1 Introduction

With the progress of computer,numerical simulations are widely applied in the studies of flow dynamics. Numerical simulations can provide the nonlinear evaluation of disturbances in boundary layers,and are widely used in the research areas such as boundary layer recep- tivity and transition mechanism. Although numerical simulations can give the evaluation of disturbances,what we concern more is the evaluation of each disturbance mode and their in- tention relationship. For example,in the study of receptivity,the interesting disturbance is the unstable wave which is excited in the boundary layer by the background disturbances in the free-stream^{[1, 2]}. The amplitude of the unstable wave is smaller than that of the back- ground disturbance,and it is covered in the numerical results. Therefore,a method of mode decomposition is proposed to analyze the numerical results. To our knowledge,the solutions of the linearized Navier-Stokes equations can be expressed in terms of their eigenfunctions^[3]. According to the previous work,the eigenfunctions are divided into discrete modes and continuous spectra^[4]. Because of the non-self-adjointness of the linearized Navier-Stokes equations, the adjoint equations are introduced to realize the mode decomposition. The eigenfunctions of the linearized Navier-Stokes equations and their adjoint equations comprise the biorthogonal eigenfunction system^[5]. Fedorov and Tumin^[6] and Tumin^{[7, 8]} gave the formal solutions to the linearized Navier-Stokes equations by using the Fourier and Laplace transformations. Chiquete and Tumin^[9] found that these formal solutions could be recast as the eigenfunction expansion. Tumin et al.^[10] studied the receptivity of wall blowing-suction by a numerical simulation and gave a detailed description of the mode decomposition in analyzing the numerical results.

For the spatial mode,the eigenvalue problem is a polynomial nonlinear eigenvalue problem. The nonlinear eigenvalue problem can be transformed to a linear eigenvalue problem by defining new eigenfunctions^{[7, 8, 10]}. However,this may enlarge the order of the equations and make the form complexity. It is bad for computing. Therefore,the operator used in the inner product must be carefully considered when mode decomposition needs to be done. For nonlinear eigenvalue problems,how to choose the operator and its form is still a problem. In this paper,the orthogonal relationship is obtained from the eigenvalue problem of the original equations. The operator used in the inner product can be directly obtained from the orthogonal relationship. Then,the mode decomposition can be done for the arbitrary function. The method can be extended to an Nth-degree polynomial eigenvalue problem. Finally,the method is verified by the numerical simulations and the linear stability theory.

In this paper,the orthogonal relationship is obtained from the eigenvalue problem of the original equations. The operator used in the inner product can be directly obtained from the orthogonal relationship. Then,the mode decomposition can be done for the arbitrary function. The method can be extended to an Nth-degree polynomial eigenvalue problem. Finally,the method is verified by the numerical simulations and the linear stability theory.

2 Nonlinear eigenvalue problem in flow stability

The governing equations of the nonlinear eigenvalue problem originating from the flow sta- bility are the linearized Navier-Stokes equations. Considering the supersonic boundary layer, the normal coordinate is y,and the streamwise and spanwise coordinates are x. and z,respec- tively. The form of the equations is,where L₀,L₁,and L₂ are 5×5 matrices,which are also differential operators about y,∂x is the derivative about x,and.

The disturbances have the following form:

where ω is the frequency,β is the spanwise wavenumber,and α is the streamwise wavenumber. Substitute the above parameters into the linearized Navier-Stokes equations. Then,theequations about Ø can be obtained as follows:

where λ = iα. L(λ) is a second-order differential operator,which is also a quadratic polynomial about λ. The detail forms are in Appendix A. The boundary conditions are

Equation (1) and the boundary conditions in (2) are both homogeneous. For the spatial mode, ω and β are parameters,and λ is the eigenvalue. They constitute a quadratic polynomial nonlinear eigenvalue problem. The unstable wave in the boundary layer is one solution of it.

If the boundary conditions are changed to

then Eqs.(1) and (3) describe the effects of the acoustic wave in the free stream on the boundary layer.

3 Orthogonal relationship of nonlinear eigenvalue problem

According to the theory of eigenvalue problem,the eigenfunctions of the eigenvalue problem of an ordinary differential equation and its corresponding adjoint equation comprise a biorthog- onal eigenfunction system. The arbitrary function satisfying the boundary conditions can be decomposed as the sum of a series of eigenfunctions. The decomposition coefficients are avail- able by the inner product between the arbitrary function and adjoint eigenfunctions. In the complex vector function field,the inner product is defined as follows:

For the linear eigenvalue problem,the operator form is assumed as

where L₀ and L₁ are operators independent of λ. Assume that the arbitrary different eigenvalues λ_n and λ_m ,the eigenfunction φ_m ,and the adjoint eigenfunction ϕ_n are orthogonal,and

The inner product in Eq.(5) is related to the operator L₁ in the eigenvalue problem (4).

For the polynomial eigenvalue problem,L(λ) is a polynomial operator about λ. Especially, when the order is second,the problem is called a quadratic eigenvalue problem,such as Eq.(1). Equations (1) and (2) comprise the eigenvalue problem,and their adjoint equation is

The corresponding homogeneous boundary conditions are in Appendix A. According to the definition of the adjoint equations,the arbitrary complex vector functions f and g satisfy

Assume that λ_n and λ_m are eigenvalues,and λ_n ≠ λ_m . Their eigenfunctions are φ_n and φ_m,and the adjoint eigenfunctions are ϕ_n and ϕ_m . The process of deducing the orthogonal relationship is given below. By definition,

Then,

According to the above formula,the following equations can be obtained:

The result of Eq.(6) subtracting Eq.(7) is

Because λ_n ≠ λ_m ,

Equation (8) is the orthogonal relationship between the eigenfunction system [φ_n] and the adjoint eigenfunction system [φ_n]. For linear eigenvalue problem,L₂ ≡ 0,and Eq.(8) degenerates the orthogonal relationship (5) of the linear eigenvalue problem.

4 Eigenfunction expansion theorem for general functions

With the orthogonal relationship (8),the eigenfunction expansion theorem for general functions can be given. Assume that the arbitrary disturbance f′ can be expanded in the eigenfunction system [φ_n]. Then,the disturbance can be expressed in the following form:

The expansion theorem is the method of solving the expansion coefficients. According to the orthogonal relationship,we can give the method as follows.

Assume

Then,we arrive at the following formula:

According to the orthogonal relationship (6),Eq.(9) can be rewritten as follows:

Where

According to these coefficients,the function f₀ can be expanded in the eigenfunction system [φ_n]. Compared with the linear eigenvalue problem,Eq.(10) is related not only to f₀ but also to f₁ . It indicates that the expansion coefficients a_n are related not only to the local function but also to the variation of the function in x. This is the fundamental difference between the linear and nonlinear eigenfunction expansions.

The method can be expanded to an Nth-degree polynomial eigenvalue problem. Here,we give an example about quartic polynomial eigenvalue problems. Sometimes,for convenience, the linearized Navier-Stokes equations need to be eliminated,e.g.,the Orr-Sommerfeld (O-S) equation in incompressible flow ^[11] ,which is a quartic polynomial eigenvalue problem for the spatial mode. By elimination,the general form of the O-S equation is

where λ_n and λ_m are eigenvalues,and λ_n ≠ λ_m . φ_m is the eigenfunction,and ϕ_n is the eigenfunction of the adjoint equations. By use of the method in Section 2,the orthogonal relationship of the quartic polynomial eigenvalue problem can be written as follows:

According to the orthogonal relationship,for the decomposed function f₀ ,we define

Then,the decomposition coefficients are

5 Results and discussion

Various combinations of the unstable wave,fast acoustic wave,and slow acoustic wave are imposed at the inlet. Numerical simulations provide the evaluation of the disturbance. The obtained numerical results will be compared with those obtained by the linear stability theory.

In this paper,the free-stream Mach number Ma is 4.5,the gas parameters are chosen to be those at the 5km altitude,the free-stream temperature T_∞ is 255.7K,and the Reynolds number Re is 10⁵ . For the fast and slow acoustic waves,the frequency ω is 2.2,the normal wavenumber k is 2,and the spanwise wavenumber β is 0. Then,the streamwise wavenumbers of the fast and slow acoustic waves are

The wavenumber of the unstable wave with the frequency 2.2 is α_TS = (2.404,−0.047).

fast acoustic wave,and slow acoustic wave. The unstable wave belongs to the discrete modes which are the solutions to the eigenvalue problem of Eqs.(1) and (2),while the acoustic waves are continuous spectra which are the solutions to the boundary value problem of Eqs.(1) and (3). Because the feature of the solutions is different between the discrete modes and the continuous spectra,the behaviors of the solutions out of the boundary layer are quite different. The unstable wave tends to zero out of the boundary layer,while the fast and slow acoustic waves are oscillational.

Four different disturbances are imposed at the inlet,which are noted as Cases 1,2,3,and 4, respectively (see Table 1). Their amplitudes are all 1 × 10⁻⁴ . S means the slow acoustic wave, F means the fast acoustic wave,and TS means the unstable wave. Direct numerical simulations are carried out for the cases.

In Case 1,the numerical simulations of small disturbances of the unstable wave can be compared with the results of the linear stability theory (LST). Figure 2 shows that they agree very well. It also illustrates that our program is right. In Cases 2,3,and 4,the amplitudes of the unstable wave and the acoustic wave have the same order of magnitude. Besides,the unstable wave is covered in the numerical simulations. Therefore,the comparison between the numerical simulations and the LST cannot be done directly. With the help of the mode decomposition, the amplitude of the unstable wave can be easily extracted,and then the comparison can be easily done. Figure 2 shows that the amplitude of the extracted unstable waves obtained by the mode decomposition agrees well with that obtained by the LST. It indicates that the method of the mode decomposition which we simplify is available.

6 Conclusions

In this paper,the orthogonal relationship of the polynomial eigenvalue problem is given, and the mode decomposition is done for an arbitrary function. The decomposition coefficients are solved by the inner product between the eigenfunction of the adjoint equations and the decomposed function. Especially,the operator used in the inner product is given in a simple form. The examples indicate that the mode decomposition which we simplify is available to analyze direct numerical simulation results.

Appendix A

The linearized Navier-Stokes equations are as follows:

where

where u₀ and w₀ are the velocities of the mean flow in the x- and z-directions, respectively, and ρ₀ ,T₀ , µ₀ , κ₀ , and C v0 are the density, the temperature, the viscosity, the thermal, and the specific heat at constant-volume of the mean flow, respectively. The subscript “y” stands for the derivative of the variables in the y-direction. µ_T = dµ/dT, and p_e = 1/(γMa 2 ). γ is the specific heat ratio.

The adjoint equation is

where A_T , B_T , and C_T are the transposes of A, B, and C.

The boundary conditions satisfy