1 Introduction

Recently, the hypersonic boundary layer transition prediction has been a significant and difficult problem in the air dynamics design for the next generation aircraft. The transition process on the surface of a hypersonic flight vehicle has a very important effect on some phenomena, e.g., drag, heating, and noise. Therefore, the reliable prediction on the transition onset is necessary. The e^N method is a theoretical prediction method for the transition based on the linear stability theory (LST). It depends on the correct prediction of the disturbances growth. In general, the LST is used to predict the linear evolution of a disturbance in two-dimensional (2D) flows. However, most boundary layers encountered in engineering are three-dimensional (3D). It means that conventional methods in 2D boundary layers are required to be developed and validated in 3D flows, which plays a great significant role in the theoretical analysis and engineering problems.

There are three common approaches for studying disturbance propagation, i.e., the direct numerical simulation (DNS), the parabolized stability equation (PSE)^[1], and the LST^[2-3]. The DNS is usually adopted in the research on scientific problems. Due to the high precision and low efficiency, the model configuration in the DNS is often simple. The results obtained from the DNS are often used as the benchmark of other methods. An alternative method is the PSE. Compared with the DNS, the PSE improves the efficiency of the DNS. It ignores the secondary variation of the disturbance shape function in the streamwise direction. The PSE is usually adopted as a perfect tool to predict the evolution of the disturbances in 2D boundary layers. Nevertheless, for 3D boundary layers, the PSE is complex in the implement, and a further development is required for the practical problems. The LST is the simplest among the three methods. It has been widely adopted in engineering applications and theoretical research because of its advantages in efficiency and implement. It is easy to be adopted in the e^N method with the saddle point method for transition prediction.

In 2D boundary layers, both the LST and the PSE can give excellent results. Unfortunately, the external flows of most flight vehicles are often 3D. The diversity and complexity make it extremely difficult in predicting small disturbance propagation in 3D boundary layers. According to the characteristics of 3D boundary layers in the spanwise direction, complex 3D boundary layers can be categorized into four types as follows:

(ⅰ) The first type is the 3D boundary layers where the mean flow is homogeneous in the spanwise direction. In such 3D boundary layers, the spanwise velocity is allowed but the derivative of the flow quantities in the spanwise direction must be zero, e.g., the flow on an infinite swept wing or flat plate. The characteristic of such a boundary layer makes the spanwise wavenumber fixed when a disturbance is propagating. As a result, fixing the spanwise wavenumber is a proper method. The conventional LST and PSE can provide perfect results by fixing the spanwise wavenumber^[4-6].

(ⅱ) The second type is the 3D boundary layers where the variations of the flow in the spanwise and streamwise directions are in the same order. In this case, the instability is still local as 2D boundary layers, but the spanwise wavenumber of the disturbance is variable. There are often two problems in predicting the evolution of a disturbance. The first one is how to decide the direction of the disturbance, and the second one is how to choose the spanwise wavenumber. Spall and Malik^[7] analyzed the stability over a prolate spheroid at an angle of attack (AOA) of 10°. They pointed out that the amplitude could be integrated along the group velocity direction, which agreed excellently with the experiment by Meier and Kreplin^[8]. Chang^[9] extended the PSE to 3D boundary layers in LASTRAC. In his code, the local line-marching approach was adopted along the inviscid flow direction or the group velocity direction. Without a method to decide the direction of a disturbance in a 3D boundary layer, the spanwise wavenumber is still a difficult problem. The ray tracing method (RTM) is a considerable selection, which is derived for conservative system. By extending the RTM to non-conservative system, Yu^[10] and Zhao et al.^[11] tried to predict the small disturbance propagation in a practical model by combing the LST with the RTM, and the results were satisfied.

(ⅲ) The third type is the 3D boundary layers where the variations in the spanwise and wall-normal directions are in the same order. Under this condition, the global stability theory is a good choice. In this method, the shape function of the disturbance is considered as a global one in the normal and spanwise cross planes^[12-14]. The global spatial growth rate can be obtained in a spatial problem. However, a very large-scale eigenvalue problem must be solved to catch the result. Therefore, it is not suitable for the applications in engineering.

(ⅳ) In addition, there is also a kind of boundary layers where the mean flow in the spanwise direction varies much more slowly than that in the normalwise direction but more quickly than that in the streamwise direction. It can be considered as the gap between the second and third type 3D boundary layers. These boundary layers are common in engineering, such as the complex structure in the lee-side of a blunt cone at small AOAs^[15]. In these boundary layers, the local assumption in the second type is not proper, but the global analysis is an expensive choice. It is urgently required to establish an efficient method to predict the evolution of the disturbance in such boundary layers.

This paper aims to establish and validate a new method for predicting the linear development of a disturbance in the last kind of boundary layers. Based on our numerical experiment, the LST with an equivalent spanwise wavenumber correction (ESWC) according to the characteristic of the basic flow is proposed. This new ESWC method fills in the gap between the second and third type boundary layers.

2 Methodology and results 2.1 Origin of the basic concept on the ESWC

The basic concept of the ESWC was firstly performed in the research on the stability analysis of a hypersonic blunt cone at a small AOA. In the following, we call it Model A for short.

Figure 1 shows the schematic diagram of Model A, which is a blunt cone with a small AOA. The half-angle is 5°, and the nose radius is 1.0mm. The unit Reynolds number is 10⁷ m^-1. The freestream Mach number is 6, and the AOA α is 1°. φ denotes the azimuthal angle measured from the windward to the leeward. The x-axis is along the centerline of the cone, and it is measured from the nosetip. The y-axis is perpendicular to the x-axis in the meridian plane of φ=180°. ξ, η, and ζ, denoted as body-fitted coordinates, represent the axial (generating line of the cone), wall-normal, and azimuthal directions, respectively. More details of the mean flow computation, including grids and discretization methods, can be found in Ref. [15]. All of the physical quantities are normalized by the freestream quantity, and the reference length scale is selected by the nose radius. Due to the existence of the AOA, the surface streamlines concentrate toward the leeward ray. It can be observed in Fig. 2 that a complex structure, which has a sharp variation in the spanwise direction, can be found in the leeward ray. Obviously, the boundary layer on the leeward ray is 3D.

Figures 3 and 4 depict the growth rate and neutral curve obtained by the LST, respectively. It is shown clearly that the growth rate and the neutral curve vary more seriously from φ=180° to φ=150° than in any other region, which also implies that there are significant 3D characteristics in this region.

In order to validate the prediction of the LST, a numerical investigation is performed. It is required to point out that the numerical investigation is to solve the disturbance equation, which is common in simulating the disturbance evolution. The inlet disturbance is given by the LST, denoted by where ϕ' denotes the disturbance quantity, and α, β, ω, and represent the streamwise wavenumber, spanwise wavenumber, frequency, and shape function, respectively. A is the amplitude of the disturbance normalized by the streamwise velocity disturbance, and c.c. represents the complex conjugate. The inlet disturbance can be obtained by the LST. In the spanwise direction, the extrapolation condition is imposed at the boundaries. In the streamwise and spanwise directions, there must be sufficient points in a wavelength scale to capture the wave crest accurately. According to the experience of Yu^[10], 20 points per one wave length are fine enough for simulating the small disturbance evolution. In the wall-normal direction, there are about 80 points in the boundary layer region, which is enough for the stability analysis. The total number of the points in the wall-normal direction is 201. In the following, if there is no special instruction, the numerical investigation about disturbance is to solve the disturbance equations. The grid size is (20× l_x)× 201 × (20× l_z), where l_x and l_z are the wavelength scales in streamwise and spanwise directions, respectively.

Figure 5 shows the streamwise disturbance u'-velocity contours in the (x, φ)-plane with the disturbance of ω=0.6 and β=0 at the inlet by the DNS. It can be seen that the disturbance has larger amplitudes downstream, indicating that the disturbance can be amplified on the leeward ray. This implies that the qualitative analysis by the LST is effective though the boundary-layer varies seriously in the spanwise direction.

However, comparing the evolution of the amplitude between the DNS and the LST, there is an obvious divergence (see Fig. 6). It suggests that the sharp boundary-layer flow variation in the lee-ray region has an amplified effect on the disturbances, and the conventional LST prediction tool should be corrected.

In our DNS, the disturbance induced at the inlet is homogeneous in the spanwise direction. Due to the instability of the mean flow at different meridians, the envelope of amplitude becomes a wave-packet downstream. It means that the development of the disturbance results from the variation of the basic flow in the spanwise direction. This effect needs to be considered in the conventional LST. Otherwise, there is an obvious divergence between the DNS and the conventional LST (see Fig. 6). It is obvious that a characteristic scale can be determined by selecting the spanwise length of the structure in Fig. 2 or the neutral curve, denoted by L, indicating that the mean flow varies seriously. Since L is equal to the spanwise wavelength of the disturbances on the leeward ray, an equivalent spanwise wavenumber can be determined by =2π/L, and it should be selected in the instability analysis by the LST. According to this idea, is equal to 0.457 72 in this case, and the results are shown in Fig. 7. In addition, this length scale almost never varies along the streamwise direction downstream. The results suggest that the LST can make a reliable predication near the leeward ray with a proper correction. Because the flow varies slowly along the streamwise direction, and can be considered to be constant approximately.

In the example above, it is obviously indicated that the 3D boundary layer with a rapid variation in the spanwise direction has an amplified or damping effect on the evolution of the disturbances. In order to predict the disturbances evolution in this kind of boundary layers, the concept of the ESWC is proposed, i.e., equivalent spanwise wavenumber can be determined by the characteristic scale. Then, the LST with correction is used to predict the disturbance evolution. This new method improves the ability of the LST on predicting disturbance evolution in the boundary layers and provides satisfied results.

2.2 Formation of the ESWC

In practical engineering, the boundary layers with a vital variation of mean flows in the spanwise direction are general for the complex geometry. Then, the instabilities of these 3D boundary layers should be analyzed. Because of efficiency, the LST with the ESWC gradually becomes one not bad choice. One example, which shows the windward side of an aircraft model, is given in the following, and the illustration of the ESWC is made at the same time. Figure 8 plots the model configuration of the aircraft, the bottom of which is like a delta wing. The freestream Mach number is 12.1. The AOA in flight is 10°, and the wall temperature is 1 000 K with an isothermal boundary condition. The values of the gas parameters are chosen to be at 42.5 km attitude. All of the physical quantities are normalized by the freestream quantity, and the reference length scale is selected by a fixed value 0.007 407 2 m. The Reynolds number is 4 860.1. The mean flow is calculated by the CFL3D software^[16-17]. Roe's scheme is adopted for the flux-difference splitting and the upwind-biased third-order scheme with a smooth limiter for the spatial difference for the Euler flux terms. Actually, this scheme makes the state variable to the third-order accuracy in the one-dimensional (1D) case in smooth regions of the flow and interpolates without oscillations near discontinuities.

A traditional LST analysis is adopted at first. Figure 9(a) shows N contours calculated by the e^N method on the delta wing. From the figure, it can be seen that there is a significant 3D characteristic in the delta wing region though it seems to be a flat plat. It should be noticed that there is a long narrow region with a larger N value near the symmetry plane.

Figure 9(b) shows the unstable region with a 1D frequency ω=1.76 with the conventional LST. It can be seen that the direction of the group velocity seems parallel to the border of the unstable region. It could be inferred that the unstable disturbances may be amplified within a long path, and the N value becomes larger. However, one issue must be pointed out, i.e., the mean flow is 3D, especially in the boundary layer near the symmetry plane region. Therefore, it is very necessary to validate the accuracy of e^N and the conventional LST. In the following, the disturbance evolution with ω=1.76 will be investigated by the DNS, and the results will be compared with the LST. Moreover, the LST with the ESWC will be developed based on the results by the DNS.

Firstly, an analysis is performed in the unstable region downstream. The computational domain is shown in Fig. 10 (block box region). The disturbance introduced at the inlet is obtained by the LST. The eigenvalues of the disturbance at the inlet are ω=1.76, α_r=2.007, -α_i=0.061, and β=0.0, where ω is the frequency, α_r is the streamwise wavenumber, -α_i is the growth rate, and β is the spanwise warenumber.

The amplitude of the disturbance is 10^-5. It means that the linear effect is completely dominant downstream.

Because of the divergence between the eigen-functions and the real ones of the flow, there is an adjusting region near the inlet of the computational domain. As a result, the results by the LST and the DNS are compared from x=440 to avoid this adjusting effect. The results by the LST and the DNS are denoted by the LST with β_LST=0 (solid line) and the DNS (solid line with symbols), respectively, in Fig. 13. As shown in Fig. 13, generally, two results are in good agreement. But there are also some divergences. Then, the ESWC is introduced. It can be seen in Fig. 11 that the disturbances are amplified in a narrow region, and the amplitudes of disturbance distribution can be seen as a wave packet in the spanwise direction. According to the method mentioned previously, the spanwise wave length can be determined by the unstable region length in the spanwise direction. That is to say, we can define a length scale which is the length of the unstable region in the spanwise direction. It is shown from Fig. 11 that the length scale varies along the streamwise direction. Hence, the length scale can be denoted by L(x). The equivalent spanwise wavenumber β(x) displayed in Fig. 12 can be obtained by β(x)=2π/L(x). With this method, the corrected LST calculates the growth rates and integrates the amplitude with β_LST=β(x). The comparison suggests that the results obtained by the LST with the ESWC are in excellent agreement with those from the DNS (see Fig. 13).

Secondly, another part near the symmetry plane is analyzed. The selected computational domain is shown in Figs. 14(a) and 14(b). Here, the unstable region has two branches. It can be seen that the above one has a greater growth rate. The directions of the group velocity are shown in Fig. 14(b).

As the case above, the disturbance at the inlet is induced by the LST result. The amplitude of the disturbance is set to be 10^-5. The eigenvalues are shown in Table 1. The wall pressure disturbance distribution by the DNS is shown in Fig. 15. The disturbance grows in the unstable region downstream, which is visible in Fig. 15.

Figure 16 plots the amplitude of the disturbance along the streamwise direction. The stability analysis is performed with different LST methods, i.e., the conventional LST (β_LST=0) and the LST with the ESWC (β=β(x)). In this case, the equivalent wavenumber β(x) is also determined by the variable unstable region scale in the spanwise direction. It can be observed that there is a huge divergence between the conventional LST and the DNS but a little one between the corrected LST and DNS. This indicates that the LST with the ESWC is much better than the conventional one but not satisfied.

What makes the divergence between the LST with the ESWC and the DNS? Figure 17 shows the streamwise velocity evolution. It can be seen that the wave crests of the disturbances are not perpendicular to the streamwise direction, i.e., the disturbance wavenumber varies downstream, and it is not as the same as the wavenumber induced at the inlet. This indicates that the scale of the unstable region in the mean flow can amplify or damp the disturbances and the wavenumber variation of the disturbance itself has an effect on the disturbance evolution.

In order to analyze the characteristics of the disturbances in the spanwise direction, especially the spanwise wavenumber, the distributions of the streamwise velocity disturbance along the spanwise direction in different streamwise locations are displayed in Fig. 18. As illustrated in Fig. 18, before x=259, there seems to be a single dominant wave crest in the unstable region. It suggests that the disturbance evolution is influenced by the instability of the unstable region or mean flow. After x=259, there are several wave crests in one wave-packet. It suggests that the disturbance has its own evolution rule, and it is not only affected completely by the scale of the mean flow. Namely, the fact that a spanwise wavenumber varies downstream should be considered in the spanwise wavenumber correction.

Fortunately, the wavenumber variation of a disturbance can be predicted by the RTM. Then, the various spanwise wavenumber can be obtained. Lighthill^[18] has proposed the theory in a conservative system. Zhao et al.^[11] extended the theory into general 3D boundary layers to predict the spanwise wavenumber. The equation for the RTM in the spanwise direction is given as , where is the directional derivative of the spanwise wavenumber along the group velocity direction, and U_g is defined as the group velocity defined by U_g=^[19]. By integrating the formula, the spanwise wavenumber can be obtained, called β_RTM, which represents the spanwise wavenumber variation of the disturbance without the mean flow effect. If the initial spanwise wavenumber β is given and we integrate the above equation, the variation of the spanwise wavenumber can be determined.

Figure 19 shows the tracing path obtained by different methods. In Fig. 18, the black solid line is the path along the group velocity direction obtained by the RTM, and the black dashed line is the trace of the maximum values of the computational amplitudes in the spanwise direction at every streamwise location, i.e., the tracing path given by the DNS. It seems that the two lines agree excellently with each other in a long region upstream. In the region downstream, there is a divergence because the solid line goes through the unstable region and the dashed line tends to the unstable area border in fact.

Figure 20 plots the spanwise wavenumbers obtained by the DNS and the RTM. It is indicated that the spanwise wavenumber in the evolution region is neither 0 at the inlet nor determined by the length of the unstable region in the spanwise direction. It is variable with a linear variation in the streamwise direction. It can be seen in Fig. 20 that the results obtained by the RTM are in excellent agreement with those obtained by the DNS, which suggests that the RTM can describe the wavenumber variation in the process of the disturbance evolution.

As mentioned above, β_RTM represents the spanwise wavenumber of a disturbance evolution by its dispersion relation without the mean flow effect, while β(x) is decided by the unstable region of the mean flow in the spanwise direction. Both of the two wavenumbers should be considered in the stability analysis. Here, an assumption is introduced: if β_RTM is smaller than β(x), i.e., the spanwise wavelength by the RTM is larger, the mean flow will have a dominant effect on the disturbance evolution. Then, β_RTM is selected. On the contrary, the wavenumber variation of the disturbance itself may be dominant, as a result, β(x) is a better choice. Therefore, the spanwise wavenumber is determined by selecting the maximum value between β(x) and β_RTM, denoted by β_LST=max(β(x), β_RTM). It is shown in Fig. 20 that there is a point of intersection between β(x) and β_RTM when x≈ 259, which is in agreement with that displayed in Fig. 18. Namely, β(x) is dominant when x < 259, while β_RTM is dominant when x > 259.

Figure 21 shows the evolution predicted by different methods. It seems that the results by the LST with the ESWC (denoted as LST_max(β(x), β_RTM) in Fig. 21) are in excellent agreement with those obtained by the DNS. It indicates that the LST with the ESWC has a higher accuracy than other LST methods. Therefore, the new method can broaden the ability of the LST in predicting the linear evolution of the disturbances in complex 3D boundary layers.

As shown above, the LST with the ESWC is developed. The dispersion relation of the disturbance and the effect from the unstable region of the mean flow in the spanwise direction are both considered. It is an effective tool on the prediction of disturbance evolution in 3D boundary layers. Then, another case in the delta wing is given in the following content as a further application of the ESWC.

In the end, a further application of the ESWC is preformed and the DNS is adopted as the benchmark. The computational domain is selected (see Fig. 22).

Figure 23 shows the contours of the amplitude of the streamwise velocity disturbances by the DNS. It can be still seen that there are two traces in the figure. One is the path that the amplitude of disturbance reaches the maximum value in the spanwise direction at one x location by the DNS (dashed line). The other is the group velocity direction obtained by the RTM (solid line). It is shown that the two traces coincide with each other.

Figure 23 shows the contours of the amplitude of the streamwise velocity disturbances by the DNS. It can be still seen that there are two traces in the figure. One is the path that the amplitude of disturbance reaches the maximum value in the spanwise direction at one x location by the DNS (dashed line). The other is the group velocity direction obtained by the RTM (solid line). It is shown that the two traces coincide with each other.

Figure 24 displays the amplitude evolution obtained by the LST with the ESWC and the DNS. It can be observed in Fig. 24 that the evolution obtained by the LST with the ESWC is in excellent agreement with the DNS, which means that the LST with the ESWC can give a very excellent prediction on the linear evolution of disturbances with satisfaction in 3D boundary layers.

3 Conclusions

In this paper, the LST with the ESWC is proposed to accurately predict the linear development of the disturbance in a kind of boundary-layer flow with significant variations in the spanwise direction, which is commonly encountered in engineering.

In complex 3D boundary layers, the conventional LST cannot predict small disturbance evolution accurately. Therefore, a necessary correction is required to be made in predicting the linear evolution of disturbance with the LST. Actually, the disturbance evolution in 3D boundary layers is mainly affected by two factors. One is the scale of the mean flow with the rapid variation or the scale of the unstable area in the spanwise direction. The other is the wavenumber variation of the disturbance itself, which can be predicted by the RTM. Therefore, the two factors should be considered together when predicting the linear evolution of the disturbance in 3D boundary layers. Then, the LST with the ESWC is proposed as the correction of the conventional LST after considering the two factors above. It is shown from the results that the amplitude evolution of the disturbance of the new approach is in excellent agreement with that of the DNS. As an effective approach on the prediction of the disturbance evolution in 3D boundary layers, the LST with the ESWC improves the prediction of the LST in the applications to complex 3D boundary layers greatly.

Actually, though the LST with the ESWC has a great effect on the prediction of the linear evolution, there is still some divergence between the prediction and the DNS results. The non-parallel effect may be the reason which is not accounted for in the LST type procedure and should be studied in future.