SCIENCE CHINA Physics, Mechanics & Astronomy

SCIENCE CHINA Physics, Mechanics & Astronomy Article May 2014 Vol.57 No.5: 950 962 doi: 10.1007/s11433-014-5428-y Physical implication of two problems in transition prediction of boundary layers based on linear stability theory SU CaiHong * 1 Department of Mechanics, Tianjin University, Tianjin 300072, China; 2 Tianjin Key Laboratory of Modern Engineering Mechanics, Tianjin 300072, China Received December 20, 2013; accepted January 21, 2014; published online March 14, 2014 Up to now, the most widely used method for transition prediction is the one based on linear stability theory. When it is applied to three-dimensional boundary layers, one has to choose the direction, or path, along which the growth rate of the disturbance is to be integrated. The direction given by using saddle point method in the theory of complex variable function is seen as mathematically most reasonable. However, unlike the saddle point method applied to water waves, here its physical meaning is not so obvious, as the frequency and wave number may be complex. And on some occasions, in advancing the integration of the growth rate of the disturbance, up to a certain location, one may not be able to continue the integration, because the condition for specifying the direction set by the saddle point method can no longer be satisfied on the basis of continuously varying wave number. In this paper, these two problems are discussed, and suggestions for how to do transition prediction under the latter condition are provided. boundary layer, transition prediction, linear stability theory, DNS PACS number(s): 47.15.Fe, 47.20.Ib, 47.27.Cn Citation: Su C H. Physical implication of two problems in transition prediction of boundary layers based on linear stability theory. Sci China-Phys Mech Astron, 2014, 57: 950962, doi: 10.1007/s11433-014-5428-y Boundary-layer transition prediction is important for many practical problems, yet it is still not a fully solved problem [1]. Up to now, the only method, which is considered to have a more or less sound physical basis, and is also widely used in engineering practices for natural transition, is a method based on linear stability theory (LST). The basic idea is based on searching for, among all possible small disturbances in the boundary layer, the one which would trigger transition, by a certain criterion, at the most upstream location. And the so found position is the transition location. In the traditional formulation of the method, a factor N, serving as the criterion for transition, should be determined empirically. Su & Zhou [2,3] made some essential *Corresponding author (email: su_ch@tju.edu.cn) Recommended by ZHOU Heng (CAS academician) improvements on the method, making it be more rational and much less dependent on experiments and experience. However, there are still two inter-related problems which are physically not clear enough and practically may cause the method inapplicable in certain cases. The first one is, as theoretically, the number of possible disturbances with a given frequency is infinity, thus a criterion choosing the most meaningful disturbance and determining its direction of propagation is necessary. Then what is the physical meaning of the criterion. The second one is that in certain cases of transition prediction for supersonic/hypersonic and three-dimensional boundary layers, in advancing the integration of the growth rate of the disturbance, up to a certain location, the proposed criterion may no longer be satisfied on the basis of continuously varying wave number. Then what can one do in those cases? In this paper, we are mainly concerned with these two problems. Science China Press and Springer-Verlag Berlin Heidelberg 2014 phys.scichina.com link.springer.com

Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 951 In linear stability theory, a small disturbance in boundary layers is assumed to be in the form of a travelling wave as: ˆ i( x z t) (,,, ) q( y)e, (1) q x y z t where q ( x, y, z, t) is the disturbance variable, qy ˆ( ) is the shape function, and the wave numbers in x and z directions respectively, and the angular frequency. In spatial problems, both and can be complex, and their imaginary parts determine the growth rate in x and z directions respectively. In the linear stability theory, eigen-value problem is posed for finding the possible waves for a given frequency, and the number of solutions is infinity, as in the eigen-value problem. Either or can be arbitrarily given, and then the other one is to be solved as an eigen-value. Although the number of possible eigen-values is again infinity, the number of those eigen-values corresponding to unstable modes is limited, usually only one or two, or even none existing. In transition prediction, one has to search for, among all unstable disturbances with different frequencies and wave numbers, the critical one which determines the location of transition under a certain criterion. Thus, in transition prediction, the first step is, for a given frequency, to find the critical one, and then find the most critical one among all critical ones for different frequencies. Malik [4] proposed to solve the first problem by applying temporal mode approach, in which both and are real. Denote as the angle between wave propagation direction and x axis, then = tan 1 (/ ). For a given frequency and, one can find a corresponding eigen-value. Vary and search the wave having maximum growth rate i, then this wave is the critical one for the given frequency. Dividing i by the group velocity of the wave, the spatial mode growth rate i can be obtained. Then one can integrate the growth rate i in the direction determined by the wave angle to obtain the total amplification factor as required in the transition prediction method. Mack pointed out that in certain practical cases, can be assumed to be real and constant as the wave propagates [5]. Accordingly, Arnal [6] suggested that the critical wave for a given frequency should be the wave having maximum growth rate under the variation of, and the integration should be carried out in the direction of the free stream velocity at the edge of the boundary layer. Cebeci and Stewartson [7,8] used the saddle point method in the theory of function with complex variables to investigate the evolution of a wave packet consisting of waves having the same frequency. The result is that, far away from the source, the dominant wave should be the one whose wave number satisfies the condition ( / ) i 0, and it would appear in the direction from the source having an angle with the stream-wise direction, and = 1 tan { ( / ) } 0 (subscripts r and i denote real and r imaginary parts, respectively), so the integration of the growth rate should be carried out along this direction. Notice that when the span-wise wave number is real and constant, the criterion for searching for the critical wave suggested by Cebeci and Stewartson is equivalent to those of Arnal s, but the integration direction, or equivalently the integration path, is different in the two methods. From the above statements, one can see that the argument put forward by Cebeci and Stewartson is mathematically more sensible than those by other authors. However, its physical meaning is not clear enough. Besides, in some cases, in carrying out the integration of the growth rate along the propagation path of the wave, whether defined by Arnal or Cebeci & Stewartson, people may suddenly find that they can no longer find the wave satisfying the criterion under the condition that wave numbers are varied continuously. What is the physical meaning for such cases and how to deal with them are also problems to be clarified. 1 Problem I: Physical implication of Cebeci- Stewartson s criterion As mentioned above, Cebeci and Stewartson used the saddle point method to deal with the wave packet propagation problem. The wave packet consists of waves with the same frequency. Accordingly, we first do a direct numerical simulation for the propagation of such a wave packet in a boundary layer on a flat plate. 1.1 Computational model and basic flow The oncoming flow has a Mach number 3, and the temperature is 79 K. The wall is assumed to be adiabatic. First, the basic laminar flow is to be computed. A computational domain is chosen far downstream from the leading edge of the flat plate. The Reynolds number is 5 10 4, based on the displacement thickness of the boundary layer at the inlet of the computational domain, obtained from the Blasius solution, and the velocity, temperature, density and viscosity coefficient of the oncoming flow. The computational domain is shown in Figure 1 as bounded by dashed lines. x, y, z are coordinates in stream-wise, normal and span-wise directions, respectively. The basic flow is a 2-D flow, so its computational domain is also 2-D, with extents in x and y directions, i.e. x L and y L, are 600 and 50. The grid number in x direction is 601, and in y direction is initially chosen to be 81, 201 or 401 for testing the accuracy of the result, and then is fixed to be 201 as will be explained later. The governing equations are compressible Navier-Stokes equations. A 5 th order upwind scheme is used for the split convective terms, and a 6 th order central scheme is used for viscous terms. A 3 step and 3rd order Runge-Kutta scheme is used for the time advancing. A simple non-reflection

952 Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 Figure 1 A diagram of computational domain. boundary condition suggested by Fu et al. [9] is used at the upper boundary. That is, the flux coming into the computational domain is simply set to be zero, while the flux going out of the domain is computed from the one-sided difference using the interior and boundary grids. The outflow boundary condition is a second order extrapolation one. The basic flow is two-dimensional and Blasius similarity solutions on a flat plate are used as the initial condition. Blasius profiles are maintained at the inlet. The computation is carried out until the whole flow field reaches a steady state. The comparison of the profiles of the basic flows at x=200, obtained by using 81, 201 and 401 grid points respectively in normal direction, is shown in Figure 2. The stream-wise velocity and temperature profiles are almost identical, while the normal velocity profiles do show a slight difference. In order to finally decide which grid number we should take for the normal direction, results of eigen-value calculations for stability analysis are compared. For =0.14, =0.485, the resulting eigen-values are shown in Table 1. Clearly, the eigen-value is not accurate enough for grid number 81, while results for grid number 201 and 401 are very close to each other. So we finally fix the grid number in normal direction to be 201, otherwise the total grid number would be too large as will be seen later. The pressure contour is shown in Figure 3. There are weak disturbances issuing from the inlet of the computational domain, which results from the adjustment of the flow, because the inflow Blasius profile is not an exact solution to N-S equation. In the following, we discard the flow field between locations x=0 and x=200, and use the remaining part as our basic flow. Figure 2 Comparison of the basic flow at x=200. (a) Stream-wise velocity; (b) normal velocity; (c) temperature. 1.2 Evolution of a wave packet computed by DNS In applying linear stability theory to predict the transition location, we calculate the growth factor of individual T-S waves. But in reality, disturbances hardly manifest themselves as regular traveling waves. Instead, disturbances in the boundary layer are more likely to be triggered locally by certain external disturbances. Hence, wave packets are more natural to be observed. If the triggered disturbance is very localized, then its initial form can be expressed by a delta function in time and space, and by Fourier transform, it is equivalent to an infinite wave system with waves of all possible frequencies and wave numbers but with the same shape function. After a short time period of evolution, the Figure 3 Pressure contour of the basic flow. initial form of a delta function would gradually be smoothed. As waves with different wave numbers have different growth rate, the wave packet would soon center around the

Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 953 Table 1 Comparison of growth rate for different normal grids Normal grids r i 81 0.214096 0.011874 201 0.214214 0.011091 401 0.214218 0.011088 most amplified one. Hence, as the first choice, we construct a wave packet as a span-wise modulation wave packet of the most amplified T-S wave as follows. At x 0 =200 (henceforth x 0 denotes this location for convenience), the inlet of the computational domain, we introduce a wave packet as follows: q y, z, t G z q( y)e cc.., i( zt) G z A e az, 0 (2) where A 0 represents the initial amplitude of the wave packet, which is chosen to be very small as 1.5 10 8, so that the wave packet evolves linearly, compatible with linear stability theory. is a real number 0.48, =0.14, a=8 10 3. The shape function q( y) is solved together with the unknown stream-wise wave number from the eigen-value problem of the O-S equation under given and and = 0.214214 0.01109i. In fact, parameters of the wave actually satisfy the condition that is real, as set by Cebeci and Stewartson. The distributions of stream-wise velocity, temperature and pressure along y direction, as denoted by q( y), are shown in Figure 4. The extent of 3-D computational domain in span-wise direction is (196, 196), which equals 30 times of wave length of the basic T-S wave, i.e. 30 ( 2 /). The grid number is 401 201 601, nearly equal to 50 million. That s why the grid number in y direction is chosen to be 201, otherwise the computation would be too costly. Periodic boundary condition is adopted in span-wise direction. The computation is performed in National Supercomputer Center in Tianjin, using 200 CPU. Figure 5 shows the distribution 2 of steam-wise velocity along z direction at y=0.921 at the inlet of the computational domain. The computation is performed up to the extent that the flow field becomes totally periodic with time. Figure 6 shows, by gray scale, the instantaneous stream-wise disturbance velocity in the plane y=0.921. For a fixed streamwise location, the shape of wave packet can be obtained by searching for the maximum of stream-wise disturbance velocity for each given z within a time period, and the result is shown in Figure 7. One can see that the amplitude of the wave packet grows steadily as it propagates downstream, and the span-wise location of its peak drifts slightly in z direction. 1.3 Linear stability analysis According to Cebeci & Stewartson, in using linear stability theory to predict transition location, the integration of the growth rate of the disturbance wave should be carried out along the direction of wave propagation, which is in turn 1 determined by the condition tan { ( / ) }, where Figure 5 Distributions of stream-wise disturbance velocity along z direction. r Figure 4 Distributions of eigen-functions along y direction at x=200. Figure 6 Gray scale figure for stream-wise disturbance velocity u in the plane of y=0.921.

954 Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 Figure 7 Form of wave packets at different stream-wise locations. Figure 8 Variations of i with at different stream-wise locations. is the angle between the propagation direction and the x axis. In case is real, as in the present case, the condition ( / ) 0 as suggested by Cebeci & Stewartson i actually corresponds to the extreme points of the curve of i versus, determined from the eigen-value problem of linear stability theory. The curve of i versus of our caseis shown in Figure 8 for x=200, 400, 600 and =0.14. In this figure, point A, with = 0.48, is the one we are searching for, which corresponds to a wave having maximum growth rate under the variation of, i.e. the most unstable one, and is the one we used above to form the wave packet in the DNS. According to the idea of the saddle point method the propagation path and the amplitude of the wave should be determined as follows, Figure 9 Comparison of the amplitude of the wave packet with the amplitude of the critical unstable wave under Cebeci-Stewartson s criterion. x L 0 d, r x 0 z z x x L idx e x0 A A, 0 (3) where A 0 =1.5 10 8 and z 0 denotes the peak location of the wave packet at x 0. i and r are to be solved from the eigen-value problem of the O-S equation at each stream-wise location under the condition i 0. Figure 9 shows the amplitude curve of the wave obtained by applying eq. (3) and the amplitude curve of the wave packet obtained by DNS (labeled by SPM and DNS, respectively), and the agreement is very good. Figure 10 shows the propagation path of the wave obtained by applying eq. (3) and the propagation path of the peak of the wave packet obtained by DNS. Again the agreement is good. Therefore, in applying the linear stability theory for transition prediction, the criterion suggested by Cebeci & Stewartson for choosing the critical wave for a given frequency actually implies that the chosen wave is the most unstable one, and the angle given by the criterion corresponds to the propagation direction of the peak of a wave packet with the most unstable wave as its central wave. Figure 10 Comparison of paths of the wave packet. From the above analysis, one can see that Cebeci-Stewartson s method is physically sensible. 1.4 Further analysis for wave packet Expand the right hand side of eq. (2) for t=0, i.e. the initial wave packet, into a Fourier series in z as expressed in eq.

Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 955 (4), in which 0 2 zl and C n is a complex number. 2 C n represents the amplitude of the component whose span-wise wave number is n. On the right hand side of eq. (4), n ranges only from 19 to 41. The remaining terms are discarded because they are sufficiently small that C n <10 13 for those terms. Figure 11 shows the relationship between the amplitude and span-wise wave number n n0. 41 n in0 z (4) n19 q y, z C q( y)e cc... Notice that initially all components in eq. (4) have the same shape function qy (), which corresponds to a T-S wave with span-wise wave number, and hence only the term in eq. (4) with n=30 corresponds to a pure T-S wave, while other terms are not. Since in transition prediction by LST, we are dealing with only T-S waves, there is a slight disparity between objects being compared. For 2-D, incompressible and parallel flows, and under temporal mode formulation, it has long been proven that for a given wave number, the eigen-value problem of Orr-Sommerfeld equation yields a complete set of eigen-functions. Any sufficiently smooth function in y, satisfying the same boundary conditions as the eigen-functions, can be expanded as an infinite series of eigen-functions, which is absolutely and uniformly convergent [10]. That is, any wave with a given wave number can be expanded as a series of T-S waves, having the same wave number. As all T-S waves in the expansion, except the first one, would die out fairly quickly, the wave would soon appear as a single T-S wave with the eigen-value as the first eigen-mode, and its initial amplitude can be obtained by projecting the initial shape function of the wave onto the direction of the first eigen-mode in function space. However, there are no corresponding mathematical theorems for compressible boundary layer flows under spatial mode formulation. Therefore, we would rather construct the initial wave packet composed of only T-S waves by a plausible method. For given frequency and span-wise wave number n we can find a corresponding T-S wave, and denote its eigen-function by qˆ n () y, and then we can compose another wave packet as:, 41 in ˆ ( )e n n... n19 yz Cq y cc (5) The terms in eq. (5) are different from the corresponding terms in eq. (4), except for n=30. However, all the differences are small, as shown in Figure 12. In solving the eigen-value problem of the O-S equation, if we normalize the eigen-function by the condition that qˆ n( yn) 1, where y n is the location that q ˆn takes its maximum value, then the distribution of stream-wise disturbance velocity of the wave packet, as defined by eq. (5), in the plane y=0.921 is shown in Figure 13(a) labeled by method 1. On the other hand, if we normalize the eigen-functions q ˆn by the condition that max qˆ n ( y) 1, while qˆ n (0.921) are real, that is, they take real value at the same location y=0.921 as the function qy () in eq. (4), then another distribution of stream-wise disturbance velocity in the same plane y=0.921 is obtained, which is labeled by method 2 as shown in Figure 13(b). We can see that the curve yielded by method 2 coincides with the curve given by eq. (4) a little better than that of method 1. Compute the evolution of each T-S wave in eq. (5) by LST and then add them together, and the evolution of the wave packet is obtained. Figures 14(a) and (b) show the comparison of the stream-wise disturbance velocity along the line y=0.921, z=0, obtained by DNS and those obtained from LST under methods 1 and 2 respectively. It can be seen that the difference between results from DNS and those of method 2 is bigger, notwithstanding that their initial difference is smaller. Figure 15 shows the comparison of the evolution of the amplitude of the wave packet. Curves obtained by both Figure 11 Amplitude of Fourier components as a function of their wave numbers. Figure 12 Eigenfunctions given by linear stability theory at x 0.

956 Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 Figure 15 Comparison of the amplitude growth. Figure 13 Comparison of disturbance velocities obtained from eqs. (4) and (5). (a) Method 1; (b) method 2. methods 1 and 2 are the same, both slightly below that from DNS. Figure 16 compares the propagation paths of the wave packets. The result given by method 1 coincides with those from DNS almost exactly, while the result given by method 2 shows a slight difference. Of course, total coincidence is not expected in either methods, because eqs. (4) and (5) represent two different initial disturbances. But in the DNS, which is essentially linear due to the smallness of each term, the initial shape qy () of each term will quickly adjust itself to approach their respective shape qˆ n () y, so the difference between results of DNS and LST should not be big. Thus it can be concluded that, the wave packet in DNS, which is obtained by span-wise amplitude modulation of a single T-S wave at the inlet, is practically equivalent to a wave packet consisting of a group of T-S waves. 1.5 Another case of validation Here we compute another case with different Mach number and Reynolds number to provide further evidence that linear stability theory coupled with Cebeci-Stewartson s criterion would result in a wave which evolves in the same way as Figure 14 Comparison of disturbance evolutions between those obtained from LST and DNS. (a) Method 1; (b) method 2. Figure 16 Comparison of the propagation path of wave packets.

Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 957 the peak of a wave packet. Let the Mach number of the oncoming flow be 2, the Reynolds number be 5 10 4, and the temperature condition at the wall be adiabatic. The computational domain for the basic flow x L y L is 1595 50, and the corresponding mesh number is 320 320. The method to obtain the basic flow is the same as in sect. 1.1. The disturbance, given by eq. (2), which is tuned to have a shape of a wave packet is introduced at x 0 =200, A 0 =1.5 10 8, a=7.4 10 5. The eigen-value =0.04650.0056i, which is solved from O-S equations by setting =0.0146 and =0.1355. qy ( ) is the corresponding eigen-function of the T-S wave. At location x 0, the above T-S wave satisfies the Cebeci-Stewartson s criterion that is real. The span-wise computational domain is (647,647), distributed by 451 uniformly. By the same procedure as in sect. 1.2, the shape of the wave packet at different stream-wise locations can be obtained as shown in Figure 17. Also, we can obtain the evolution curve of its peak amplitude and the path of its peak propagation. The comparison with the corresponding results obtained by LST under Cebeci-Stewartson s criterion is shown in Figures 18 and 19, respectively. Again, the agreement is satisfactory. However, if we perform Fourier transform to the wave packet in span-wise direction as shown in eq. (4), then we will find that among all waves, the wave satisfying Cebeci-Stewartson s criterion has the biggest initial amplitude. So its later dominance is not 100% convincing. To avoid this, we construct another wave packet as follows: t q x, y, z, t G z q( y)e cc.., i 0 2 z zsin( ) (ln 0.01) c z 0 e M A, z 0, Gz z zc A0, z 0, N nn in z G z C cc 0 ne.., where q( y) represent the eigen-functions obtained from O-S equation with 0.0146, 0.1355 satisfying the Cebeci-Stewartson s criterion at x 0 =200. Actually, for a fixed frequency 0.0146, the eigen-functions of O-S equation with different span-wise wave numbers between 0.1 and 0.2 have only very slight difference, as shown in Figure 20. In eq. (6), A 0 =1 10 4, M=300, c 0.18. Perform Fourier decomposition for G(z) in span-wise direction, and we get eq. (7), where 0 2 zl. C n is the amplitude of the Fourier component with span-wise wave (6) (7) Figure 17 Form of wave packets at different stream-wise locations. Figure 18 Comparison of the amplitude of the wave packet with the amplitude of the critical unstable wave under Cebeci-Stewartson s criterion. Figure 19 Comparison of paths of propagation. number n =n 0. Figure 21 shows the distribution of G(z) along span-wise direction. Figure 22 shows C n for each Fourier component versus span-wise wave number. It can be seen that for the wave packet constructed in such a way, in a certain range of span-wise wave number, all components have nearly the same initial amplitude. Since the wave satisfying the Cebeci-Stewartson s criterion has a span-wise

958 Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 Figure 20 Comparison of eigen-functions obtained by different spanwise wave numbers. each mesh point is totally periodic with time, the isocontour of stream-wise disturbance velocity in (x,z) plane at y=1.59 can be plotted as shown in Figure 23. It can be seen that far far downstream of the inlet, on each side of z=0, a peak appears. We can obtain the peaks location and their amplitude of the wave packet at each stream-wise location by the similar way to that in above sections, and compare it with the propagation path and amplitude given by linear stability theory under Cebeci-Stewartson s criterion, and their comparison of amplitudes is shown in Figure 24. Figure 25 shows the comparison of their paths of propagation. Again, the results are satisfactory. The distribution of disturbance velocity û in the spectral space can be found by performing Fourier transform to the stream-wise disturbance velocity. The iso-contour of û in (x,) plane is plotted in Figure 26 by seeking the maximum û along y direction. It can be seen that far downstream of the inlet, the wave having span-wise wave 0.14 becomes dominant, which is exactly what Cebeci-Stewartson s saddle point method claims. The dominance Figure 21 Distribution of G(z) along span-wise direction. Figure 23 Contour of stream-wise disturbance velocity u in the (x,z) plane at y=1.59. Figure 22 C n for each Fourier components. wave-number 0.14, which is within the range, it no longer has the advantage of initial dominance, or alternatively, it competes with other components on an equal basis, in accordance with the idea of transition prediction by LST. Again, the disturbance is introduced at x 0 =200, the computational domain in span-wise direction is (647,647), and is discretized with 601 uniform grids. When the flow at Figure 24 Comparison of amplitudes of wave packet.

Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 959 2.2 Linear stability analysis Figure 25 Comparison of paths of propagation. For a fixed frequency =0.84, linear stability analysis is performed for basic flow profiles at different stream-wise locations. And the i -curve is obtained as shown in Figure 27. It can be seen that the shape of i -curve of different locations changes significantly, not only quantitatively, but also qualitatively. At x=100, there are three extreme points, which means there are three waves satisfying the Cebeci-Stewartson s condition i 0. One is A, corresponding to a two-dimensional wave with span-wise wave number 0, and the other two correspond to threedimensional waves with wave numbers having the same magnitude but opposite signs. The difference between the latter two only lies in their direction of propagation, so we only need to consider one of them, say, the wave whose wave number > 0, corresponding to B. A, B are used to denote the two-dimensional and three-dimensional waves satisfying the condition i 0, respectively. Notice that for x=100, point A corresponds to a local minimum of growth rate, while B corresponds to a local maximum. But beyond the location x=125, A becomes also a local maximum, while two other extreme points appear, both corresponding to the local minimum of growth rate. Obviously at x 0, B corresponds to the most unstable wave, whose Figure 26 Iso-contour of û in (x,) plane. would occur at a place far from the original wave packet, no matter what the relative magnitude of waves composing the wave packet is. 2 Problem II: Cebeci-Stewartson s condition cannot be satisfied continuously Again, we use DNS and linear stability analysis to study this problem. 2.1 Computational parameters for the DNS A boundary layer on a flat plate is considered. The freestream Mach number is 6 and the temperature is 79 K. The wall is adiabatic, and the Reynolds number is 10 4, based on the displacement thickness of boundary layer at the inlet of computational domain, the oncoming velocity, temperature, density and viscous coefficient. The numerical method is the same as for problem I. The computational domain x L y L for the basic laminar flow is 300 25, having 1201 151 grids. The inlet of our DNS x 0 is located at x=100. Figure 27 Curves of i -for different stream-wise locations. (a) x= 100 135; (b) x=140 200.

960 Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 0.797. But with the increase of x, A not only becomes a point corresponding to the local maximum of growth rate, but also eventually becomes the most unstable wave beyond the location x=131. On the other hand, the corresponding growth rate of B decreases gradually, and even no longer is an extreme point for x148, i.e. it ceases to satisfy the condition i 0. And also, the two new extreme points disappear when x=148. Figure 28 depicts the eigenfunctions u of A and B at x 0. Although Cebeci and Stewartson only proposed to use the condition i 0 for choosing the critical unstable wave, and did not explicitly specify that it should also correspond to a local maximum of the growth rate, but from the physical nature of the problem, i.e. transition prediction, the latter condition is obviously necessary. If there are more than one maximum, people should choose the most unstable one. Consequently, wave B should be our first choice. However, as stated before, in following B, we would not be able to proceed further when x148, as no neighboring point of B satisfying i 0 can be found. As a result, the dashed line B ceases at x=148 as shown in Figure 29. Figure 30 depicts the variation of the span-wise wave number of B as a function of its stream-wise location. Its span-wise wave number at x=148 is 0.64. On the other hand, if we follow the wave represented by A, which is not the most unstable one initially, its integrated amplification factor N would be smaller than those of B for their initial stage, but soon it would surpass B and keeps growing further in a faster pace, as shown also in Figure 29. We can also do a corresponding DNS for the evolution of wave packets to show more clearly the situation. 2.3 Direct numerical simulation for the evolution of wave packets Two wave packets, originated by span-wise modulation, with the same modulation function, on T-S waves corresponding to A and B, are introduced simultaneously at the Figure 28 Shapes of eigenfunctions u. Figure 29 Amplification factor N versus downstream distance. Figure 30 Variation of the wave number of B in stream-wise direction. inlet of our computational domain, i.e. at x 0 =100. The expression for the wave packets is iz it q( y, z, t) G( z)[ qa( y) qb( y)e ]e cc.., (8) azz 2 e 2 L G z A. 0 Here, A 0 =1.5 10 8, a=3 10 3. qa( y), qb( y) represent eigen-functions of A, B, respectively. =0.84, A =0, B = 0.797. The computational domain is 200 25 200, having 801 151 381 grids. When the whole flow field becomes periodic with time, we can obtain the shape of wave packet at different stream-wise locations, as shown in Figure 31. As can be seen, the amplitude of the wave packet grows steadily and it grows slower near the inlet while faster further downstream. However, it is not clear alone with the above figure what roles are played by wave packets originated from A and B respectively. For a given stream-wise location x, span-wise Fourier transform is done for the disturbance velocity at each y location. Then seek the maximum value of each Fourier component along y direction, resulting in a value as a function of x and span-wise wave number, as shown in

Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 961 Figure 31 Shape of wave packet at different stream-wise locations. Figure 32. As can be seen, wave packet B amplifies faster than wave packet A not far from x 0. However, starting from x=130, wave packet A amplifies significantly, while B almost ceases to amplify. So A becomes the dominant wave, having much bigger amplitude than B. Therefore, for transition prediction, A is more likely to be the one that eventually would trigger transition, unless that B is already sufficiently large for triggering transition before x=148. There is a subtle difference in LST analysis and DNS for wave packets. In LST analysis, wave B becomes meaningless when x148, as it no longer satisfies Cebeci-Stewartson s criterion, while in DNS, the computation can keep going on, whether the criterion is satisfied or not. In applying LST, wave A may even be totally neglected, as in the initial stage, since it does not correspond to a local maximum of amplification rate. One has also to keep in mind that here we have only compared the relative amplification factor of the unstable waves, while in real situations, the final amplitude of unstable wave may play a more important role than simply the amplification factor. The final amplitude not only depends on amplification factor, but also depends on the initial amplitude of the unstable wave, which in turn depends on the receptivity mechanism and the nature of background disturbances. Moreover, since at first, A does not correspond to a local maximum of growth rate, rather, it corresponds to a local minimum of growth rate, so its neighboring waves are in fact growing faster initially. Therefore, if wave A is chosen to be the critical wave under Cebeci-Stewartson s criterion, one needs to check whether waves having span-wise wave numbers close to those of A can have larger amplification factor before A triggers the transition by a certain criterion. In a word, under the above situation, i.e. there are more than one wave satisfying Cebeci-Stewartson s criterion. As the situation is rather complicated, more analysis needs to be done for transition prediction. Figure 32 Evolution of fluctuating amplitude in spectral space. (a) x=100 110; (b) x=130 220. 3 Conclusions 1) For transition prediction methods based on linear stability theory, the one proposed by Cebeci & Stewartson has more sensible physical basis. Their criterion of choosing the critical wave under a given frequency is actually to choose the wave whose growth rate is a local maximum, and the paths of the integration of the growth rate agree very well with the propagation path of the peak of a wave packet with the chosen wave as its central wave. 2) If initially, there are more than one disturbance satisfying Cebeci-Stewartson s criterion, then one needs to deal with them more carefully. Each of them should be tested to see if it can trigger transition, not just to choose the most unstable one, because, as indicated in the above example, the initially less unstable one may become the most unstable one latter. Therefore, in such a case, one really needs to do a more comprehensive stability analysis at different locations first, as shown in Figure 27, to see the relative variation of the growth rate of waves with different span-wise wave numbers. Then test all possible candidates, which may trigger transition by a certain criterion. And if the chosen candidate wave does not keep having the local maximum growth rate as when x131 for the wave A in Figure 27(a),

962 Su C H Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 then its neighboring wave, i.e. wave with a very close wave number should also be tested to make sure that the wave triggering transition is indeed the most amplified one. The author is grateful to Professor ZHOU H. for his constant encouragement and meaningful discussions on problems that the author encountered during her work. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11002098 and 11332007). 1 Bertin J J, Cummings R M. Critical hypersonic aerothermodynamic phenomena. Annu Rev Fluid Mech, 2006, 38: 129 157 2 Su C H, Zhou H. Transition prediction of a hypersonic boundary layer over a cone at small angle of attack-with the improvement of e-n method. Sci China Ser G-Phys Mech Astron, 2009, 52(1): 115 123 3 Su C H, Zhou H. Transition prediction for supersonic and hypersonic boundary layers on a cone with angle of attack. Sci China Ser G-Phys Mech Astron, 2009, 52(8): 1223 1232 4 Malik M R. COSAL: A black box compressible stability analysis code for transition prediction in three-dimensional boundary layers. 1982, NASA CR 165925 5 Mack L M. Stability of transition prediction in three-dimensional boundary layers, on swept wings at transonic speeds. In: IUTAM Symposium, Toulouse, France, 1989 6 Arnal D, Casalis G, Juillen J C. Experimental and theoretical analysis of natural transition on infinite swept wing. In: IUTAM Symposium, Toulouse, France, 1989 7 Cebeci T, Stewartson K. Stability and transition in three-dimensional flows. AIAA J, 1980, (18): 398 405 8 Cebeci T, Shao J P, Chen H H, et al. The preferred approach for calculating transition by stability theory. In: Proceeding of International Conference on Boundary and Interior Layers, ONERA, Toulouse France, 2004 9 Fu D X, Ma Y W, Li X L, et al. DNS of Compressible Flow (in Chinese). Beijing: Science Press, 2010. 244 10 Zhou H, Li L. The eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation. Appl Math Mech, 1981, 2(3): 295 305