1 Introduction

The instability of laminar flow and transition to turbulence have maintained constant interest in fluid mechanics problems because transition controls important aerodynamic quantities, e.g., drag or heat transfer^[1]. The e^N method based on the linear stability theory (LST) has been widely used to predict transition. It was initially developed for two-dimensional (2D) flows and then extended to more complex problems. The elementary task of the e^N method is to integrate the growth rates of disturbances and obtain a logarithmic amplification ratio of amplitude. This amplification ratio, termed the N factor, is the key criterion to predict the transition location.

In 2D incompressible flows, the most unstable disturbance is 2D. Therefore, 2D waves are considered frequently. The dispersion relation of unstable waves established by the LST can be written as ω = f(α) with ω denoting the complex frequency and α the streamwise complex wavenumber. The N factor can be obtained by integrating the spatial growth rate at a specified real ω. Early theoretical study and numerical calculation were usually based on the temporal mode instability. In order to compare the calculated temporal growth rate with the spatial growth rate measured in stability experiments, Gaster^[2-3] established the link between the eigenvalues of temporal and spatial instability. Therefore, the N factor can also be obtained by integrating the temporal growth rate.

The extension of the e^N method to 3D flows is not straightforward. The dispersion relation of the general 3D disturbance is written as f(α, β, ω) = 0. For spatial mode instability, a disturbance can amplify in both streamwise and spanwise directions. Therefore, how to determine the growth rate is the first puzzle of 3D e^N method. For a disturbance with specified frequency ω and spanwise wavenumber β_r, the solution of the dispersion relation is not unique unless the spanwise growth rate β_i is set beforehand. Some solutions have been proposed to assign or compute β_i. For instance, it is possible to use the wave packet theory and to impose the ratio to be real^[4]. This solution is termed the saddle-point method (SPM) which has been the most popular in the 3D e^N method^[5-7]. β_i is determined by iterative calculations in the SPM. Aiming to avoid the iterative calculations, Yu and Luo^[8] proposed a simpler solution by assuming that the disturbances amplify along the group velocity direction. They obtained identical results with the SPM. In the case of infinite swept wings, it is often assumed that there is no amplification in the spanwise direction^[9-10], i.e., β_i=0. There is a contradiction between this solution and the one by Yu and Luo^[8]. Zhao et al.^[11] eliminated this contradiction by concluding that the variation of β_i does not change the growth rate in the group velocity direction, and hence the N factor. Their conclusion is undoubtedly above rubies to the application of 3D e^N method. However, their work lacked strict theoretical proof and systematic numerical validation. Necessary error estimate was not provided either. Malik^[12] proposed another solution based on temporal mode instability that the N factor can be obtained by integrating along the group velocity direction, where U_g is the magnitude of the group velocity. Although the different solutions mentioned above have been applied to calculate the disturbance growth rate in the 3D e^N method, a unified understanding has not been reached. How to determine the growth rates of disturbances in 3D flows is still experiential knowledge to some extent.

In order to eliminate the confusion about calculating the growth rate in 3D flows, the relation among the wave parameters of the disturbance is theoretically studied in this paper. It is organized as follows. In Section 2, a brief introduction on the dispersion relation of disturbance is given and the theoretical derivation of the generalized growth rate (GGR) conservation is presented mathematically. Numerical validation and error estimation of the GGR conservation are given in Section 3. The e^N method and GGR from a physical point of view are also discussed. Finally, a summary and some concluding remarks are given in Section 4.

2 Methodology 2.1 Dispersion relation of disturbances in boundary layers

The dispersion relation of disturbances can be established based on the LST. The flow quantities can be decomposed into steady baseflow and perturbation quantities in the form ϕ = ϕ₀ + ϕ'. Here, ϕ' = (ρ', u', v', w', T') are perturbation quantities of density, velocity components, and temperature, respectively. In addition, the pressure perturbation can be obtained through ρ' and T' by the equation of state. The parallel hypothesis is adopted, and the perturbation quantities in boundary layers can be presented as the form of travelling waves,

(1)

where represents the shape function, α, β, and ω are the wave parameters of the disturbance ϕ', and c.c. is the complex conjugate of the previous term. All these three wave parameters can be complex. Their real parts ω_r, α_r, and β_r) represent the frequency, streamwise, and spanwise wavenumbers of disturbances, respectively. Their imaginary parts ω_i, - α_i, and - β_i) are the temporal growth rate and spatial growth rates in streamwise and spanwise directions, respectively. The dispersion relation of disturbances can be written as

(2)

For temporal mode instability, α and β are specified real numbers. The complex ω can be solved by (2). For spatial mode instability, ω is real, but both α and β can be complex. Generally, the imaginary parts of wave parameters are far smaller than the real parts for disturbances in boundary layers^[13].

2.2 Gaster transformation

Gaster transformation provides a link between temporal and spatial growth rates for 2D disturbances in 2D flows. This classical transformation relation is above rubies, and its derivation procedure is also the foundation of this paper. Therefore, it is necessary to review the derivation procedure of Gaster transformation.

The dispersion relation of 2D disturbances can be simplified as ω = f₁(α). With the assumption that ω is an analytic function of α, the Cauchy-Riemann relations hold,

(3)

The two cases of most interest labelled (T) and (S) are considered in the article^[2]. Case (T) Temporal mode instability, α_i^T = 0,

(4)

Case (S) Spatial mode instability, ω_i^S =0,

(5)

If we now integrate the Cauchy-Riemann relations (3) with respect to α_i from state (T) to state (S), with keeping α_r = const. = α_r^T, the Cauchy-Riemann relations (3) follow

(6)

(7)

Generally, for disturbances in boundary layers, and α_i^S = O(ω_im), where ω_im is the maximum value of ω_i. Then, (6) follows

(8)

where O(ω_imα_i^S) ~ O(ω_im²) ~ 10^-2~-4. Neglecting terms of order ω_im², we have a very good approximation,

(9)

If we expand in a Taylor series at any point α_i^* in the range 0 to α_i^S and substitute (7), we get

(10)

Because ω_i^S = 0, (10) can be written as

(11)

Again neglecting the terms of order (α_i^S)² and provided that is non-zero, we have

(12)

where can be evaluated at any state between (T) and (S), meaning that is a constant in a certain accuracy.

(9) and (12) are termed Gaster transformation which provides a link between the values of the parameters existing in the temporal and spatial mode instability. A similar analysis can also be carried out if the frequency is kept constant. Then, α_r^S = α_r^T, and (12) can also be obtained.

2.3 GGR of disturbance and conservation of GGR

In this subsection, we discuss the relation among wave parameters from a new perspective. Three cases, i.e., 2D waves in 2D flows, 3D wave of spatial mode, and general 3D waves, are considered. The GGRs in different forms are defined for each case. After deducing the conservation relations of GGR for all three cases, the consistency among the GGRs of these three cases is proved.

2.3.1 Case of 2D wave in 2D flow

One general state between (T) and (S) is considered. This state labelled (P) is as follows. Case (P) Time- and spatially-increasing,

(13)

where α_r^P=α_r^T=α_r^S= const., and ω_i^P and α_i^P are arbitrary within the limitation that they are far smaller than their real parts, respectively.

Replacing the state (S) with the state (P) in the derivation procedure of Gaster transformation, (9) and (11) follow that

(14)

where can be evaluated at any state between (T) and (P) and is a constant in a certain accuracy.

We introduce the generalized growth rate for 2D wave marked as σ_g and neglect the terms of order (α_i^P)². Then,

(15)

Here, U_g = is the magnitude of the group velocity for 2D waves. (15) suggests that keeping α_r^P = const. = α_r^T, the variation of α_i^P does not change the values of ω_rP and σ_g, which is termed the GGR conservation. That is to say, for an arbitrary state (P) between (T) and (S), σ_g and ω_r^P are constants, whose values are equal to and ω_r^T, respectively. This indicates that the GGR of a disturbance only depends on the real parts of wave parameters and is independent of the imaginary parts.

2.3.2 Case of 3D wave in spatial mode

The disturbance is always spatially-increasing in the boundary layers, e.g., first/second mode and crossflow mode instability. For a single frequency disturbance of spatial mode in steady flows, the frequency keeps constant in the propagation of the disturbance. However, the disturbance can increase in both streamwise and spanwise directions. Consider the following two spatial instability states with the same frequency labelled (X) and (Q).

Case (X) Increase in the x-direction, ω = ω_r^X,

(16)

Case (Q) Increase in both the x- and z-directions, ω = ω_r^Q = ω_r^X,

(17)

The dispersion relation of disturbances can be written as α = f₂(β), and α is assumed to be an analytic function of β. An analogous analysis can be carried out. Keeping β_r^Q = β_r^X =const., one obtains an analogous formula to (14),

(18)

Introducing another form of the GGR, (18) can be rewritten as

(19)

where can be evaluated at any state between (X) and (Q) and is a constant in a certain accuracy, and θ is the direction angle of the group velocity. The GGR represents the growth rate of disturbance in the direction of group velocity. (19) also suggests that the GGR is conservative at specified real parts of wave parameters.

2.3.3 Case of general 3D wave

Now, a general 3D wave increasing in temporal and two spatial directions is considered. The previous results will be utilized straightly for simplifying the derivation procedure.

Two states labelled (T) and (Q) are redefined.

Case (T) Temporal mode instability, α_i^T=0, β_i^T = 0,

(20)

Case (Q) Temporal and spatial mode instability,

(21)

These two states satisfy the condition of α_r^Q = α_r^T and β_r^Q=β_r^T. The dispersion relation of disturbances can be written as

(22)

and ω is assumed to be a binary analytic function of α and β. The following two couples of Cauchy-Riemann relations hold

(23)

(24)

We first integrate the Cauchy-Riemann relations (23) with respect to α_i from state (T) to α_i=α_i^Q with β = β_r^T = const., and the upper limit of integral is marked as state (M). Then, we integrate the Cauchy-Riemann relations (24) with respect to β_i from state (M) to state (Q) with α = α_r^Q + iα_i^Q = const., and analogies of (8) and (10) can be obtained,

(25)

(26)

Neglecting the high order small quantities and introducing the general form of the GGR, we have

(27)

where represents the magnitude of group velocity of the 3D disturbance, and is the direction angle of group velocity. Both U_g and θ are constants in a certain accuracy.

(15) is the particular case of (27) for the 2D wave. For 2D flows, ω is a symmetric function of β for β = 0. Hence, = 0. Then, (27) is back to (15). Now, we prove that (19) is another particular case of (27) for 3D waves in spatial mode instability. (22) can be rewritten as

(28)

The total differential of (28) yields

(29)

For a single frequency disturbance of 3D waves in the spatial mode, the frequency is a real constant^[11]. Hence, dω ≡ 0, and then . With considering , , and , it follows

(30)

Substituting the formula 1/(1 + t) ≈ 1 - t + t² into (30), taking the real part of the equation, and neglecting the terms of order and , we have

(31)

Then, (27) degenerates to (19).

In this subsection, the concept of GGR is introduced, and the conservation relation of GGR is deduced for three cases. The conservation of GGR for these cases will be validated numerically in the next section.

3 Numerical results

The conservation of GGR is numerically validated for first/second modes and crossflow modes in a flat plate flow and a swept blunt plate flow, respectively. The first-mode instability, which is analogous to the Tollmien-Schlichting waves in incompressible flows, is most unstable at oblique angles. The second-mode instability, however, is planar-component dominant^[14]. In addition, crossflow instability is a kind of inviscid instability.

3.1 GGR conservation of T-S wave

A flat plate boundary layer flow with Mach number 4.5 is adopted. The flow parameters are set according to the gas parameters of 30 km altitude, and the temperature of free stream is T_∞ = 226.5 K. The base flow is determined by the similar solution of flat plate with no-slip and adiabatic wall boundary condition. T-S wave instability is analyzed at the location of 2.25 m downstream from the leading edge, and the Reynolds number based on the displacement thickness is 17 000.

First and second (Mack) mode waves coexist in the Mach 4.5 flat plate flow, and Fig. 1 shows the spatial growth rate contours of these two modes. The higher frequency area is the unstable region of second mode, and the lower frequency area is the one of first mode. The black dot labelled A and B mark the most unstable waves of the second and first modes, respectively. The wave parameters at the locations of A and B are shown in Table 1.

Since the most unstable wave of second mode is 2D, the location A is chosen to validate (15). It should be noted that just α_r and β_r are chosen and kept invariant. Changing α_i from –0.037 9 (state S) to 0 (state T) (in this process, ω_i changes from 0 (state S) to 0.028 (state T)), one can obtain ω_r and ω_i according to the dispersion relation. Then, the GGR can be calculated by (15) at different combinations between ω_i and α_i. Figure 2 shows the variations of ω_r, GGR, and U_g. Their relative deviations compared with the GGR in the reference state (T) are also plotted in this figure. It is shown that ω_r is almost a constant and the relative deviations of GGR and U_g are smaller than 1.6% and 3%, respectively, which means that the GGR is approximately conservative with the specified real parts of wave parameters for the second mode.

We choose the most unstable T-S wave of first mode to validate (27). There are three growth rates in this equation. For convenience, only the results with specified β_i=0 are provided. Figure 3 shows the variations of ω_r, σ_g, U_g, and θ, and their relative deviations compared with a reference state (T) are also plotted. It is shown that the relative deviations of ω_r and σ_g, are both smaller than 0.02%. The relative deviations of U_g and θ are smaller than 0.1% and 0.4%, respectively. It seems that the GGR conservation is more accurate for the first mode than that for the second mode. The direction angle θ is about 1.65°, which means that the group velocity direction is almost in line with the x-direction, i.e., the potential flow direction.

3.2 GGR conservation of crossflow wave

Transition on the surface of a vehicle is most likely to be dominated not only by T-S waves but also by crossflow waves. Hence, it is worth validating the GGR conservation for crossflow waves. A swept blunt plate model (shown in Fig. 4) is applied to analyze crossflow instability. The mean flow over the swept blunt plate is obtained by solving the Navier-Stokes equations^[15-16]. The radius of the blunt nose is r₀=35 mm. The free stream Mach number is taken to be 6, and the swept angle θ_SA is set to be 45°. The flow parameters are set according to the gas parameters of 30 km altitude. The temperature of free stream is T_∞=226.5 K, and the unit Reynolds number is 2 260 000 m^-1. No-slip and adiabatic wall boundary conditions are used.

The stability analysis is carried out at the location of x/r₀ = 6 downstream from the leading edge. Crossflow instability contains both stationary mode (the frequency is zero) and travelling mode unstable waves. Contours of growth rates in the x-direction of crossflow waves are shown in Fig. 5. The black dots C and D represent the most unstable stationary and travelling crossflow waves, respectively. The wave parameters at the locations of C and D are shown in Table 2.

From the transition prediction perspective, we are most interested in spatial instability. Hence, just (19) is discussed in this subsection. Figure 6 shows the variations of GGR and θ versus β_i. Their relative deviations compared with a reference state (β_i=0) are also plotted. It is shown that the variations of GGR are smaller than 0.1% and 0.2% for stationary and travelling mode crossflow waves, respectively, which also suggests that the GGR is conservative for stationary and travelling mode crossflow waves. The relative variations of θ are smaller than 0.4% and 0.6%, respectively.

3.3 More convenient method of calculating GGR and error estimation

In order to obtain the growth rate in the group velocity direction in the spatial mode, one can set β_i = 0 straightly according to the GGR conservation. It provides great convenience for the calculation of GGR. However, would tend to infinity if θ is close to 90°, which contradicts the condition that the imaginary part is far smaller than the real part of the wave parameters in the theoretical derivation of GGR conservation. This may lead to failure of calculation. Therefore, it is necessary to study the effect of the direction angle of the group velocity.

In a general 3D boundary layer flow, the group velocity directions of disturbances are generally close to the potential flow direction. Therefore, the value of the direction angle is mainly determined by the direction of coordinate axis x. As shown in Fig. 6, the group velocity direction angles in the original coordinate system are 47.6° and 49.6° for stationary and travelling waves, respectively. The group velocity direction is marked as l, and the angle between x and l is marked as θ. Now, we change the direction of x-axis gradually. Then, the GGR is calculated in new coordinate systems by (19) with ω_i = 0 and β_i = 0. Figure 7 presents the values of the GGRs and their relative errors versus the angle θ for stationary and traveling modes. It can be shown that the calculation errors of σ_g increase rapidly when θ is greater than 80° for both stationary and travelling crossflow waves. In fact, -α_i reaches the level of about 2 with θ=80° for both stationary and travelling crossflow waves. The relative errors of σ_g are smaller than 1% for both waves with θ = 70° and smaller than 0.3% with θ = 60°. It suggests that the GGR can be accurately calculated with θ being in a wide range of 0~60°. The condition of θ ≤ 60° can be easily guaranteed in practical stability analysis and transition prediction.

3.4 Discussion about e^N method and application of GGR

The e^N method has been widely used to predict transition of boundary layers. The first step in the e^N method is to obtain the N factor by integrating the growth rate. Generally, the integration is carried out for each specified frequency. Then, the maximum of N factors for different frequencies can be obtained. A consensus is that the growth rate should be integrated along the direction of group velocity^[17-18]. However, there are three different classical methods to determine the growth rate. Now, by the GGR conservation, it can be easily clarified that the three methods to determine the growth rate are essentially equivalent.

In the first method, Malik^[12] proposed that one should integrate along the direction of group velocity in the temporal mode. The integration formula follows

(32)

That is to say, the complex ω is solved with the specified α and β according to the dispersion relation. Aiming to keep ω_r = const., an iterative calculation process is needed for growing boundary layers.

In another method, Arnal and Casalis^[1] and Yu and Luo^[8] suggested that the growth direction of disturbance is specified as the group velocity direction. Hence, the growth rate in the perpendicular direction of the group velocity direction is assumed to be zero. Then, the N factor is calculated by the formula

(33)

This method is based on the spatial mode calculation, i.e., α is solved with specified ω and β according to the dispersion relation. In their calculations, ω_r can be set a constant straightly, and an iterative calculation process is needed to satisfy the condition of α_isinθ - β_icosθ ≡ 0. In fact, this condition is not necessary. According to the GGR conservation, σ_g=-α_icosθ - β_isinθ is a constant, and it is independent of β_i.

For infinite spanwise flows (e.g., swept wing), Mack^[9] proposed the third method. It is to integrate the spatial growth rate along the x-axis with ω_i = 0 and β_i = 0 (the x-axis is along the chordwise direction). The N factor is calculated by the formula,

(34)

This method is convenient for infinite spanwise flows because no iterative calculation process is needed. However, it is not suitable for general 3D boundary layers.

Through the discussion in this paper, a more convenient calculation method for general 3D boundary layers follows

(35)

That is to say, one can simply set ω_i and β_i to be zero in the spatial mode calculation, and then integrate the GGR along the group velocity direction according to (35) except the condition that the direction angle of group velocity is closed to 90°. In addition, (34) is a special case of (35) for infinite spanwise flows because of the relation dx = cosθ ds.

Actually, the integrals in (32), (33), and (35) are equivalent because all the integrands are the special cases of GGR. Although the essential equivalence among (32)–(35) can be concluded from the theoretical derivation and numerical validation above, the N factors and integration path are shown by different formulae ((32)–(35)) in Fig. 8 for the swept blunt flow above. Just the travelling crossflow wave with a specified frequency ω_r = 0.4 is considered. Because the flow is uniform in the spanwise direction, β_r sustains a constant in the propagation of disturbance (β_r=3). It can be shown in Fig. 8 that the N factors of different ways agree very well. The direction angles (corresponding the integration path) also agree well, and the deviations among them are smaller than 0.6°.

3.5 Discussion about GGR from physical point of view

We consider the most complicated case that a disturbance amplifies spatially and temporally. This disturbance propagates from A(x₁, z₁, t₁) to B(x₂, z₂, t₂). Then, the N factor can be calculated from the following material integration formula:

(36)

The path of integration from A to B should be the group velocity line as discussed previously. The group velocity line can be presented as and , where U_gx and U_gz are the components of group velocity in the streamwise and spanwise directions, respectively. We differentiate the formula above and obtain dx = U_gxdt and dz = U_gzdt. Then, (36) leads to

(37)

In the group velocity direction, one has ds=U_gdt. Then,

(38)

σ_g^S can be referred to as the material spatial growth rate of a disturbance. Therefore, the GGR introduced in this paper is the material spatial growth rate in physical nature. In addition, by (37), one can also introduce the material temporal growth rate,

(39)

Now, the conservation relation of GGR can be described from a physical point of view. With the precondition of keeping the real parts of wave parameters invariant, small changes of α_i, β_i, or ω_i do not change the material growth rate of a disturbance. The physical reason why the material growth rate has the conservative characteristic is that the imaginary parts of wave parameters are far smaller than the real parts.

4 Conclusions

In order to unify the methods for determining growth rates of disturbances in the 3D e^N method, the relation among the wave parameters of the disturbance in boundary layer is studied theoretically. The derivation procedure of Gaster transformation is reviewed, and the GGR in the group velocity direction is defined for three kinds of disturbances, i.e., 2D wave in 3D flow, 3D wave of spatial mode, and general 3D. It is strictly deduced in theory that for the three cases, the GGR is conservative for boundary layers.

This conservation relation is also validated numerically for first/second modes and crossflow modes which are the most important instability waves on surfaces of practical vehicles. The GGR conservation manifests that the existing ways of calculating disturbance growth rate in the e^N method can be unified, and all of them are particular cases of GGR in essence. Because of the GGR conservation, one is no longer confused with the determination of the disturbance growth rate in 3D flows. In addition, a convenient method of calculating disturbance growth rate is proposed. Guaranteeing that the direction angle of group velocity is smaller than 60°, one can straightly set β_i = 0 and then calculate the GGR accurately. Finally, it is pointed out that the GGR is the material spatial growth rate of a disturbance. The knowledge about the GGR of the disturbance is helpful to perfect the theoretical foundation of 3D e^N method.

Acknowledgements

The authors are grateful to Ph. D. candidate Dongdong XU of Tianjin University for valuable discussion and suggestions.