1 Background

1.1 Turbulent Boundary Layer

Turbulent boundary layer (TBL) plays an important role in many engineering applications and geophysical flows. For example, the development and separation of turbulent boundary layer can significantly affect the lift, drag, and also the instability of the aircraft and vehicle. Passive scalar is also important to be investigated for studying turbulent boundary layer, which is a diffusive contaminant in a fluid flow. Understanding the behavior of passive scalar is a necessary step in understanding turbulent mixing, chemical reaction, or combustion [41]. For example, passive scalar can show the mixing degree of two different flows in a reaction flow.

1.2 Turbulent/Non-turbulent Interface

Over the past few decades, a large number of studies [15, 27, 38] have been devoted to understanding TBL from various points of view. Turbulent boundary layers are known as highly intermittent flows, where both turbulent and non-turbulent (laminar) fluids coexist. The studies show that the turbulent and non-turbulent flows are separated by an apparent boundary. In 1928, Prandtl [26] firstly pointed out the existence of this sharp interface between turbulent and non-turbulent flows in the intermittent region, which is called turbulent/non-turbulent interface (TNTI). After decades, the existence of TNTI was firstly examined in a free shear layer by Corrsin and Kistler [6], and recent studies [1, 29] have revealed that the TNTI is a thin layer with finite thickness. The turbulent and non-turbulent flow regions are separated by this TNTI layer, where flow properties, such as enstrophy, kinetic energy dissipation, and scalar concentration, sharply change in this layer so that they are adjusted between the turbulent and non-turbulent flows [28]. This layer is also important for the exchanges of substance, energy, and heat between turbulent and non-turbulent flow and is also related to the spatial development of turbulence [10]. Therefore, it is very important to understand the characteristics of TNTI.

The spatial distribution of turbulent fluids also plays an important role in scalar mixing in TBLs because turbulence can create small-scale fluctuating scalar fields, which enhances turbulent mixing at the molecular level. Modeling of turbulent mixing is crucial in numerical simulations of reacting flows [8] and combustions [35]. One of the key quantiles in the modeling of turbulent reacting flows is scalar dissipation rate, which strongly depends on the characteristics of turbulence [34]. Many models developed for simulating turbulent reacting flows contain the scalar dissipation rate as an unknown variable [5, 7, 17, 24]. The TNTI often appears near the interface that separates two streams with different chemical substances, thus, the mixing process near the TNTI can be important in chemically reacting flows [9, 42, 44], where the chemical reaction rate is strongly affected by turbulence. Therefore, it is also important to investigate the characteristics of scalar mixing near the TNTI.

2 Current Researches and Our Objectives

2.1 Current Researches

The TNTI appears in many canonical flows such as jets, wakes, and boundary layers. Recently, with the improvement of supercomputer resources and laser-based measurement techniques, many numerical simulations and experiments have been conducted to investigate the TNTI in canonical turbulent flows [29]. The flow properties near the TNTI in these flows have been investigated with the conditional statistics computed as a function of the distance from the TNTI [1]. The TNTI layer is found to consists of two (sub) layers with different dynamical characteristics as shown in Fig. 1. The outer part is called viscous superlayer (VSL), where viscous effects dominate vorticity evolution, while the region between the VSL and turbulent core region is called turbulent sublayer (TSL) [36], where the inviscid effects, such as vortex stretching, become important.

Fig. 1
figure 1

The concept of inner structures in TNTI layer

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The phenomenon of mass transferred from the non-turbulent region to the turbulent region is called the turbulent entrainment process, by which turbulence spatially grows. As introduced in previous study [22], the entrainment caused by large-scale eddies is called engulfment, and the entrainment process caused by small-scale eddies near the TNTI is referred to as nibbling. The nibbling-type entrainment is aroused by the viscous diffusion of vorticity near the TNTI layer while the engulfment is described as the non-turbulent flow which is drawn into the turbulent side by large-scale eddies before acquiring vorticity. The dominant mechanism for the entrainment process has been argued for many years. Recent studies have suggested that the nibbling process is responsible for the entrainment mechanism, and large-scale features of turbulence impose the total entrainment rate [10, 22, 47].

The geometry of TNTI is also an important issue for understanding the entrainment process: large pockets structures on the TNTI interface can indraft the non-turbulent fluids into the turbulent region before acquiring vorticity (engulfment) if the TNTI interface is intensely folded [37]. It is doubtless that the complex geometry of the interface is highly related to the total entrainment rate because the total entrainment rate can be expressed as the surface integral of the local entrainment velocity (nibbling). Therefore, it may need more information to know the relation between nibbling and engulfment as mentioned by Borrell and Jiménez [2]. Some recent studies have revealed the influence of large-scale structures on the geometry of TNTI in the boundary layer [20, 32], which can make differences in the entrainment process between the TBLs and free shear flows because the large-scale motions depend on flow types.

Furthermore, the turbulent flow under the TNTI layer contains eddies with a wide range of scales, and all length scales can affect the properties and geometry of the TNTI layer. Therefore, motions from the smallest to the largest scales need to be captured in measurement or simulations. Especially in direct numerical simulations (DNS), all scales should be resolved, and insufficient resolution can directly affect computational results. With the DNS, researchers are able to access three-dimensional data of all quantities, which is difficult to obtain in experiments especially in high-speed flows. However, the resolution of TNTI in DNS for TBLs has not been investigated.

Recently, the TNTI in TBLs have been studied in experiments [4, 25, 31, 32, 48] and direct numerical simulations [2, 11, 20, 32]. Some characteristics of the TNTI in TBLs are found to be similar to the ones in the free shear flows [30], e.g., vorticity jump in the TNTI layer, and fractal features of the TNTI. There are still few studies on the TNTI in the turbulent boundary layer compared with in free shear flows. Especially, TNTI studies in TBL have been done in incompressible flows.

In many aerospace engineering applications, the TBLs often develop in a transonic or supersonic free-stream, where compressibility is no longer negligible [33]. Compressibility effects on the TNTI have been studied in compressible mixing layers [12, 13, 23, 40]. There are only a few experimental studies on the TNTI in high-speed TBLs [48, 49], in which they conducted fractal analysis on the TNTI of supersonic turbulent boundary layer with the experimental data. However, the flow measurement near the TNTI is very limited and difficult especially in high-speed flows so that many characteristics about TNTI in compressible TBLs have not been investigated.

2.2 Objectives

Even though there are already some DNS studies on the TNTI in TBL [2, 11, 20], grid setting has not been evaluated with consideration of the resolution on the TNTI in TBL. This is because most DNS studies for TBL focus on the near-wall region, where the grid spacing is carefully considered in usual DNS for TBL, while the intermittent region has not been fully considered.

The TNTI has been extensively studied in recent studies on free shear flows and some similar characteristics of the TNTI are also found in the TBL [30]. However, there are few studies on the TNTI in the TBL than in free shear flows, and many issues remain unclear for the entrainment process near TNTI layer. Although a high-speed regime is of great importance in realistic aerospace engineering applications, most studies on the TNTI have been done in incompressible flows. However, the TNTI in compressible turbulence is still less understood compared with the one in incompressible flows.

Understanding the characteristics of the TNTI is greatly important in modeling and predicting the spatial development of turbulence as well as the flow control based on the turbulent structures near the TNTI. As described above, it is important to investigate the TNTI in compressible TBLs. In this study, direct numerical simulations with two types of grid setting are performed for both the subsonic and supersonic turbulent boundary layers in order to investigate the spatial resolution effects on TNTI, compressibility effects, entrainment process, as well as the development of the high-speed turbulent boundary layers.

The main objectives in the present study are to

  1. a.

    Develop the DNS code for the compressible TBLs with two types of grid setting.

  2. b.

    Evaluate a reasonable grid setting for the TNTI study in compressible TBLs.

  3. c.

    Investigate the compressibility effects on TNTI in compressible TBLs.

  4. d.

    Elucidate the physical mechanism of the entrainment in compressible TBLs.

3 Main Achievements

3.1 High Resolution Simulation

DNS of subsonic and supersonic temporally evolving turbulent boundary layers are performed for studying the TNTI. Two different setups of the DNS are considered, where in one case the grid spacing is determined solely based on the wall unit (case C001 and C002) while the other case uses the computational grid small enough to resolve both the turbulent structures underneath the TNTI and near the wall (case F001 and F002). The global statistics are compared between the present DNS and previous studies, showing that the DNSs with both grids reproduce well the first- and second-order statistics of the fully-developed turbulent boundary layers. The visualization of temporally evolving turbulent boundary layer with high resolution for \(M=0.8\) (case F001) is shown in Fig. 2.

Fig. 2
figure 2

Visualization of temporally evolving turbulent boundary layer for \(M=0.8\) case F001. Color represents passive scalar \(\phi \)

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However, the spatial distribution of vorticity in the outer region is found to be very sensitive to the spatial resolution near the TNTI. At the present Reynolds number (\(Re_{\theta }\approx 2200\)), the DNS based on the grid size determined by the wall unit does not have sufficient resolutions near the TNTI. The lack of resolution results in spiky patterns of the enstrophy isosurface used for detecting the outer edge of the TNTI layer (Fig. 3) and thicker TNTI layer thickness. This problem can be solved by increasing the number of the grid points, where a smoother enstrophy isosurface is similar to the previous studies of incompressible free shear flows obtained in the DNS with the grid small enough to resolve Kolmogorov scale in the turbulent core region below the TNTI.

Fig. 3
figure 3

Visualization of irrotational boundary forming at the outer edge of the TNTI layer for \(M=0.8\) (case F001, case C001) and \(M=1.6\) (case F002, case C002). Color represents dilatation \(\nabla \cdot {\boldsymbol{u}}\)

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3.2 TNTI in Compressible Turbulent Boundary Layers

Based on the 3D high-resolution DNS, the structure of the TNTI layer in compressible turbulent boundary layers can be investigated. The outer edge of TNTI layer, namely, irrotational boundary, is detected as an isosurface of vorticity as shown in Fig. 3a and c. The present results show that the thickness of the TNTI layer, defined with a large gradient of conditional mean vorticity magnitude, is about 15 times of the Kolmogorov scale \(\eta _{TI}\) in turbulence near the TNTI layer. The inner (sub)layers of the TNTI layer are detected based on the vorticity dynamics, where the TSL and VSL are found to have a thickness of 11–\(12\eta _{TI}\) and \(4\eta _{TI}\), respectively.

Even though the compressibility effects increase with Mach number, the conditional statistics confirm that the direct influences of compressibility are small near the TNTI layer, and the profiles of the conditional statistics are qualitatively similar between incompressible and compressible turbulent boundary layers. These structures of the TNTI layer and their thicknesses divided by the Kolmogorov scale are very similar to those found in incompressible free shear flows. The compressibility effects at the Mach numbers \(M=0.8\) and 1.6 are very small within the TNTI layer, which appears in the outer intermittent region.

The local entrainment process is studied with the propagation velocity of the enstrophy isosurface, which represents the speed at which non-turbulent fluids cross the outer edge of the TNTI layer. It has been shown that the compressibility effects are almost negligible for the propagation velocity, which is dominated by the viscous effects rather than a dilatational effect or baroclinic torque. The mean downward velocity is found in the non-turbulent region in the intermittent region, which is consistent with spatially evolving boundary layers [3, 20]. The mass entrainment rate per unit horizontal area of the temporal TBLs is consistent with the theoretical prediction  [33] for the spatial compressible TBLs. This confirms that the dominant mechanism for the momentum transport, which is related to the TBL thickness growth, is not different between spatial and temporal compressible TBL as also found in incompressible TBLs [18]. Furthermore, the mass entrainment rate normalized by \(u_\tau \rho _0\) at \(M=0.8\) also agrees well with experiments of spatially developing incompressible TBLs at various Reynolds numbers. Furthermore, the entrainment process across the TNTI layer is studied with the mass transport equation in the local coordinate system (\({\boldsymbol{x}}^\mathrm{I}, t'\)) which is moving with the outer edge of the TNTI layer. The statistics of the mass flux show that the mass within the VSL is transferred toward the TSL in the direction normal to the TNTI while the TSL is dominated by a tangential transfer. These mass fluxes within the VSL and TSL are compared with the single vortex model for the entrainment within the TNTI layer, which was proposed for incompressible flows [45] because of very small effects of the compressibility in the outer region of the turbulent boundary layer, the entrainment model given by a single vortex predicts the mass flux within the TNTI layer fairly well, which strongly suggests the connection between the entrainment process within the TNTI layer and the small-scale vortical structures found underneath the TNTI layer of the turbulent boundary layers.

The irrotational boundary detected by an isosurface of passive scalar is also shown in this study, and the detected irrotational boundary shows an excellent agreement with the one detected by vorticity in visualization. It indicates passive scalar is also a good marker of turbulent fluids, which is easy to measure in experiments compared with vorticity. Conditional mean passive scalar also exhibits a sharp jump within the TNTI layer, and the highest conditional mean scalar dissipation rate appears near the boundary between the VSL and TSL. This indicates that the fluid locally entrained from non-turbulent side encounters the fluid coming from the turbulent side, where the difference in the passive scalar between these fluids creates large scalar gradients. It is also shown that the production rate of scalar gradient and enstrophy within the TNTI layer is as high as in the turbulent core region, and peaks in conditional averages of these quantities appear within the TNTI layer. Both visualization and conditional statistics show the dependence on the TNTI orientation for the scalar dissipation rate and the production rate of scalar gradient as shown in Fig. 4, both of which have a large value near the leading edge facing the downstream direction than the trailing edge facing the upstream direction. The production rate of scalar gradient within the TNTI layer of the trailing edge is comparable to the non-turbulent value, which causes a lower scalar dissipation rate near the trailing edge. These tendencies are explained from the difference in streamwise velocity between turbulent and non-turbulent fluids in a similar way to the TNTI orientation dependence of enstrophy production (rate) given for incompressible planar jets [43, 46].

Fig. 4
figure 4

Instantaneous profile of scalar dissipation rate (color contour) and irrotational boundary (white line) on x-y plane for case F001 (\(M=0.8\))

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4 Real-World Data Circulation

Real-World Data Circulation (RWDC) is a recent advanced new research field, which is often conducted for the real-world products or services in society. The information in RWDC is represented and analyzed in the form of data. By analyzing these data, people can create new designs or improve the old ones.

RWDC exists in virtually all fields related to our lives, including business, medical treatment, economics, education, and industry, and is highly related to the development and globalization of the world [16, 39]. In the real world, even though people get data or information from the fields they are interested in, it is difficult to know how to use these data to contribute to our life or improve our knowledge. That is the reason why we need to study RWDC.

Industries are of great importance in our daily life, from the plastic bag for food to the aircraft by which people can travel everywhere on this earth. However, manufacturers cannot unilaterally create valuable products. In other words, techniques and user requirements are necessary for creating valuable products or services. The recent proposed fourth industrial revolution (Industry 4.0) [19], which has been developing with Information Technology (IT), has attracted a lot of attention. This industrial revolution tries to connect many fields together, e.g., the Internet of Things (IoT), Cyber-Physical System (CPS), information and communications technology (ICT), and Enterprise Architecture (EA) [21]. RWDC is one of the key points in the realization of Industry 4.0.

4.1 Relation Between RWDC and the Present Study

Due to technological limitations, it is still difficult for industry to create desired products or services based on current techniques. For advancing the techniques, it is necessary to deeply understand the fundamental aspects, namely, practical or industrial related science, which can help engineers essentially achieve a new stage of the technique.

Fig. 5
figure 5

The general concept of RWDC in fluid dynamics

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Fluid dynamics plays a significant role in the performance of many industrial applications, such as aircraft and automobile. How to use the practical or industrial related science in fluid dynamics to improve industries can be studied by RWDC as shown in Fig. 5, which is a general concept of RWDC in fluid dynamics for industrial products design. Some products in industries may have some problems or users may not be satisfied with the products, but engineers may not be able to solve the problems and need more information from the theory or fundamental characteristics. Then, researchers conduct some simulations or experiments to acquire data for fundamental studies. Thirdly, these fundamental studies can be done with the acquired data, and it can contribute to the development of theories or empirical laws. Finally, theories or empirical laws can be directly used for industrial design. By these steps, RWDC builds bridges for scientific study and real-world product design.

Computational fluid dynamics (CFD) is a widely used method for studying fluid dynamics, which has been using to design many parts of the aircraft. The usage of CFD is growing rapidly due to the increases of computational resource [14], which also decreases the time and financial cost of aircraft design, because aerodynamics studies of aircraft highly depended on the wind tunnel experiments in the past, that takes a lot of time, human resources, and financial cost. Commercial CFD software (CFX, Fluent, Fastran and Comsol, and so on) are frequently used for flow simulations in industrial applications, including many complex flows, e.g., combustion flow, high-speed flow, and flow with a heating wall. However, the turbulent flow simulations for industrial applications can only be done with the turbulent model due to the limited computational resource in the current stage. Even though these turbulent models excellently reduce the computational cost, it also brings some accuracy problems, especially in complex flows. Until now, a turbulent model is still one of the biggest problems in CFD community.

As mentioned in da Silva et al. [29], a better understanding of the TNTI layer is needed for predicting the chemical reaction flows or combustion flows, in which the properties essentially depends on the position and inner structures of TNTI layer. New computational algorithms and turbulent models near the TNTI layer should also be further considered because this layer separates the turbulent region from the non-turbulent region where the grid mesh used in applications near the TNTI is generally larger than the thickness of this layer.

The present study focuses on the TNTI layer in compressible turbulent boundary layers, which can be found in many engineering applications as described in Sect. 1 of this chapter, especially in aerospace engineering. The relation between RWDC and the present study can be explained by four steps as shown in Fig. 6. Firstly, the development of compressible TBLs can significantly affect the efficiency and instability of the aircraft and vehicles, and the many fundamental mechanisms about the development of compressible TBLs are still unclear as described in Sect. 1. These issues need to be investigated more. Then, direct numerical simulations of compressible TBLs are conducted for studying the fundamental characteristics of the TNTI layer and the entrainment process in compressible TBLs, which are strongly related to the development of compressible TBLs. Direct numerical simulations are conducted in the present study because it directly solves all the turbulence motions with different scales without any turbulent model. Thirdly, the results about the TNTI layer in the present study can contribute to the understanding of the development of compressible TBLs, which provides information for the CFD algorithms and turbulent model development near the TNTI. Finally, these improvements in understanding of the development of compressible TBLs can be directly used in the design of real-world industrial products such as aircrafts and turbines.

Fig. 6
figure 6

Connection of RWDC and TNTI study in compressible TBLs

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4.2 Contributions to the Society

The present work can contribute to the understanding of the development of compressible TBLs, which can significantly contribute to design of industrial products from three points:

  1. a.

    It can be used to improve the turbulent model and new CFD algorithms near the TNTI for compressible TBLs as described in Sect. 4.1. This will be a very big step if a suitable turbulent model or new CFD algorithms can be developed for industries because an improvement of CFD is an improvement for all the products designs which are related to the high-speed boundary layer. For example, the improvement of CFD in compressible TBLs can exactly improve the prediction of the flow around a high-speed vehicle, and it can be used to design a better vehicle with less drag and higher stability, subsequently better efficiency.

  2. b.

    Besides, understanding the development of compressible TBLs is also helpful for developing related theories or empirical laws, that can be directly used in the industrial product design.

  3. c.

    Furthermore, it also gives some ideas for flow control, e.g., it is possible to control the development of compressible TBL by controlling the large structures in the boundary layer because the large structures can affect properties of the TNTI in TBLs.

The work in the present study mainly focuses on the first and second steps of RWDC shown in Fig. 6, and contributes to the third step.