Biorthogonal Eigenfunction System Research Articles

Biorthogonal decomposition of the disturbance flow field generated by particle impingement on a hypersonic boundary layer

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.

Receptivity and stability of hypersonic leading-edge sweep flows around a blunt body

Abstract

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.

Numerical Simulation and Theoretical Analysis of Perturbations in Hypersonic Boundary Layers

Direct numerical simulations of receptivity in a boundary layer over a flat plate and a sharp wedge were carried out with two-dimensional perturbations introduced into the flow by periodic-in-time blowing-suction through a slot. The free stream Mach numbers are equal to 5.92 and 8 in the cases of adiabatic flat plate and sharp wedge, respectively. The perturbation flow field was decomposed into normal modes with the help of the multimode decomposition technique based on the spatial biorthogonal eigenfunction system. The decomposition allows filtering out the stable and unstable modes hidden behind perturbations having another physical nature.

Coupled dynamic thermoviscoelasticity problem

We construct a closed analytic solution of a coupled dynamic thermoviscoelasticity problem for bodies of canonical shape. The solution is represented in the form of spectral expansions in a biorthogonal eigenfunction system of the nonself-adjoint pencil of differential operators generated by the problem under study. The spectral expansions are obtained with the help of a special class of nonsymmetric integral transforms.

Three-dimensional spatial normal modes in compressible boundary layers

Three-dimensional spatially growing perturbations in a two-dimensional compressible boundary layer are considered within the scope of linearized Navier–Stokes equations. The Cauchy problem is solved under the assumption of a finite growth rate of the disturbances. It is shown that the solution can be presented as an expansion into a biorthogonal eigenfunction system. The result can be used in a decomposition of flow fields derived from computational studies when pressure, temperature, and all the velocity components, together with some of their derivatives, are available. The method can also be used if partial data are available when a priori information may be utilized in the decomposition algorithm.

Direct numerical simulation and the theory of receptivity in a hypersonic boundary layer

Direct numerical simulation of receptivity in a boundary layer over a sharp wedge of half-angle 5.3degrees is carried out with two-dimensional perturbations introduced into the flow by periodic-in-time blowing-suction through a slot. The freestream Mach number is equal to 8. The perturbation flow field downstream from the slot is decomposed into normal modes with the help of the biorthogonal eigenfunction system. Filtered-out amplitudes of two discrete normal modes and of the fast acoustic modes are compared with the linear receptivity problem solution. The examples illustrate how the multimode decomposition technique may serve as a tool for gaining insight into computational results.

Three-dimensional wave packets in a compressible boundary layer

A three-dimensional wave packet generated by a local disturbance in a two-dimensional hypersonic boundary layer flow is studied with the aid of the previously solved initial-value problem. The solution to this problem can be expanded in a biorthogonal eigenfunction system as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. A specific disturbance consisting of an initial temperature spot is considered, and the receptivity to this initial temperature spot is computed for both the two-dimensional and three-dimensional cases. Using previous analysis of the discrete and continuous spectrum, the inverse Fourier transform is computed numerically. The two-dimensional inverse Fourier transform is calculated for two discrete modes: Mode F and Mode S. The Mode S result is compared with an asymptotic approximation of the Fourier integral, which is obtained using the Gaussian model as well as the method of steepest descent. It is shown that the method of steepest descent provides an excellent approximation to the more computationally intensive numerical evaluation of the inverse Fourier transform. Additionally, the three-dimensional inverse Fourier transform is found using an asymptotic approximation of the Fourier integral. A main feature of the resulting three-dimensional wave packet is its two-dimensional nature, which arises from an association of Mode S with Mack’s second mode.

Biorthogonal Eigenfunction System in the Triple‐Deck Limit

The solutions of receptivity problems for a periodic‐in‐time actuator placed on the wall in a two‐dimensional boundary layer and for a two‐dimensional hump are discussed within the scope of the biorthogonal eigenfunction expansion technique in the limit of high Reynolds number when the triple‐deck scaling is imposed. It is shown that the solutions obtained with the help of the biorthogonal eigenfunction system are equivalent to the solutions derived within the scope of the triple‐deck theory.

Receptivity of a boundary-layer flow to a three-dimensional hump at finite Reynolds numbers

The receptivity of boundary-layer flow to a three-dimensional hump (an array of humps) at a finite Reynolds number is solved with the help of an expansion of the linearized solution of Navier-Stokes equations into the biorthogonal eigenfunction system. There are two counterrotating vortices behind the roughness element that bring the high-speed fluid down into the wake region. Depending on the geometry, two relatively high-speed streaks could be observed in the wake downstream from the hump. The results for the flow-field structure are in qualitative agreement with available computational data. The quantitative discrepancy is attributed to the nonlinear character of the receptivity mechanism at the parameters considered in the computational studies.

Multimode Decomposition in Compressible Boundary Layers

Two-dimensional spatially growing perturbations in a two-dimensional compressible boundary layer are considered within the scope of linearized Navier-Stokes equations. Because a spatially growing solution can be expanded into a biorthogonal eigenfunction system, the latter can be utilized for decomposition of flow fields derived from computational studies when pressure, temperature, and all velocity components, together with their derivatives, are available. The method can be used also in a case where partial data are available when a priori information leads to consideration of a finite number of modes. Nomenclature (, , ) x y t A = vector-function of 9 components j A = j-th component of vector A D = d dy 2/ 3 e = ratio of the bulk viscosity to the dynamic viscosity , ij ij

Multimode decomposition of spatially growing perturbations in a two-dimensional boundary layer

Three-dimensional spatially growing perturbations in a two-dimensional incompressible boundary layer are considered within the scope of linearized Navier–Stokes equations. The Cauchy problem is solved under the assumption of a finite growth rate of the disturbances. It is shown that the solution can be presented as an expansion into a biorthogonal eigenfunction system. The result can be utilized for decomposition of flow fields derived from computational studies when pressure and all velocity components, together with their derivatives, are available. The method can be used also in a case where partial data are available when a priori information leads to consideration of a finite number of modes. In the case of a continuous spectrum, the problem of decomposition based on partial information is ill-posed, but the method might be applied under additional assumptions about the perturbations.

Instability and Receptivity of Laminar Wall Jets

Results of eigenvalue analysis based on global and local eigenvalue considerations are presented. A collocation method with the Chebyshev polynomial approximation has been used for the global eigenvalue analysis. The results explain the appearance of a second unstable mode. In the case of real frequencies with Reynolds number R 381 they become separated by interchannging their branches, then the second unstable mode occurs. The receptivity problem has been considered with respect to perturbations emanating from a wall. The results illustrate that high-frequency modes have a stronger response than low-frequency modes. It is shown that the method of expansion in a biorthogonal eigenfunction system and the method used by Ashpis and Reshotko are equivalent with regard to the receptivity problem solution.

A normal multimode decomposition method for stability experiments

A method for decomposition of an experimental signal that contains several normal modes with a prescribed frequency, is proposed. The method is based on an expansion of solutions for small-amplitude spatially growing disturbances in a biorthogonal eigenfunction system. It is shown that data of one velocity component at a given cross-stream section of the flow are in some cases, sufficient for the decomposition. The method is applied successfully to analyze small-amplitude periodic disturbances in a transitional wall-jet flow.

Receptivity of pipe Poiseuille flow

The receptivity problem is considered for pipe flow with periodic blow–suction through a narrow gap in the pipe wall. Axisymmetric and non-axisymmetric modes (1, 2, and 3) are analysed. The method of solution is based on global eigenvalue analysis for spatially growing disturbances in circular pipe Poiseuille flow. The numerical procedure is formulated in terms of the collocation method with the Chebyshev polynomials application. The receptivity problem is solved with an expansion of the solution in a biorthogonal eigenfunction system, and it was found that there is an excitation of many eigenmodes, which should be taken into account. The result explains the non-similar character of the amplitude distribution in the downstream direction that was observed in experiments.

Three-wave non-linear interaction in a three-dimensional compressible boundary layer

A perturbation method for analysis of non-linear wave interaction in three-dimensional compressible boundary layer is developed. The method is based on a biorthogonal eigenfunction system for three-dimensional compressible boundary layers. It is assumed that characteristic space and time scales of the disturbances are much less than space and time scales of non-linear development and an averaging technique in intermediate scales may be applied. The method is a generalization of Zelman's results for two-dimensional incompressible boundary layer. As an example, a three-wave interaction is considered. A numerical example for so-called Tollmien — Schlichting wave interaction shows the possibility of amplification for rather broad packets without an exact resonance synchronizm for three-wave interaction.