [52]. Hence this system has a total of four attractors, each with equal work output: (a) anti-clockwise vortex centre and upwelling plume (as shown in figure 4a), (b) downwelling plume and anti-clockwise vortex centre, (c) clockwise vortex centre and downwelling plume, and (d) upwelling plume and clockwise vortex centre.

Figure 4.

Typical flow structure of a convective 2-obstacle flow in which the obstacles are separated by Δ_o = H/2, with different inputs: (a) [m_u1 m_u2] = [0 0], (see animation of this simulation (https://www.youtube.com/watch?v=CVTo8zuPS_w)), (b) [m_u1 m_u2] = [0 1], (see animation of this simulation (https://www.youtube.com/watch?v=Jtyu8zI46Bs)). Obstacles are shown as white crosses (actual size is only approx. 0.8% of the system height). Colour shows the normalized temperature of the fluid. (Online version in colour.)

With the input of [m_u1 m_u2] = [0 1], the second obstacle becomes attractive to convection plumes, as shown in figure 4b. This case thus has two attractors comprising: (a) alignment with anti-clockwise vortex centre and upwelling plume (as shown in figure 4b) and (b) alignment with clockwise vortex centre and downwelling plume.

The third input: [m_u1 m_u2] = [1 0], is equivalent to the previous case except that now the first obstacle is attractive to convection plumes, and the second obstacle is attractive to vortex centres. Hence the two attractors are now: (a) Upwelling plume and clockwise vortex centre (as shown in figure 5a) and (b) Downwelling plume and anti-clockwise vortex centre.

Figure 5.

Typical flow structure of a convective 2-obstacle flow in which the obstacles are separated by Δ_o = H/2, with different inputs: (a) [m_u1 m_u2] = [1 0], (see animation of this simulation (https://www.youtube.com/watch?v=cv1jQWLiEpg)), (b) [m_u1 m_u2] = [1 1], (see animation of this simulation (https://www.youtube.com/watch?v=eUq6LgCENyU)). Obstacles are shown as white crosses (actual size is only approx. 0.8% of the system height). Colour shows the normalized temperature of the fluid. (Online version in colour.)

For the final input, [m_u1 m_u2] = [1 1], there is competition between the obstacles, since both are now attractive to convection plumes. Since both obstacles are equivalent, neither can ‘win’ this competition, and hence the steady state consists of either vortex centre being placed equidistant between the two obstacles (figure 5b). A second stable attractor consists of the clockwise vortex centre being poised between the two obstacles.

Note that one might expect a second pair of attractors for this system, wherein either convection plume is poised halfway between the two obstacles. However, such a state is unstable to perturbations, because any deviation from a plume situated precisely halfway between the obstacles will cause a net horizontal force on the flow field. The flow then migrates such that a plume moves towards one of the obstacles. However, when the plume reaches the obstacle, the flow field overshoots and continues moving, eventually reaching a state in which a vortex centre is poised between the obstacles (figure 5b, or the equivalent mirror reflection).

Let us now consider the input–output relationship of this system, shown in table 2. The highest work output is exhibited by the two systems in which only one obstacle extracts horizontal momentum. This is due to the co-alignment of the ‘on’ obstacle and a convection plume, coupled with the fact that both components of momentum are being extracted from the ‘on’ obstacle. The first input [0 0] produces slightly less work because neither obstacle extracts horizontal momentum. However, the co-alignment between plume and the obstacle that extracts vertical momentum provides a mid-level work output. The lowest extracted work corresponds to the [1 1] input, since in this case, neither obstacle can align with a convection plume.

Table 2.

Truth table for double obstacle convection flows with separation Δ_o = H/2. Steady-state kinetic energy extraction W_o is given in arbitrary lattice units.

Δo	mu1	mu2	Wo/10−7	Wo > 3.5 × 10−7	Wo > 4.5 × 10−7
H/2	0	0	4	1	0
H/2	0	1	5	1	1
H/2	1	0	5	1	1
H/2	1	1	3	0	0

We can again consider the output bit for this system to be whether the extracted work is greater than a threshold value. For W_o > 3.5 × 10⁻⁷, the system implements a NAND gate. If the threshold is instead W_o > 4.5 × 10⁻⁷, then the system implements an XOR gate (this threshold was used for the OR gate in §3a).

If these three gates could be endowed with a means of information transfer between arbitrary combinations (to allow circuit construction) and a memory system, then convective obstacle flows could in principle be used to implement a Turing machine. This would confer such systems with computational universality. Our recent work showed that a convective system can store at least one bit of information [52], so the memory might also be provided by convective systems, in theory.

The present work is intended purely as a proof of concept. We anticipate the engineering and implementation of these ideas to follow in future work, perhaps by other researchers. Nonetheless, in terms of physical instantiation, the obstacles could consist of very thin, narrow metal blades, inserted perpendicular to the plane of the flow system. If such a blade was oriented perpendicular to the vertical direction, it would not be possible for any vertical fluid momentum to pass through it (no upward or downward flow could occur in its vicinity). A second blade oriented parallel to the vertical direction, placed directly below or above the first blade, could implement horizontal momentum extraction when required. The vertical momentum blade could be mechanically linked to the next gate in the circuit, such that its deflection (a proxy for momentum extraction) controls the presence or the absence of the horizontal blade in the next gate.

4. Conclusion

This work demonstrated that convective, double-obstacle flows can implement at least three logical operations. Since convection can also exhibit a primitive memory, it may be possible to construct computationally universal convective systems. The computational power of thermally driven fluid flows could offer a new mode of natural computing, with potential applications in flow control and microfluidic logic. The low thermal driving force places our system firmly in the laminar regime.

Given the simplicity of our system, it is an exemplar of information processing by pure physical processes. This suggests an alternative perspective for large-scale convective flow systems, in particular the atmospheres of planetary bodies. Such flows are highly turbulent and it has long been known that they exhibit very complex energy landscapes, with features such as strange attractors [69–74]. Could it be that there is a complex process of computation underlying the phase space trajectories of such systems? And if so, perhaps there is a compact informational description of thermal flows that can reveal some of their large-scale behaviour.

Our current and future endeavours seek means by which multiple gates can be combined into circuits. In fact, the bistability that confers nonlinear logic operations can also be exhibited by flows without obstacles, as long as they have a non-integer aspect ratio such as 1.25:1. Coupling between gates could occur using thermally conductive coupling, using the non-uniformity of the boundary heat flux profiles. Efforts to translate these ideas into working gates and circuits are currently underway.

Data accessibility

The simulation code for generating all results can be found in the electronic supplementary material.

Authors' contributions

S.J.B. first noticed that convective systems might have computational abilities, he wrote the code for the simulations, analysed the results and wrote the paper. Y.L.Y. provided guidance and advice.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the Caltech Division of Geological and Planetary Sciences Discovery Fund. Y.L.Y. was supported in part by an NAI Virtual Planetary Laboratory grant from the University of Washington to the Jet Propulsion Laboratory and California Institute of Technology.