Applied Mathematics and Mechanics (English Edition) Conservation relation of generalized growth rate in boundary layers

Transcription

1 Appl. Math. Mech. -Engl. Ed., 39(12), (2018) Applied Mathematics and Mechanics (English Edition) Conservation relation of generalized growth rate in boundary layers Runjie SONG 1, Lei ZHAO 2, Zhangfeng HUANG 1, 1. Department of Mechanics, Tianjin University, Tianjin , China; 2. Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang , Sichuan Province, China (Received Jan. 29, 2018 / Revised Jul. 8, 2018) Abstract The elementary task is to calculate the growth rates of disturbances when the e N method in transition prediction is performed. However, there is no unified knowledge to determine the growth rates of disturbances in three-dimensional (3D) flows. In this paper, we study the relation among the wave parameters of the disturbance in boundary layers in which the imaginary parts of wave parameters are far smaller than the real parts. The generalized growth rate (GGR) in the direction of group velocity is introduced, and the conservation relation of GGR is strictly deduced in theory. This conservation relation manifests that the GGR only depends on the real parts of wave parameters instead of the imaginary parts. Numerical validations for GGR conservation are also provided in the cases of first/second modes and crossflow modes. The application of GGR to the e N method in 3D flows is discussed, and the puzzle of determining growth rates in 3D flows is clarified. A convenient method is also proposed to calculate growth rates of disturbances in 3D flows. Good agreement between this convenient method and existing methods is found except the condition that the angle between the group velocity direction and the x-direction is close to 90 which can be easily avoided in practical application. Key words generalized growth rate (GGR), boundary layer stability, e N method Chinese Library Classification O357.4, O Mathematics Subject Classification 76E09 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 Citation: SONG, R. J., ZHAO, L., and HUANG, Z. F. Conservation relation of generalized growth rate in boundary layers. Applied Mathematics and Mechanics (English Edition), 39(12), (2018) The authors SONG, R. J. and ZHAO, L. contributed equally to this work Corresponding author, hzf@tju.edu.cn Project supported by the National Natural Science Foundation of China (Nos and ), the National Key R&D Plan (No. 2016YFA ), and the FengLei Youth Innovation Fund of China Aerodynamics Research and Development Center (No.KT-FLJJ ) c Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

2 1756 Runjie SONG, Lei ZHAO, and Zhangfeng HUANG 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 α β 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 ωi U g 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 = φ 0 + 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

3 Conservation relation of generalized growth rate in boundary layers 1757 the perturbation quantities in boundary layers can be presented as the form of travelling waves, φ = φ(y)e i(αx+βz ωt) + c.c., (1) where φ(y) 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 f(α, β, ω) = 0. (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 = f 1 (α). With the assumption that ω is an analytic function of α, the Cauchy-Riemann relations hold, = ω i α i, α i = ω i. (3) The two cases of most interest labelled (T) and (S) are considered in the article [2]. Case (T) Temporal mode instability, α T i = 0, Case (S) Spatial mode instability, ω S i = 0, α = α T r + iα T i (α T i = 0), ω = ω T r + iω T i. (4) ω = ω S r + iωs i ( ω S i = 0 ), α = α S r + iαs i. (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 T r, the Cauchy-Riemann relations (3) follow α S ωr S ωt r = i ω i dα i, (6) 0 ω S i ω T i = α S i 0 dα i. (7) Generally, for disturbances in boundary layers, ωi = O(ω im ) and α S i = O(ω im ), where ω im is the maximum value of ω i. Then, (6) follows ω S r = ω T r + O(ω im α S i ), (8) where O(ω im α S i ) O(ω2 im ) Neglecting terms of order ωim 2, we have a very good approximation, ω S r = ω T r. (9)

4 1758 Runjie SONG, Lei ZHAO, and Zhangfeng HUANG If we expand ωr in a Taylor series at any point α i in the range 0 to α S i and substitute (7), we get ωi S ωi T = α S ω ( r 1 ) i (α i α ) + r 2 (αs i )2 α S 2 ω r i α i (α i α ) +. (10) i Because ω S i = 0, (10) can be written as ω T i = α S i (α i ) + O((α S i ) 2 ). (11) Again neglecting the terms of order (α S i )2 and provided that ωr is non-zero, we have ω T i = α S i, (12) where ωr can be evaluated at any state between (T) and (S), meaning that ωr 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 S r = αt r, 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 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, α = α P r + iα P i, ω = ω P r + iω P i, (13) where α P r = α T r = α S r = const., and ωp i and α P i 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 ω P r = ωt r, ωp i ω T i = α P i + O((α P i )2 ), (14) where ωr 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 P i )2. Then, ω P r = ωt r, σ g = ωp i U g α P i = ωt i U g. (15) Here, U g = ωr is the magnitude of the group velocity for 2D waves. (15) suggests that keeping α P r = const. = α T r, the variation of α P i does not change the values of ω P r 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 P r are constants, whose values are equal to ωt i U g and ω T r, 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.

5 Conservation relation of generalized growth rate in boundary layers 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 X r, α = α X r + iα X i, β = β X r + iβ X i (β X i = 0). (16) Case (Q) Increase in both the x- and z-directions, ω = ω Q r = ω X r, α = α Q r + iαq i, β = βq r + iβ Q i. (17) The dispersion relation of disturbances can be written as α = f 2 (β), and α is assumed to be an analytic function of β. An analogous analysis can be carried out. Keeping β Q r = β X r =const., one obtains an analogous formula to (14), α Q r = α X r, αq i α X i = β Q i. (18) Introducing another form of the GGR, (18) can be rewritten as α Q r = α X r, σ g = α Q i cosθ β Q i sin θ = α X i cosθ = const., (19) where tan θ = αr 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 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 T i = 0, β T i = 0, Case (Q) Temporal and spatial mode instability, α = α T r, β = βt r, ω = ωt r + iωt i. (20) α = α Q r + iα Q i, β = βq r + iβ Q i, ω = ω Q r + iω Q i. (21) These two states satisfy the condition of α Q r = α T r and βr Q = βr T. The dispersion relation of disturbances can be written as ω = f 3 (α, β), (22) and ω is assumed to be a binary analytic function of α and β. The following two couples of Cauchy-Riemann relations hold = ω i α i, = ω i β i, α i = ω i, (23) β i = ω i. (24)

6 1760 Runjie SONG, Lei ZHAO, and Zhangfeng HUANG We first integrate the Cauchy-Riemann relations (23) with respect to α i from state (T) to α i = α Q i with β = βr 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 Q r + iαq i = const., and analogies of (8) and (10) can be obtained, ω Q r = ω T r + O(ω imα Q i ) + O(ω imβ Q i ), (25) ω Q i ω T i = α Q i + β Q i + O((α Q i ) 2 + (β Q i ) 2 ). (26) Neglecting the high order small quantities and introducing the general form of the GGR, we have ωr Q = ωr T, σ g = ωq i (α Q i cosθ + β Q i sinθ) = ωt i = const., (27) U g U g where U g = ( ωr ) 2 + ( ωr ) 2 represents the magnitude of group velocity of the 3D disturbance, and θ = arctan( ωr / ωr ) 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, ωr = 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 The total differential of (28) yields dω = ω = f 3 [α(β), β]. (28) ( ω α α β + ω ) dβ. (29) β For a single frequency disturbance of 3D waves in the spatial mode, the frequency is a real constant [11]. Hence, dω 0, and then α β = ω β / ω α α. With considering β = αr + i αi, ω β = ωr + i ωi, and ω α = ωr + i ωi, it follows + i α i = ( ω / r ωr )( 1 + i ω / i ωr )/( 1 + i ω / i ωr ). (30) Substituting the formula 1/(1 + t) 1 t+t 2 into (30), taking the real part of the equation, and neglecting the terms of order O( ωi ω i ) and O( ωi ) 2, we have = / ωr. (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.

7 Conservation relation of generalized growth rate in boundary layers 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 30km altitude, and the temperature of free stream is T = 226.5K. 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 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. α β Fig. 1 Spatial growth rate contours of first-mode and second-mode T-S waves in βω-plane Table 1 Wave parameters at locations of A and B Location α r α i β r β i ω r ω i A B 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 (state S) to 0 (state T) (in this process, ω i changes from 0 (state S) to (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.

8 1762 Runjie SONG, Lei ZHAO, and Zhangfeng HUANG Fig. 2 Values (solid lines) of ω r, σ g, and U g of arbitrary state (P) and their relative deviations (dashed lines) compared with state (T) for wave at location of A 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 r 0 = 35mm. 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 30km altitude. The temperature of free stream is T = 226.5K, and the unit Reynolds number is m 1. No-slip and adiabatic wall boundary conditions are used. The stability analysis is carried out at the location of x/r 0 = 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.

9 Conservation relation of generalized growth rate in boundary layers 1763 /( ) Fig. 3 Values (solid lines) of ω r, σ g, U g, and θ of arbitrary state (Q) and their relative deviations (dashed lines) compared with state (T) for wave at location of B θ Fig. 4 Sketch of swept blunt plate α Fig. 5 Spatial growth rate contours of crossflow instability in βω-plane

10 1764 Runjie SONG, Lei ZHAO, and Zhangfeng HUANG Table 2 Wave parameters at locations of C and D Location α r α i β r β i ω r ω i C D /( ) β β /( ) β β Fig. 6 Values (solid lines) of σ g and θ of arbitrary state (Q) and their relative deviations (dashed lines) compared with reference state (β i = 0) for waves at locations of C and D 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, α i = σg cos θ 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

11 Conservation relation of generalized growth rate in boundary layers % 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 The condition of θ 60 can be easily guaranteed in practical stability analysis and transition prediction. /( ) /( ) Fig. 7 Variations of GGR versus angle θ and their relative error compared with reference state (θ = 0) for stationary (ω = 0) and travelling crossflow waves (ω = 1.17) 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 ωi U g along the direction of group velocity in the temporal mode. The integration formula follows N = s2 s 1 ω i U g ds, α i = 0, β i = 0. (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 N = s2 s 1 ( α i cosθ β i sinθ)ds. (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 i sin θ β i cosθ 0. In fact, this condition is not necessary. According to the GGR conservation, σ g = α i cosθ β i sinθ 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, N = x2 x 1 α i dx. (34)

12 1766 Runjie SONG, Lei ZHAO, and Zhangfeng HUANG 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 s2 N = α i cosθds. (35) s 1 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 = cosθds. 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. /( ) Fig. 8 Comparison about N factors and direction angles θ (corresponding integration path) obtained by different methods for travelling crossflow wave (ω = 0.4) 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(x 1, z 1, t 1 ) to B(x 2, z 2, t 2 ). Then, the N factor can be calculated from the following material integration formula: N = B A α i dx β i dz + ω i dt. (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 x = x 1 + T t 1 U gx dτ and z = z 1 + T t 1 U gz dτ, 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 gx dt and dz = U gz dt. Then, (36) leads to N = B A ( α i U gx β i U gz + ω i )dt = B A (U g ( α i cosθ β i sin θ) + ω i )dt. (37)

13 Conservation relation of generalized growth rate in boundary layers 1767 In the group velocity direction, one has ds = U g dt. Then, σ S g = α i cosθ β i sin θ + ω i U g. (38) σ S g 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, σ T g = U g σ S g. (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. References [1] ARNAL, D. and CASALIS, G. Laminar-turbulent transition prediction in three-dimensional flows. Progress in Aerospace Sciences, 36(2), (2000) [2] GASTER, M. A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. Journal of Fluid Mechanics, 14(2), (1962) [3] GASTER, M. The role of spatially growing waves in the theory of hydrodynamic stability. Progress in Aerospace Sciences, 6, (1965) [4] CEBECI, T. and STEWARTSON, K. On stability and transition in three-dimensional flows. AIAA Journal, 18(4), (1980) [5] CEBECI, T., SHAO, J. P., CHEN, H. H., and CHANG, K. C. The preferred approach for calculating transition by stability theory. Proceeding of International Conference on Boundary and Interior Layers, ONERA, Toulouse (2004)

14 1768 Runjie SONG, Lei ZHAO, and Zhangfeng HUANG [6] SU, C. H. and ZHOU, H. Physical implication of two problems in transition prediction of boundary layers based on linear stability theory. Science China Physics, Mechanics and Astronomy, 57(5), (2014) [7] SU, C. H. and ZHOU, H. Stability analysis and transition prediction of hypersonic boundary layer over a blunt cone with small nose bluntness at zero angle of attack. Applied Mathematics and Mechanics (English Edition), 28(5), (2007) [8] YU, G. T. and LUO, J. S. Application of e N method in three-dimensional hypersonic boundary layers (in Chinese). Journal of Aerospace Power, 29(9), (2014) [9] MACK, L. M. Stability of three-dimensional boundary layers on swept wings at transonic speeds. Symposium Transsonicum III, Springer, Berlin/Heidelberg, (1989) [10] MALIK, M. R. and BALAKUMAR, P. Instability and transition in three-dimensional supersonic boundary layers. The 4th Symposum on Multidisciplinary Analysis and Optimization, American Institute of Aeronautics and Astronautics, Reston (1992) [11] ZHAO, L., YU, G. T., and LUO, J. S. Extension of e N method to general three-dimensional boundary layers. Applied Mathematics and Mechanics (English Edition), 38(7), (2017) [12] MALIK, M. R. COSAL-A Black Box Compressible Stability Analysis Code for Transition Prediction in Three-Dimensional Boundary Layers, NASA, CR (1982) [13] NAYFEH, A. H. and PADHYE, A. Relation between temporal and spatial stability in threedimensional flows. AIAA Journal, 17(10), (1979) [14] CHEN, X., ZHU, Y. D., and LEE, C. B. Interactions between second mode and low-frequency waves in a hypersonic boundary layer. Journal of Fluid Mechanics, 820, (2017) [15] ZHAO, L. Study on Instability of Stationary Crossflow Vortices in Hypersonic Swept Blunt Plate Boundary Layer (in Chinese), Ph. D. dissertation, Tianjin University, (2017) [16] ZHAO, L., ZHANG, C. B., LIU, J. X., and LUO, J. S. Improved algorithm for solving nonlinear parabolized stability equations. Chinese Physics B, 25(8), (2016) [17] KUEHL, J., PEREZ, E., and REED, H. JoKHeR: NPSE simulations of hypersonic crossflow instability. The 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, American Institute of Aeronautics and Astronautics, Reston (2012) [18] CHANG, C. L. LASTRAC. 3d: transition prediction in 3D boundary layers. The 34th AIAA Fluid Dynamics Conference and Exhibit, American Institute of Aeronautics and Astronautics, Reston (2004)