Two-dimensional Compressible Boundary Layer Research Articles

Linear stability analysis of compressible boundary layer over an insulated wall using compound matrix method: Existence of multiple unstable modes for Mach number beyond 3

The linear spatial stability of a parallel two-dimensional (2D) compressible boundary layer on an adiabatic plate is investigated by considering both 2D and three-dimensional (3D) disturbances. The compound matrix method is employed here, for the first time, for compressible flows, which, unlike other conventional techniques, can efficiently eliminate the stiffness of the equations governing the spectral amplitudes. The method is first validated with published results in the literature corresponding to spatial and temporal instability of flows ranging from low subsonic to high supersonic Mach numbers (M), which shows a good match depending upon the proper choice of free-stream temperature and the wall dispersion relation. Subsequently, flow compressibility effects and the spanwise variation of disturbances are also investigated for M ranging from low subsonic to high supersonic cases (from M = 0.1 to 6). Mack (AGARD Report No. 709, 1984) reported the existence of two unstable modes for M > 3 from viscous calculations (the so-called “second mode”) that subsequently fuse to create only one unstable zone when M increases. Our calculations show a series of higher-order unstable modes for M > 3 in addition to the findings of Ma and Zhong [“Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions,” J. Fluid Mech. 488, 31–78 (2003)], where such higher-order modes for supersonic boundary layers are all noted to be spatially stable. The number and the frequency extent of the corresponding unstable zones increase with an increase in M beyond 3 while propagating downstream at a higher speed than those corresponding to incompressible, subsonic, and low supersonic (M < 2) cases.

Compressible Falkner–Skan–Cooke boundary layer on a flat plate

The incompressible Falkner–Skan–Cooke (FSC) boundary layer is extended to the compressible regime on a flat plate, aiming at establishing a generalized model to facilitate the crossflow-related investigations. The compressible FSC flow is also compatible with the two-dimensional compressible Blasius boundary layer characterized by a self-similarity solution. A family of boundary layer with local self-similarity is obtained from an ordinary differential equation system. Pressure-gradient parameter and local swept angle are introduced to parameterize the crossflow, which can vary in both direction and magnitude. The results reveal that the crossflow is oriented toward the curvature center of the inviscid streamline, regardless of the direction of pressure gradient. The existence of crossflow results in moderate directional deviation of skin friction, while tiny difference is observed for the magnitude. In comparison, much less effect is observed on the thermal properties of boundary layer. Crossflow also leads to the distortion of vorticity inside boundary layer, accompanied by the production of helicity. In addition, it is revealed that cross-flow strength varies monotonously with the pressure-gradient parameter, while it varies non-monotonously with the local swept angle. The effects of Mach number and wall temperature on crossflow are investigated as well.

Transient growth of Görtler vortices in two-dimensional compressible boundary layers. Application to surface waviness

This work is related to the numerical calculation of the maximum spatial growth of Görtler vortices using a set of linear partial differential equations derived to take curvature effects into account. The method is based on the calculation of the most amplified perturbation using a direct/adjoint iterative procedure. A new method for computing the optimal perturbation which attain largest growth at the shortest streamwise location is presented. The resulting neutral curve is shown to restore the concept of neutral curve for the Görtler problem, defining an envelope for the unstable domain. The present paper also highlights the potential for transient growth in the development of Görtler vortices. Compressible boundary layers over curved surfaces are considered in the framework of studying the effects of surface imperfections on laminar–turbulent transition. Non-negligible values of N-factor are shown to arise over the first streamwise locations due to interactions between non-normal modes. This result emphasizes the need to take transient growth into account when predicting transition location over wavy walls.

Linear stability analysis in compressible, flat-plate boundary-layers

The stability problem of two-dimensional compressible flat-plate boundary layers is handled using the linear stability theory. The stability equations obtained from three-dimensional compressible Navier–Stokes equations are solved simultaneously with two-dimensional mean flow equations, using an efficient shoot-search technique for adiabatic wall condition. In the analysis, a wide range of Mach numbers extending well into the hypersonic range are considered for the mean flow, whereas both two- and three-dimensional disturbances are taken into account for the perturbation flow. All fluid properties, including the Prandtl number, are taken as temperature-dependent. The results of the analysis ascertain the presence of the second mode of instability (Mack mode), in addition to the first mode related to the Tollmien–Schlichting mode present in incompressible flows. The effect of reference temperature on stability characteristics is also studied. The results of the analysis reveal that the stability characteristics remain almost unchanged for the most unstable wave direction for Mach numbers above 4.0. The obtained results are compared with existing numerical and experimental data in the literature, yielding encouraging agreement both qualitatively and quantitatively.

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.

Laminar diffusion flame in the reacting mixing layer

Laminar flows in which mixing and chemical reactions take place between parallel streams of reactive species are studied numerically. The governing equations for laminar flows are from two-dimensional compressible boundary-layer equations. The relevant chemistry is a finite rate single step irreversible reaction with Arrhenius kinetics. Ignition, premixed flame, and diffusion flame regimes are found to exist in the laminar reacting mixing layer at high activation energy. At high Mach numbers, ignition occurs earlier due to the higher temperatures in the unburnt gas. In diffusion regimes, property variations affect the laminar profiles considerably and, thus, need to be included when there are large temperature differences. The maximum temperature of a laminar reacting mixing layer is in the almost linear relationship with the adiabatic flame temperature at low heat release, but is only weakly linearly-correlated at high heat release.

Effect of a space-time source structure simulating a dielectric barrier discharge on the laminar boundary layer

The time-dependent pulse-periodic action of a surface electric discharge on a flat-plate laminar boundary layer is simulated theoretically. The effect of the discharge is estimated within the framework of the numerical solution of the boundary value problem for the time-dependent two-dimensional compressible boundary layer with additional terms in the momentum and energy conservation equations simulating the force and thermal action of the discharge on the gas flow with allowance for the pressure gradient across the boundary layer induced by the corresponding body force component. The effect of certain parameters of the problem formulated above on the gas velocity induced by the discharge in the boundary layer is also estimated.

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

Compressible boundary layer flow with non-uniform slot injection (or suction) over i) a cylinder and ii) a sphere

Non-similar solutions of steady two-dimensional and axi-symmetric compressible laminar boundary layer flows with non-uniform slot injection (or suction) have been obtained, from the starting point of the stream-wise co-ordinate to the exact point of separation. The difficulties arising at the starting point of the stream-wise co-ordinate, and at the edges of the slot and at the point of separation have been overcome by applying the method of quasi-linear implicit finite difference scheme, along with an appropriate selection of finer step size along the stream-wise direction. It is observed that the separation can be delayed by a non-uniform slot suction and also by moving the slot downstream but the effect of non-uniform slot injection is just the opposite. The increase of total enthalpy at the wall causes the separation to occur earlier while cooling delays it. The increase of Mach number shifts the point of separation upstream due to the adverse pressure gradient.

Choked flow of low density gas in a narrow parallel-plate channel with adiabatic walls

The continuum and slip choked flows of low-density air through a narrow parallel-plate channel with adiabatic walls are investigated by means of finite-difference numerical calculation, experiment and analytical solution. In the numerical calculation based on two-dimensional compressible viscous boundary layer equations, the previous method is improved to achieve a higher precision of the choked state by decoupling the pressure term from momentum equation. The numerical results of discharge coefficient, pressure distribution and surface temperature distribution agree well with the experimental data. As the upstream tank pressure drops, the numerical results of friction coefficient and recovery factor approach the asymptotic values for the incompressible developed flow.

Analysis of the secondary stability of compressible boundary layer flows over a two-dimensional plate

This paper used the Floquet's three-dimensional linear stability theory in the analysis of two-dimensional compressible boundary layer, a set of stability equations is constructed, the effect of three dimensional linear small perturbation on the two-dimensional compressible boundary layer transition is studied, and the effect of coming flow Ma number on growth and development of the subharmonics is calculated. It can be seen from the calculations, the effect caused by the interaction of two-dimensional and three-dimensional perturbation waves on the development of two-dimensional compressible laminar boundary layer.

Analysis and Computation of Extremum Points with Application to Boundary-Layer Stability

A method for analysis and computation of derivatives and extremum points of variable-coefficients differential eigenvalue problems is presented. The method utilizes the orthogonality of the adjoint eigenfunctions to the homogenous part of the once or more differentiated problem to derive an analytical expression for the rate of change of eigenvalue with respect to a free parameter. The extremum point can be analyzed and computed by setting and driving, respectively, the first rate of change of the eigenvalue with respect to the free parameters to zero. Higher order derivatives can be computed by solving, sequentially, sets of inhomogeneous two-point boundary value problems. The method is applied to analyze and compute the most amplified inviscid instability wave in two-dimensional compressible boundary layers and the most amplified viscous instability wave in three-dimensional incompressible boundary layers. It is shown analytically that while the most-amplified spatial instability wave in two-dimensional incompressible boundary layer is two dimensional, the corresponding most amplified wave in three-dimensional boundary layer is generally oblique. It is also shown analytically that the most-amplified disturbance in three-dimensional boundary layer is generally a traveling disturbance. Furthermore, it is shown analytically that the inviscid growth rate is an extremum point.

Laminar Diffusion Flame in the Reacting Mixing Layer

Laminar flows in which mixing and chemical reactions take place between parallel streams of reactive species are studied numerically. The governing equations for laminar flows are from two-dimensional compressible boundary-layer equations. The chemistry is a finite rate single step irreversible reaction with Arrhenius kinetics. Ignition, premixed flame, and diffusion flame regimes are found to exist in the laminar reacting mixing layer at high activation energy. At high Mach numbers, ignition occurs earlier due to the higher temperatures in the unburnt gas. In diffusion regimes, property variations affect the laminar profiles considerably and need to be included when there are large temperature differences. The maximum temperature of a laminar reacting mixing layer is almost linear with the adiabatic flame temperature at low heat release, but only weakly at high heat release.

Multiple Unstable Modes in the Reacting Mixing Layer

This paper investigates the linear stability of reacting mixing layers with special emphasis on the existence of multiple unstable modes. The governing equations for laminar flows are from two-dimensional compressible boundary-layer equations. The chemistry is a finite rate single step irreversible reaction with Arrhenius kinetics. For the incompressible reacintg mixing layer with variable density. A necessary condition for instability has been derived. The condition requires that the angular momentum, not the vorticity, to have a maximum in the flow domain. New inflectional modes of instability are found to exist in the outer part of the mixing layer. For the compressible reacting mixing layer, supersonic unstable modes may exist in the abscence of a generalized inflection point. The outer modes at high Mach numbers in the reacting mixing layer are continuations of the inflectional modes of low Mach number flows. However, the generalized inflection point is less important at supersonic flows.

Mean flow study of two-dimensional subsonic turbulent boundary layers

We present some mean flow data for subsonic compressible boundary layers obtained during the calibration of a newly built boundary layer wind tunnel. The measurements were carried out at zero pressure gradient and at subsonic speeds. The freestream Reynolds number ranged from 11×10 6 to 44×10 6 /m. Different boundary-layer profiles and parameters were examined and compared with the universally accepted data for two-dimensional compressible equilibrium turbulent boundary layers. The van Driest transformation was used to compare the data with the well-known incompressible correlations. The results together with the estimates of the wake component suggest that the tested turbulent boundary layer is fully developed, two dimensional, and in equilibrium condition

Nonlinear evolution of a first mode oblique wave in a compressible boundary layer: I. Heated/cooled walls

The nonlinear stability of an oblique mode propagating in a two-dimensional compressible boundary layer is considered under the long wavelength approximation. The growth rate of the wave is assumed to be small so that the ideas of unsteady nonlinear critical layers can be applied. It is shown that the spatial/temporal evolution of the mode is governed by a pair of coupled unsteady nonlinear equations for the disturbance vorticity and density. Expressions for the linear growth rate show clearly the effects of wall heating and cooling, and in particular how heating destabilizes the boundary layer for these long wavelength inviscid modes at O(1) Mach numbers. A generalized expression for the linear growth rate is obtained and is shown to compare very well for a range of frequencies and wave angles at moderate Mach numbers with full numerical solutions of the linear stability problem. The numerical solution of the nonlinear unsteady critical layer problem using a novel method based on Fourier decomposition and Chebyshev collocation is discussed and some results are presented

Effect of suction on the subharmonic instability of compressible boundary layers

The effect of self-similar suction distributions on the subharmonic instability of a two-dimensional (2-D) compressible boundary layer over an insulated flat plate is analyzed using the Floquet model. The resulting problem is solved numerically by using finite differences. Results are presented for different Mach numbers. For supersonic flows results for first- and second-mode primary waves are presented. It is found that when the primary wave is a first mode, suction stabilizes the subharmonic wave at low and moderate Mach numbers. However, at high Mach numbers suction loses its effectiveness. When the primary wave is a second mode, suction is also found to stabilize the subharmonic mode. When the primary wave is a first mode merging with a second mode, the subharmonic wave is strongly destabilized by suction.

Effect of heat transfer on the subharmonic instability of compressible boundary layers

The effect of heat transfer on the subharmonic instability of a two-dimensional compressible boundary layer over a flat plate is analyzed using the Floquet model. The resulting problem is solved numerically by using both finite differences and the computer code suport. Results are presented for different Mach numbers. For supersonic flows, results for the first and second modes are presented. The results for an adiabatic flat plate are in good agreement with the results of the numerical simulations of Thumm et al. [in Laminar-Turbulent Transition, edited by D. Arnal and R. Michel (Springer-Verlag, Berlin, 1990), p. 271; (private communication)]. When the wall temperature is decreased and the primary wave is a second-mode wave, it is found that the subharmonic instability is shifted to higher frequencies and downstream locations. When the primary wave is a first-mode wave, the effect of cooling might be stabilizing or destabilizing depending on the frequency. When the primary wave is a first mode, cooling stabilizes the subharmonic wave at low spanwise wave numbers and destabilizes it at higher spanwise wave numbers. When the primary wave is either a first- or a second-mode wave, the most amplified subharmonic mode shifts toward a higher spanwise wave number as the cooling level increases.

Triple-deck structure

The asymptotic foundation of two-dimensional steady triple-deck theory is reviewed. The total flow is assumed to be the sum of a basic-flow component Q( x, y) governed by the boundary-layer equations and a disturbance component q( x, y), resulting from a sudden streamwise change (localized disturbance), such as that caused by a hump or dip, a suction or blowing discontinuity, a heating or cooling discontinuity, a trailing edge etc. Substituting the total flow Q + q into the steady two-dimensional Navier-Stokes equations and subtracting the basic-flow quantitites yields nonlinear equations describing the disturbance quantities q. The disturbance quantities are assumed to vary with the streamwise scale X = ( x − 1)Re α , where x = x ∗¦x r ∗, x r ∗ is the center of the localized disturbance, Re = U ∞ ∗x r ∗/v , U ∞ ∗ is the freestream velocity and v is the kinematic viscosity of the fluid. Introducing the stretching transformation Y = yRe β , β > 0, and scaling ψ as ψ = ψRe − χ , where ψ is the streamfunction of the disturbance, we find that there are three distinguished limits when 0 < α < 1 2 and hence three sets of least degenerate problems, resulting in a triple-deck structure. These limits correspond to β = α, β = 1 2 and β = 1 2 + 1 3 α . Matching the pressure in these decks demands that α = 3 8 and leads to the triple-deck structure. The upper- and middle-deck problems are linear, whereas the lower-deck problem is linear when χ > 3 4 and nonlinear when χ ⩽ 3 4 . When α = 1 2 , there are two distinguished limits corresponding to β = 1 2 and 2 3 . The upper-deck problem is linear, whereas the lower-deck problem is linear when χ > 5 6 and nonlinear when χ ⩽ 5 6 . The theory is self-consistent and does not depend on the problem being investigated. Application of these theories to the linear three-dimensional compressible stability of two-dimensional compressible boundary layers is described and limitations of the triple-deck theory are discussed.

Subharmonic instability of compressible boundary layers

The subharmonic instability of a two-dimensional compressible boundary layer over an insulated flat plate is analyzed using the Floquet model. The resulting problem is solved numerically by using both finite differences and the computer code suport. Results are presented for subsonic, transonic, and supersonic flows. For supersonic flows results for the first and second modes are presented. The effects of Mach number, spanwise wave number, amplitude of the primary wave, Reynolds number, and frequency are studied.