Linear stability of supersonic Couette flow of a molecular gas under the conditions of viscous stratification and excitation of the vibrational mode

The linear stability of viscous two-dimensional perturbations in the supersonic plane Couette flow of perfect and vibrationally excited gases is investigated. In both cases an alternative is considered so that the transport coefficients were taken either constant or dependent on the static flow temperature. The Sutherland model is used to take the temperature dependence of the shear viscosity into account. It is shown that “viscous” stratification increases considerably the flow stability as compared with the case of constant viscosity. At the same time, the simple constant viscosity model conserves all characteristic features of the development of viscous perturbations in the Sutherland model. The dissipation effect of excitation of the vibrational mode is conserved in taking the temperature dependence of the transport coefficients into account. For both models the corresponding increase in the critical Reynolds number is of approximately 12%.     

Critical Reynolds number of the couette flow of a vibrationally excited diatomic gas energy approach

An energy balance equation for plane-parallel flows of a vibrationally excited diatomic gas described by a two-temperature relaxation model is derived within the framework of the nonlinear energy theory of hydrodynamic stability. A variational problem of calculating critical Reynolds numbers Re_cr determining the lower boundary of the possible beginning of the laminar-turbulent transition is considered for this equation. Asymptotic estimates of Re_cr are obtained, which show the characteristic dependences of the critical Reynolds number on the Mach number, bulk viscosity, and relaxation time. A problem for arbitrary wave numbers is solved by the collocation method. In the realistic range of flow parameters for a diatomic gas, the minimum critical Reynolds numbers are reached on modes of streamwise disturbances and increase approximately by a factor of 2.5 as the flow parameters increase.     

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.     

Linear stability of the Couette flow of a vibrationally excited gas. 1. Inviscid problem

Stability of the Couette flow of a vibrationally excited diatomic gas with a parabolic profile of static temperature is studied within the framework of the linear theory. A set of explicit asymptotic estimates are obtained for inviscid disturbances described by a system of linearized equations of two-temperature gas dynamics. It is shown that the first Rayleigh condition (theorem) is satisfied for unstable modes, and the classification of inviscid modes into even and odd modes is valid. A generalized condition of the presence of an inflection point on the velocity profile, which is necessary for disturbances to evolve, is obtained. The sufficient condition in Howard’s semi-circle theorem is refined. Complex phase velocities of two-dimensional even and odd inviscid modes are numerically calculated as functions of the Mach number, degree of excitation of vibrational levels of energy, and characteristic relaxation time. In the Couette flow problem, in contrast to the case of a free shear layer, the growth rate of the most unstable second mode increases with increasing Mach number and tends to a certain limit for which an asymptotic expression in the form of an ordinary differential equation is obtained. The calculated results show that the effect of reduction of the growth rate on the background of the relaxation process is clearly expressed in the range of flow parameters considered.     

Instabilities of a gas-liquid flow between two inclined plates analyzed using the Navier–Stokes equations

The paper is devoted to a theoretical analysis of a counter-current gas-liquid flow between two inclined plates. We linearized the Navier–Stokes equations and carried out a stability analysis of the basic steady-state solution over a wide variation of the liquid Reynolds number and the gas superficial velocity. As a result, we found two modes of the unstable disturbances and computed the wavelength and phase velocity of their neutral disturbances varying the liquid and gas Reynolds number. The first mode is a “surface mode” that corresponds to the Kapitza's waves at small values of the gas superficial velocity. We found that the dependence of the neutral disturbance wavelength on the liquid Reynolds number strongly depends on the gas superficial velocity, the distance between the plates and the channel inclination angle for this mode. The second mode of the unstable disturbances corresponds to the transition to a turbulent flow in the gas phase and there is a critical value of the gas Reynolds number for this mode. We obtained that this critical Reynolds number weakly depends on both the channel inclination angle, the distance between the plates and the liquid flow parameters for the conditions considered in the paper. Despite a thorough search, we did not find the unstable modes that may correspond to the instability in frame of the viscous (or inviscid) Kelvin–Helmholtz heuristic analysis.     

Viscous and Inviscid Instabilities of Flow Along a Streamwise Corner

Here we consider the stability of flow along a streamwise corner formed by the intersection of two large flat plates held perpendicular to each other. Self-similar solutions for the steady laminar mean flow in the corner region have been obtained by solving the boundary layer equations for zero and nonzero streamwise pressure gradients. The stability of the mean flow is investigated using linear stability analysis. An eigensolver has been developed to solve the resulting linear eigenvalue problem either in a global mode to obtain an approximation to all the dominant eigenmodes or in a local mode to refine a particular eigenmode. The stability results indicate that the entire spectrum of two-dimensional and oblique viscous modes of a two-dimensional Blasius boundary layer is active in the case of a corner layer as well, but away from the cornerline. In a corner region of finite spanwise extent, the continuous spectrum of oblique modes degenerates to a discrete spectrum of modes of increasing spanwise wave number. The effect of the corner on the two-dimensional viscous instability is small and decreases the growth rate. The growth rate of outgoing oblique disturbances is observed to decrease, while the growth rate of incoming oblique disturbances is enhanced by the corner. This asymmetry between the outgoing and incoming viscous modes increases with increasing obliqueness of the disturbance. The instability of a zero pressure gradient corner layer is dominated by the viscous modes; however, an inviscid corner mode is also observed. The critical Reynolds number of the inviscid mode rapidly decreases with even a small adverse streamwise pressure gradient and the inviscid mode becomes the dominant one. Received 17 March 1998 and accepted 28 April 1999     

Inviscid instability of two-fluid free surface flow down an incline

The inviscid temporal stability analysis of two-fluid parallel shear flow with a free surface, down an incline, is studied. The velocity profiles are chosen as piecewise-linear with two limbs. The analysis reveals the existence of unstable inviscid modes, arising due to wave interaction between the free surface and the shear-jump interface. Surface tension decreases the maximum growth rate of the dominant disturbance. Interestingly, in some limits, surface tension destabilises extremely short waves in this flow. This can happen because of the interaction with the shear-jump interface. This flow may be compared with a corresponding viscous two-fluid flow. Though viscosity modifies the stability properties of the flow system both qualitatively and quantitatively, there is qualitative agreement between the viscous and inviscid stability analysis when the less viscous fluid is closer to the free surface.     

Energy Estimate of the Critical Reynolds Numbers in a Compressible Couette Flow. Effect of Bulk Viscosity

A variational problem of determining the critical Reynolds number of the laminar-turbulent transition is numerically solved within the framework of the nonlinear energy theory of stability of compressible flows. Stability of various modes in the Couette flow of a compressible gas is estimated by the method of collocations. It is demonstrated that the minimum critical Reynolds numbers in the range of the ratio of the bulk viscosity ηb to the shear viscosity η, which is realistic for diatomic gases, are reached for modes of streamwise disturbances. The critical Reynolds numbers increase as the bulk viscosity is increased in the interval η_b = 0-2η, with the maximum increase in the limit being approximately 30%.     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.     

Flutter of infinite elastic plates in the boundary-layer flow at finite Reynolds numbers

The stability of an infinite elastic plate in supersonic gas flow is investigated taking into account the presence of the boundary layer formed on the plate surface. The effect of viscous and temperature disturbances of the boundary layer on the behavior of traveling waves is studied at large but finite Reynolds numbers. It is shown that in the case of the small boundary layer thickness viscosity can have both stabilizing and destabilizing effect depending on the phase velocity of disturbance propagation.     

On Stability of Plane and Cylindrical Poiseuille Flows of Nanofluids

Stability of plane and cylindrical Poiseuille flows of nanofluids to comparatively small perturbations is studied. Ethylene glycol-based nanofluids with silicon dioxide particles are considered. The volume fraction of nanoparticles is varied from 0 to 10%, and the particle size is varied from 10 to 210 nm. Neutral stability curves are constructed, and the most unstable modes of disturbances are found. It is demonstrated that nanofluids are less stable than base fluids; the presence of particles leads to additional destabilization of the flow. The greater the volume fraction of nanoparticles and the smaller the particle size, the greater the degree of this additional destabilization. In this case, the critical Reynolds number significantly decreases, and the spectrum of unstable disturbances becomes different; in particular, even for the volume fraction of particles equal to 5%, the wave length of the most unstable disturbances of the nanofluid with particles approximately 20 nm in size decreases almost by a factor of 4.     

Investigations of time-growing instabilities in laminar separation bubbles

The occurrence of temporally growing unstable disturbances is investigated based on eigenvalues with zero group velocity from linear stability theory (LST) and compared with observations of upstream travelling disturbances obtained in a two-dimensional direct numerical simulation (DNS) of an unsteady laminar separation bubble. Numerical solutions of the Orr–Sommerfeld equation using analytically constructed base-flow velocity profiles modelled by a modified hyperbolic tangent function help to identify the role of parameters, such as maximum reverse flow, wall distance and intensity of the shear layer, as well as Reynolds number on the possibility that a true time-growing instability occurs. Then, the viscous or inviscid nature of the solutions found is classified on the basis of their eigenfunctions. For large wall distances two unstable modes are found. Apart from a low-frequency motion of the bubble the DNS exhibit high-frequency oscillations which periodically appear and disappear. Part of these disturbances travel upstream and amplify with respect to time. Their initial occurrence and their frequency are in excellent agreement with the results of the parameter study based on LST and a closer examination of the disturbances yields insight into their spatial structure.     

Effect of Heat Supply on the Stability of the Supersonic Swept Atiachment-Line Boundary Layer

The stability of a boundary layer with volume heat supply on the attachment line of a swept wing is investigated within the framework of the linear theory at supersonic inviscid-free-stream Mach numbers. The results of numerical calculations of the flow stability and neutral curves are presented for the flow on the leading edge of a swept wing with a swept angle χ=60° at various free-stream Mach numbers. The effect of volume heat supply on the characteristics of boundary layer stability on the attachment line is studied at a surface temperature equal to the temperature of the external inviscid flow. It is shown that in the case of a supersonic external inviscid flow volume heat supply may result in an increase in the critical Reynolds number and stabilization of disturbances corresponding to large wave numbers. For certain energy supply parameters the situation is reversed, the unstable disturbances corresponding to the main flow-instability zone are stabilized but another zone of flow-instability with small wave numbers and a significantly lower critical Reynolds number appears.     

Three-wave interactions of disturbances in a hypersonic boundary layer on a porous surface

Interactions of disturbances in a hypersonic boundary layer on a porous surface are considered within the framework of the weakly nonlinear stability theory. Acoustic and vortex waves in resonant three-wave systems are found to interact in the weak redistribution mode, which leads to weak decay of the acoustic component and weak amplification of the vortex component. Three-dimensional vortex waves are demonstrated to interact more intensively than two-dimensional waves. The feature responsible for attenuation of nonlinearity is the presence of a porous coating on the surface, which absorbs acoustic disturbances and amplifies vortex disturbances at high Mach numbers. Vanishing of the pumping wave, which corresponds to a plane acoustic wave on a solid surface, is found to assist in increasing the length of the regions of linear growth of disturbances and the laminar flow regime. In this case, the low-frequency spectrum of vortex modes can be filled owing to nonlinear processes that occur in vortex triplets.     

Effect of viscosity on the lift-drag ratio of a thin blunt wing at hypersonic rates of flow round it

The effect of viscosity on the carrying properties of hypersonic aircraft appears at great flight altitudes, where an important factor is the interaction of the laminar boundary layer with inviscid flow. In the present study the method of bands is used to make an approximate calculation of this effect for a regime of weak viscous interaction [1]. The results of [2] are used for conditions of inviscid flow round a body. The local coefficient of friction and coefficients of the additional pressure induced by the boundary layer are determined from the data for a plate of infinite width [3]. Simple relationships are obtained which make it possible to estimate the effect of viscosity on the magnitude of the maximum lift-drag ratio and the value of the angle of attack corresponding to it. The results are given of an experimental study of hypersonic flow round a plane triangular wing in a broad range of Reynolds numbers, and these confirm the relationships obtained and indicate the region in which they are applicable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 149–152, November–December, 1988.     

Localized disturbances in parallel shear flows

The development of localized disturbances in parallel shear flows is reviewed. The inviscid case is considered, first for a general velocity profile and then in the special case of plane Couette flow so as to bring out the key asymptotic results in an explicit form. In this context, the distinctive differences between the wave-packet associated with the asymptotic behavior of eigenmodes and the non-dispersive (inviscid) continuous spectrum is highlighted. The largest growth is found for three-dimensional disturbances and occurs in the normal vorticity component. It is due to an algebraic instability associated with the lift-up effect. Comparison is also made between the analytical results and some numerical calculations.Next the viscous case is treated, where the complete solution to the initial value problem is presented for bounded flows using eigenfunction expansions. The asymptotic, wave-packet type behaviour is analyzed using the method of steepest descent and kinematic wave theory. For short times, on the other hand, transient growth can be large, particularly for three-dimensional disturbances. This growth is associated with cancelation of non-orthogonal modes and is the viscous equivalent of the algebraic instability. The maximum transient growth possible to obtain from this mechanism is also presented, the so called optimal growth.Lastly the application of the dynamics of three dimensional disturbances in modeling of coherent structures in turbulent flows is discussed.     

Supersonic flow deceleration in plane and axisymmetric channels

Supersonic perfect gas flow in plane and axisymmetric channels with the same duct contour is studied on the basis of a numerical solution of the two-dimensional Navier-Stokes and Euler equations. The calculations were carried out at an inlet Mach number M_∞=4 for various Reynolds numbers and “bell-mouth“ half-angles. The effect of these parameters, as well as that of the flow three-dimensionality, on the flow pattern is demonstrated. In particular, the existence of viscous flow regimes providing the most effective supersonic flow deceleration and a higher degree of total pressure recovery as compared with the inviscid flow is established. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–152, March–April, 1998. The study was carried out with the support of the Russian Foundation for Fundamental Research (project No. 95-01-01129a).     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

The hypersonic Mach number independence principle in the case of viscous flow

The hypersonic Mach number independence principle of Oswatitsch is important for hypersonic vehicle design. It explains why, above a certain flight Mach number (M _∞ ≈ 4−6, depending on the body shape), some aerodynamic properties become independent of the flight Mach number. For ground test facilities this means that it is sufficient for the Mach number in the test section to be high enough, that Mach number independence exists. However, the principle was derived for calorically perfect gas and inviscid flow only. In this paper a theoretical study for blunt bodies in the case of viscous flow is presented. We provide numerical results which give insight into how attached viscous flow behaves at high Mach numbers. The flow past an axisymmetric configuration is analysed by applying a coupled Euler/second-order boundary-layer method. Wall boundaries are treated by assuming an adiabatic or radiation-adiabatic wall for laminar flow. Calorically perfect or equilibrium air is accounted for. Lift, drag, and moment coefficients, and lift-to-drag ratios are given for several combinations of flight Mach number and altitude, i.e. Reynolds number. For blunt bodies considered here, which are pressure dominated, Mach number independence occurs for the adiabatic wall, but not for the radiation-adiabatic wall assumption.     

Generalized reference enthalpy formulations and simulation of viscous effects in hypersonic flow

A generalized reference enthalpy formulation for the skin friction, heat transfer and radiation-equilibrium temperature distributions over aerodynamic surfaces in attached hypersonic / hyperenthalpic flow is proposed. The formulation, which has been extensively employed in various forms by numerous investigators in the perfect gas regime, has also been recently demonstrated to provide adequate estimates of the heat transfer distribution in thermochemically active high enthalpy flow conditions when coupled to thermochemically active Euler solutions. It is now used to reveal the relevant similitude parameters for viscous effects in hypersonic flow, and the importance of the temperature distribution across the boundary layer and of the temperature-viscosity relation. It is shown that, although reproduction of the flight total flow enthalpy as well as surface temperature is the obvious solution for full viscous simulation in (perfect gas) hypersonic flow, the hot surface testing requirement and, in a number of practical applications, also the hot flow requirement may be relaxed with reasonably small error that can be of the same order as the measurement accuracy in present-day hypersonic testing. This similitude error, however, may increase significantly in cases exhibiting strong viscous/inviscid interaction or when the laminar-turbulent transition process becomes important. In this respect, alternative full simulation solutions, which are less demanding in terms of reproduction of the high levels of flight freestream and surface temperature or even Reynolds number, are discussed. Received 6 May 1997 / Accepted 8 October 1997