Investigation on flow structure and aerodynamic characteristics over an airfoil at low Reynolds number—A review

Near-space low-speed aircrafts [high-altitude long-endurance unmanned air vehicles (UAVs) and near-space solar vehicles] and high-performance micro-air vehicles (MAVs) have been rapidly developed in recent years. A common challenge in design of the aircraft mentioned is to evaluate the effect of a low Reynolds number.^1–3 

The ratio of the inertial force to the viscous force of a fluid is used to define the Reynolds number $Re=ρVLμ$⁠, where V is the velocity of air at the inlet, ρ is the density, μ is the kinematic viscosity of the fluid, and L denotes the characteristic scale of the vehicle.^4,5 The threshold used to define the low Reynolds number is different in different fields of research. In the design of aerospace vehicles, the value of 10⁵ is commonly used to identify the range of the low Reynolds number, and conventional aircrafts operate at Reynolds numbers of up to 10⁶ and higher,^6,7 as shown in Fig. 1. According to the definition of the Reynolds number, the ratio of the inertial force of the fluid to its viscous force is lower at a lower Reynolds number, indicating the growth of viscous effects. Therefore, it is important to consider the viscous effect on flows at a low Reynolds number.^8–10 

Studies^5,11 have confirmed that viscous effects control the characteristics of the aircraft because they can dictate drag while limiting the lift–drag ratio of the wing. A low-speed aircraft operating at a high altitude (near space with low atmospheric density) and micro-air vehicles with small dimensions both lead to a low flight Reynolds number in the range of 10⁴–10⁵.^7,12 On the one hand, the lift–drag ratio of such an aircraft can be deteriorated rapidly, resulting in a degradation in overall aerodynamic performance. On the other hand, an aircraft at a low Reynolds number also exhibits nonlinear aerodynamic characteristics and unsteady flows that have adverse effects on flight performance.^13–15 Therefore, examining the aerodynamic characteristics and flow structure of an aircraft at a low Reynolds number is important for engineering and academic purposes.⁵ 

Flow at a low Reynolds number has been widely researched as a branch of aerodynamics. The theoretical basis for the aerodynamics at a low Reynolds number is the laminar separation bubble (LSB) theory proposed by Horton et al.¹⁶ It describes the separation, transition, reattachment, and other physical phenomena in the laminar separation bubble (LSB). Lissaman⁵ subsequently summarized the problems of an airfoil at a low Reynolds number based on mechanical theory, experimental research, and theoretical design and concluded that flow over the airfoil in this case is closely related to separation and transition. Mueller¹⁷ noted in a review that laminar separation bubbles have a significant influence on the aerodynamic performance of airfoils at low Reynolds numbers and it is necessary to accurately predict the occurrence and evolution of the separation bubbles when designing highly efficient low-speed airfoils. Some excellent reviews^18,19 of flows over an airfoil at low Reynolds numbers have been published. However, because such flows involve many problems, such as the mechanism of the triggering effect of laminar flow separation, the transition mechanism, and complex nonlinear aerodynamic characteristics, few systematic reviews of the issue are available.

This review provides a comprehensive understanding of common problems in flow encountered at a low Reynolds number. A compact overview of the structure of the LSB, its transition process, and aerodynamic characteristics related to a low Reynolds number are given for aviation-related applications, and the latest developments in related research are systematically discussed. In Sec. II, the structure and development of the laminar separation bubble are described. The transition process in the LSB can be divided into four stages—the receptivity, primary instability, secondary instability, and breakdown stage—which are discussed in Sec. III. The nonlinear aerodynamic characteristics of the airfoil at a low Reynolds number are presented in Sec. IV, with an emphasis on the nonlinear effects of the lift curve at a small angle of attack as well as the static hysteresis effect. Finally, the results of past research are summarized.

The prominent structure of flow at a low Reynolds number is the LSB. Laminar separation bubbles and their impact on the aerodynamics of the airfoil were first described by Jones.²⁰ The interest in thin airfoils led to a series of studies on the basic structure and characteristics of LSBs. Gaster²¹ examined the structure and characteristics of bubbles and claimed that their size and behavior depend on the Reynolds number and pressure gradient of the separated shear layer. According to Gaster’s results, a semi-empirical model of laminar flow separation bubbles (see Fig. 2) was developed by Horton.¹⁶ The laminar boundary layer can separate under an adverse pressure gradient, and the separated layer is more likely to transition to turbulent flow owing to its increased susceptibility to transition. Due to the increase in the shear force, the turbulent flow can acquire a sufficient amount of kinetic energy to make it possible for the flow to reattach and form an LSB structure.¹⁶ 

The classic two-dimensional (2D) schematic diagram of the laminar separation bubble is shown in Fig. 2. It is described by the mean dividing streamline, which intersects the surface at the separation and the reattachment points.²³ The transition in the separated shear layer is reduced to a point rather than a region. The region of recirculation flow can be divided into two sub-regions relative to the average position of transition. Combined with the pressure distribution (see Fig. 3), the distribution of plateau pressure from the separation point to the transition point can be observed, and the pressure rapidly recovers from the transition point to the reattachment point. This pressure distribution is clearly different from that of inviscid flow.^24–26 Near the separation point, the recirculation region is characterized by slow-moving reverse flow. As a result, the gradient of streamwise pressure is nearly zero and results in the plateau pressure being distributed. This region is usually called the “dead air region.” In the latter part of the bubble, the stronger circulation velocity and an abrupt reattachment of the shear layer lead to a rapid increase in surface pressure.^27,28

The classical semi-empirical model of the LSB is not perfect; for example, there is not simply one type of structure of the LSB at a low Reynolds number.²⁹ The LSB can be classified into several types (Owen and Klanfer³⁰), and “short” and “long” LSBs depend on the length of the bubble in comparison with the thickness of the displacement in it during separation. The results shown in Table I represent the relationship between different types of bubbles and the thickness of the displacement.²¹ “Short” bubbles have lengths of only a few percent points of the chord, about 10²$δs*$ to 10³$δs*$⁠, where $δs*$ is the displacement thickness at the separation, while long bubbles may cover 20%–30% of the entire airfoil, about 10⁴ $δs*$⁠, and form from the trailing edge of the airfoil. In general, “short” LSBs affect only the local pressure distribution, and “long” LSBs can affect the pressure distribution over the entire chord.²⁴ Marxen and Henningson³¹ improved the criterion used to distinguish between short and long bubbles and summarized the following characteristics of short bubbles: (1) The influence of the bubble on potential flow is limited and local. (2) The ratio of the laminar to the transition region in the LSB is 1.6–3.

The structures of the “long” LSB and “short” LSB are shown in Figs. 4(a) and 4(b), respectively. In addition to these bubbles, the trailing-edge LSB was discovered in symmetrical airfoils by Bai et al.^32,33 The trailing edge LSB structure is different from the classic laminar separation bubble in terms of structural shape and the law of evolution, as shown in Fig. 4(c). This kind of LSB is shaped like a stick while the classic LSB is shaped like a bulge. There is no clear time-averaged reattachment point for this type of separation bubble.^32,33 In addition, the main vortices are always located above the trailing edge and do not change position with the angle of attack.

One of the earliest studies on the issue was conducted by Tani,²⁴ who observed that the length of the LSB decreases with an increase in the angle of attack at a small angle until it reaches the critical state when the bubble suddenly increases in length (see Fig. 5; $Lδs*$ is the length of the bubble in terms of its displacement thickness, and $RδS*$ is the boundary -layer Reynolds number at separation). The transition from short to long bubbles is known as a bubble “burst,” and much of the early work in the area focused on predicting this^34–36 because it is directly related to the stall of the wing. The parameters governing bursting were proposed by Gaster.²¹ He developed a two-parameter bursting criterion: the Reynolds number $RδS*$ of the boundary layer and a pressure distribution in the region of the bubble of $δs2/vΔU/Δx$⁠.

Moreover, Mueller and DeLaurier³⁷ claimed that short LSBs formed earlier may burst to form long bubbles at the critical Reynolds number and argued that Horton’s separated bubble model is suitable for representing the structure of the short bubble. Other studies^38,39 have found that long separation bubbles are first formed at a small angle of attack. As the angle of attack increases, the long bubbles move to the front edge, their length gradually decreases, and they eventually evolve into a short bubble structure. Through water tunnel experiments and numerical simulations, Bai et al.^32,33 noted that as the angle of attack increases to a critical value, the time-averaged trailing-edge LSB can suddenly transform into a typical long LSB structure.

Lambert and Yarusevych²³ measured pressure fluctuations at different streamwise locations in LSBs on the NACA-0018 airfoil, as shown in Fig. 6. The results showed the distinct development of strongly periodic fluctuations in the pressure signals that could be used to detect, track, and characterize the roll-up vortices. They also proposed that the development of large-scale vortex structures in the separation bubble can affect the stability of flow, which further confirmed that the LSB is a time-averaged structure of unstable separation. With the application of new methods to study LSBs, a growing number of researchers believe that they represent unsteady flow. Research on the mechanism of transformation between different laminar flow separation bubbles is still in progress.^40–43 

For the LSB, in addition to the separation of the boundary layer caused by the strong adverse pressure gradient, the following known conditions (Reynolds number and turbulent intensity of free flow) can affect the formation of the separation bubble. To understand the mechanical factors that affect the transformation of the structure of the LSB, a series of experimental studies have been carried out. For example, Selig^44,45 and Muller¹⁷ conducted systematic tests involving multiple angles of attack at different Reynolds numbers on various airfoils. Based on a large number of experimental data, the effects of the Reynolds number and angle of attack on the structure of the LSB were summarized. Figure 7 shows a schematic diagram of the streamline of the airfoil with different Reynolds numbers at a moderate angle of attack.⁴⁶ At Reynolds numbers Re > 500 000, as shown in Fig. 7(a), laminar flow on the upper surface of the airfoil was affected by a large adverse pressure gradient near the leading edge, creating a short LSB. At Reynolds numbers 50 000 < Re < 100 000, as shown in Fig. 7(b), the thickness of the separated bubble and the turbulent boundary layer increased, creating a long LSB. At other Reynolds numbers 10⁴ < Re < 5 × 10⁴, as shown in Fig. 7(c), the LSB was closer to the trailing edge of the airfoil even at a very small angle of attack. However, this is different from the phenomena whereby at higher Reynolds numbers, the shear layer does not transition but reattaches to the airfoil to form a turbulent boundary layer. Moreover, according to the experimental data reported by Selig,^44,45 Dong et al.^47,48 captured the positions of the separation point, transition point, and reattachment point on an Fx63-137 airfoil surface by using oil film interference technology and a numerical simulation, as shown in Fig. 8. It is clear that as the Reynolds number increased, the laminar separation point moved backward while the attachment point moved forward, which means that the entire separation bubble shortened. Similarly, as the angle of attack increased, the separation point moves forward, and the length of the separation bubble decreased.

Moreover, the stability of flow at a low Reynolds number is poor, and the flow structure of the airfoil is sensitive to turbulence intensity. Mueller⁴⁹ and Burgmann et al.⁵⁰ showed that turbulence intensity has a significant effect on the mean flow field. They found that the increase in turbulence intensity led to delayed separation and thinner bubbles. It is often necessary to consider turbulence when examining the influence of turbulence intensity on the transition process. Reshotko^51,52 noted the classical Tollmien–Schichting (T–S) instability with linear disturbance growth in the case of low turbulence intensity. At higher turbulence levels, this scenario is omitted, and a bypass transition directly occurs. At the highest turbulence intensity, all linear transient growth processes may disappear. Therefore, the intensity of turbulence plays an important role in the development of flow. In addition, experimental studies^53–55 conducted in different wind tunnels and water tunnels exhibit inconsistency in the locations of separation, transition, and reattachment due to differences in the intensity of inflow turbulence.

Laminar–turbulent transitions are a common feature that plays an important role in the aerodynamics of a low Reynolds number.^56–58 The transition process within the separation bubble is shown in Fig. 9. The transition from laminar flow to turbulent flow occurs due to initial instability. A receptivity process then ensues (receptivity stage), and the disturbance generates instability waves in the laminar boundary layer.^59,60 The next stage of the process is an exponentially increasing instability wave called the primary instability that can be modeled by linear stability theory.^61–64 Further downstream of the separation, the Kelvin–Helmholtz instability caused by the separated profile of inflectional velocity begins to dominate the transition process in the LSB. Due to Kelvin–Helmholtz instability, the shear layer is forced into a vortex, and the separated shear layer is reattached by entraining a high-momentum fluid.⁶⁵ This part of the transition process is called secondary disturbance and features the exponential growth of disturbance and nonlinear interaction.⁶⁶ In the breakdown stage, further amplified disturbances cause the separated shear layer to roll up into 3D vortices.^66,67 These vortices formed in the separated shear layer are a major source of unsteadiness within the separation bubbles and can lead to fluctuating loading on the airfoils that may induce vibrations and produce noise.⁶⁸ In addition to the transition process mentioned above, the feedback interaction in the LSB⁶⁹ is attractive and is shown in Fig. (9).

Because the process of growth of the primary instability takes more time than the other stages, an approximate prediction of the transition can be made at this stage.⁵⁹ The growth of the primary instability can be predicted by using linear stability theory, where the governing equation is linearized and the viscous term is ignored.⁷⁰ Moreover, the disturbance is assumed to take the form of periodic mode disturbance, with time and the periodic in one or more spatial directions.⁷⁰ In the case of incompressible parallel flow, the wave of primary instability can be described by the eigensolution of the Orr–Sommerfeld equation.⁷¹ With an increase in complexity, the equation of parabolic stability accounts for streamwise variations in the base flow.⁷² At this stage, it is important to distinguish between two forms of primary instability: absolute instability and convective instability.^73,74 An example of convective instability is the growth of T–S waves in the boundary layer, whereas absolute instability is the main cause of the vortex shedding of the bluff body and has been observed in the free shear layer.^75,76 However, commonly used methods to predict the transition, such as the e^N method, consider only convective stability.⁷⁷ This instability has not been detected in naturally occurring separation bubbles on airfoils.⁵⁹ 

Compared with the primary instability stage, the receptivity stage is relatively difficult to understand but is important for predicting the transition process.⁷⁸ The entire receptivity process is as follows: Long-wavelength disturbances in free flow can excite the Lam–Rott disturbance near the leading edge of the wing. With an increase in the streamwise location of the free flow, the wavelength and amplitude of the disturbance decrease until it has a sufficiently small wavelength to continue to exist as an Orr–Sommerfeld disturbance.^79,80 In addition to determining the amplitude of the primary instability, receptive processes play an important role in the development of acoustic feedback loops and may affect the frequency selection of the “global” instability.^81,82 Furthermore, the global instability involving the feedback loop may lead to a transition to turbulence without increasing the disturbance explicitly. For example, Deng et al.⁸³ analyzed the pressure fluctuations at different locations on the suction surface of the NACA-0012 airfoil through a direct numerical simulation (DNS), as shown in Fig. 10. Although no external disturbances were introduced, the initial perturbations upstream may have originated from upward traveling acoustic waves that were generated in the wake. They claimed that the transition to turbulence was triggered by the sensitivity of the boundary layer to sound waves from the wake.

At present, our understanding of the transition process in the separation bubble is limited to the primary instability stage, and the role of secondary instability in the development of the 3D structure is unclear.^84–87 Marxen et al.⁸⁸ studied separation bubbles through numerical simulations and experiments and found that the 2D disturbance in separation bubbles is much more amplified than in oblique waves at the forcing frequency. They claimed that the magnitude of the 3D disturbance upstream of the transition point has a limited effect on the transition process and proposed that there exists an absolute secondary instability after the primary instability stage that leads to three dimensionality and has a major impact on the onset of the transition. In research on the development of the 3D flow in the aft portion of an LSB, Jones et al.⁶⁸ observed that there are two active secondary instabilities in the transition process. One is in the elliptical region of fluid flow (core region of the spanwise vortex), and the other is in the hyperbolic region of fluid flow, called the braid region, as shown in Fig. 11. The mechanism of instability of the core region of the vortex is similar to that of elliptical instability, and the mechanism of instability of the braided region is similar to mode-B instability or hyperbolic instability transition.^89,90 Michelis et al.⁹¹ found that the spanwise deformation of the dominant vortex structure is caused by the superposition of normal and oblique modes of instability from upstream of the separation.

In the free shear layer, once the 2D primary perturbation is sufficiently large, the 3D perturbations are significantly amplified due to the secondary instability, which may have a fundamental or sub-harmonic nature. This amplifies the disturbance in the fundamental or sub-harmonic frequency relative to the 2D frequency.^92,93 The basic type of secondary instability is related to the formation of the flow vortex structure while the sub-harmonic secondary instability is related to the merger of vortices called “vortex pairs.’^94–96 Yarusevich et al.⁹⁷ studied the formation, merging, and breakdown of vortices in separation bubbles on the airfoil and considered that the coherent structure rolled up by the separated shear layer is the main source of unsteadiness in the separation bubble (see Fig. 12). Marxen et al.⁹⁸ analyzed the formation and evolution of vortices in separated bubbles during transition and proved that the elliptical instability in the core region of the vortex leads to its spanwise deformation while another instability in the braid region causes the spanwise vortex to break into small-scale turbulent structures.

The development of vortical structures is shown in Fig. 15. From the viewpoint of dynamics of the separation bubble, unsteadiness in the separation bubble is mainly caused by the formation and evolution of vortices in the separated shear layer.⁵⁰ Research on the formation and development of shear layer vortices has attracted considerable attention in recent years.^23,67,99,100 The formation of the vortex in the LSB is similar to that in the mixing layer.⁹⁸ A characteristic of the process of formation of the spanwise vortex is the diminishing growth of a K–H instability wave and the saturation of its higher harmonics.⁹⁴ The process of vortex formation is complete at the point where the K–H instability wave reaches its maximum amplitude.⁹⁴ After its formation, the process of merging of the vortex appears in shear flow and is the main mechanism for the evolution of the decaying turbulence.¹⁰¹ In the initial stage of merging, the two vortices undergo diffusive growth when they orbit each other. When the size of the vortex core exceeds the critical value of the gap between the vortices, they begin to approach each other.^102,103 Eventually, the two vortex cores merge to form a single vortex structure that continues to expand through diffusion.¹⁰⁴ The vortices become distorted in the spanwise direction shortly after they are formed and merged and eventually break down into smaller-scale turbulent structures.¹⁰⁵ The break-up process is induced by the spanwise deformation of the vortices and accompanied by the formation of a streamwise structure.¹⁰⁴ Hain¹⁰⁶ found that 3D vortices are related to various instabilities, such as primary disturbances in the shear layer that lead to the breakdown of the vortex and the formation of different types of secondary structures. For example, the typical C-shaped vortex filaments can be further deformed into a “screwdriver” vortex pair. As the Reynolds number increases, the interaction between the vortex pairs also increases and induces the generation of a λ-shaped vortex.¹⁰⁷ A similar vortex structure often appears in separation bubbles on a flat plate, and other types of secondary structures can be observed in the reattached region of the airfoil.^108,109 In addition, the frequency of vortex shedding in the separation bubble of the airfoil is affected by the angle of attack and the Reynolds number.¹¹⁰ Some researchers have observed spanwise vortex structures downstream of the mean reattachment location, but they usually break before this.^62,111

As early as in the 1900s, researchers had noted that the aerodynamic characteristics of an airfoil are odd at a low Reynolds number compared with those at a high Reynolds number.¹¹² Schmitz¹¹³ studied its aerodynamics over a range of low Reynolds numbers of 2 × 10⁴–2 × 10⁵ by a wind tunnel test. The results showed that there is a critical Reynolds number for thick camber airfoils at which their performance changes drastically. The results for an N60 airfoil in different ranges of Reynolds numbers are shown in Fig. 13. For airfoils at a Reynolds number higher than the critical Reynolds number, the maximum lift coefficient (C_L) and maximum lift-drag ratio (C_L/C_D) increased with the Reynolds number, and the flow on the airfoil transitioned to turbulent.¹¹⁴ Since then, based on Schmitz’s research results, some studies have confirmed that the critical Reynolds number of most airfoils is between 10⁴ and 10⁶.¹¹⁵ Above the critical Reynolds number, a smooth airfoil has a higher lift-to-drag ratio than a rough airfoil. However, below it, the lift-to-drag ratio of the smooth airfoil decreases sharply while the performance of the rough airfoil is less affected.¹¹⁴ Mueller¹¹⁶ used visualization technology and aerodynamic measurement technology to determine that the laminar flow separation of bubbles on the surface of symmetrical airfoil contributes to a sharp increase in the lift coefficient at a Reynolds number of 4 × 10⁴ < Re < 4 × 10⁵ and summarized the influence of the Reynolds number on the aerodynamic characteristics. Selig¹¹⁷ conducted a series of wind tunnel tests on the aerodynamic characteristics of the airfoil at a low Reynolds number. The lift and drag of 60 airfoils at 6 × 10⁴ < Re < 3 × 10⁵ were measured by the strain–gauge force balance, and the integral of the wake was measured through a Pitot tube. This study comprehensively demonstrated the aerodynamic characteristics of different airfoils, analyzed the law of change in its lift and drag coefficients under different Reynolds numbers, and provided experimental data for subsequent studies on low Reynolds numbers.¹¹⁴ 

There is an interesting phenomenon that the aerodynamic results measured by different tests are inconsistent but the trend of change is consistent, which shows that data obtained at lower Reynolds numbers are more difficult to repeat than those at higher Reynolds numbers.^33,114 There are two main reasons for this phenomenon. On the one hand, the aerodynamic force measured at a low Reynolds number is small and cannot guarantee the accuracy of the measurement. On the other hand, the force generated is very sensitive to the free stream turbulence, and the test model and different test equipment have a significant influence on the measurement results.³³ 

The aerodynamic characteristics of the airfoil at a low Reynolds number are also affected by the level of flow turbulence and the shape of the airfoil.^118,119 The influence of the turbulence level (Tu) on its aerodynamics has received considerable attention in the literature.¹²⁰ The effect of Tu is related to the stall angle or the maximum lift coefficient. Hoffmann et al.¹²¹ measured the lift coefficient C_L and drag coefficient C_D of the NACA 0015 airfoil at a low Reynolds number and noted that the change in Tu from 0.25% to 9% resulted in an increase in the maximum value of C_L by 30%. Huang and Lee¹²² studied the effects of Tu on the aerodynamic loads and flow characteristics of the NACA0012 airfoil. The results showed that the effect of Tu on the maximum C_L became significant when it was above the critical value. Below the critical value, an increase in Tu delayed stall. Wang et al.¹²³ also measured the lift-to-drag coefficients C_L/C_D, lift coefficient C_L, drag coefficient C_D, and the flow structure of the NACA0012 airfoil when the turbulent intensity of flow Tu was between 0.6% and 6%. Tu has a more significant effect on the shear-layer separation, reattachment, transition, and formation of the separation bubble at a low Reynolds number, and the resulting C_L, C_D, and C_L/C_D on the airfoil angle of attack varies with Tu, as shown in Fig. 14. Hoerner¹²⁴ showed that the aerodynamic characteristics of a thin plate and cambered plates are better than those of the traditional airfoil at Reynolds numbers lower than 10⁵. In addition, the lift coefficient (C_L) and drag coefficient (C_D) of the plate at low Reynolds numbers are little affected by the change in the Reynolds number. Selig et al.¹¹⁷ examined the aerodynamic characteristics of different airfoils and found that the drag pole was nearly insensitive to variations in the Reynolds number once this was above 10⁵. However, below this critical value, the drag at different Reynolds numbers was nonlinear. Winslow et al.⁴⁶ simulated the aerodynamic characteristics, surface pressure, and flow field of some airfoils, including the NACA 0012 and Clark-Y, as shown in Fig. 15. They observed that below a Reynolds number of 10⁵, cambered plate airfoils had better aerodynamic characteristics than thick conventional airfoils with rounded-leading edges. The performance of the flat plate was found to be usually independent of the Reynolds number, but at a certain Reynolds number, its performance can improve with a reduction in thickness.

The most common explanation for the unusual behavior of airfoils at low Reynolds numbers is related to the development of laminar separation bubbles.¹²⁵ The most typical behavior is exemplified by their nonlinear aerodynamic characteristics, which have a great impact on the flight performance of the aircraft.¹²⁶ One of the effects is the nonlinearity of the aerodynamics at low angles of attack. According to lift line theory,^127,128 the lift curve of an airfoil at a higher Reynolds number is almost linear, with a slope of 2π, while the lift curve of a symmetrical airfoil with a near-zero angle of attack is no longer with a slope of 2π at low Reynolds numbers.⁴⁴ The “dead band” in the lift curve of the symmetrical airfoil at a low Reynolds number has been observed as shown in Fig. 16. The slope of the lift curve is initially less than 2π and then increases to a typical linear slope until stall.^129,130 This lift characteristic of a symmetrical airfoil at low Reynolds numbers is undesirable because it is detrimental to the longitudinal control performance of the aircraft. A series of airfoil tests were carried out by Selig et al.,⁴⁴ who found that the special nonlinear characteristics of the lift curve near a zero-degree angle of attack are common in symmetric airfoils. Moreover, Mueller et al.¹¹⁶ obtained similar results and found that the value of C_L of some airfoils was not only reduced but was also negative at small, positive angles of attack. They claimed that this special phenomenon was caused by the LSB. With an increase in the attack angle, the pressure gradient on the upper surface increased, which led to a larger LSB. The larger bubble increased the displacement thickness, which introduces the negative camber. However, Bai et al.^32,33 studied the nonlinear effects at low angles of attack by flow visualization and claimed that the evolution of the trailing-edge LSB and the long LSB caused them.

As a nonlinear aerodynamic characteristic, static hysteresis is relatively common for round-nosed airfoils at low Reynolds numbers.¹³¹ The aerodynamic hysteresis of the airfoil is expressed as a dependence on changes in the angle of attack.¹³² The coefficients of lift, drag, and moment of the airfoil are multivalued functions of the angle of attack rather than single value functions.¹³³ This means that the lift coefficient and the lift drag ratio are significantly different at a given angle of attack, which affects the recovery of the aircraft from the stall state.^134,135

A static hysteresis loop can be further subdivided by considering its direction, either clockwise or counter-clockwise,¹⁹ as shown in Fig. 17. A clockwise hysteresis loop is generally called stall hysteresis, which occurs at the maximum lift coefficient of the airfoil. With a further increase in the angle of attack, the lift coefficient suddenly decreases, and the laminar separation bubbles are completely separated. As the angle of attack decreases, the laminar separation bubbles cannot reattach immediately until the angle of attack decreases, and the lift coefficient suddenly increases.^44,121 A counterclockwise hysteresis is called pre-stall hysteresis, which occurs in the mid-lift range.^44,131 With an increasing angle of attack, the long bubble grows larger and merges with the wake, resulting in a stable lift coefficient. As the angle of attack further increases, the bubble moves toward and shortens into short bubbles, resulting in a significant increase in the lift coefficient of the airfoil. When the angle of attack is increased further, the bubble moves toward the leading edge and shortens into a short bubble, resulting in a significant increase in the lift coefficient. As the angle of attack decreases, the short bubble remains attached until a smaller angle of attack is obtained, forcing the lift coefficient to decrease sharply and forming a counterclockwise hysteresis loop.^44,121 In addition, any flow parameter that may affect the behavior of the LSB and the transition can introduce hysteresis.¹³⁶ Hoffmann¹²¹ studied the effect of turbulence intensity on the hysteresis of the wing and observed that hysteresis was eliminated at a large turbulence intensity. This is thought to be caused by the transition of the separated boundary layer. Marchman et al.^137,138 also observed that the span of hysteresis loops is sensitive to the turbulence intensity and acoustic interference. At the same time, for thicker wings, the stall hysteresis becomes more prominent with an increase in the aspect ratio. Mueller¹³¹ found that the hysteresis loops of the Lissaman 7769 and Miley M06-12-128 airfoils decreased and disappeared with increasing Reynolds numbers, but the Fx63-137 airfoil shows an opposite trend.¹³⁹ This shows that not only the Reynolds number but also the shape of the airfoil can affect static hysteresis. Compared with the extensive research on dynamic stall hysteresis,⁹ the phenomenon of static stall hysteresis has received much less attention in the literature.

This survey consolidated and evaluated the recent progress in research on the aerodynamics of an aircraft operating in flows at a low Reynolds number, with an emphasis on the evolution of the structure of the laminar separation bubble of the airfoil, laminar–turbulent transition in the LSB, and the nonlinear aerodynamic characteristics of the airfoil as characterized by the nonlinearity in lift at small angles of attack and static hysteresis.

A series of experiments and numerical simulations in the literature capture the structure of different LSBs at low Reynolds numbers, such as the “long” LSB, “short” LSB, and “trailing-edge” LSB. It is believed that these LSBs can evolve into one another under the influence of the flow parameters, but the mechanism of this evolution requires further research. In case of the laminar–turbulent transition in the LSB, the current understanding of the transition process generally starts from the primary instability, followed by the secondary instability and breakdown. During the transition, the development of small disturbances leads to a series of dynamic evolutions of vortices rolled up by the shear layer in the LSB, which may be directly related to the transition. However, the vortex dynamics in LSBs are far from being completely understood.⁹⁸ Thus far, a large number of tests have been carried out on airfoils at low Reynolds numbers. From the experimental data, the aerodynamic characteristics at low Reynolds numbers can be roughly understood, and the nonlinear effect can be measured. However, it is still difficult to predict the results accurately using the available tools of simulation, and the deviation in data in different experiments increases with a decrease in the Reynolds number.

It can be concluded that although considerable research has been conducted on the aerodynamics of aircrafts at low Reynolds numbers, our understanding of flow in this case is far from sufficient. The problems mentioned in this paper are just the tip of the iceberg in the context of issues pertaining to flow at low Reynolds numbers, which pose significant challenges to the design and development of aircrafts in this environment. We hope that the summary of the problem of a low Reynolds number provided in this paper can be helpful for research in this field.

This work was supported by the Foundation of National Key Laboratory Science and Technology on Aerodynamics Design and Research (Grant No. 614220119040108) and the Fundamental Research Funds for the Central Universities (Grant No. NF2020001).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.