Pointwise Estimates and Stability for Dispersive–Diffusive Shock Waves

We study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion. Along with the usual Lax shocks, such equations are known to admit undercompressive shocks. We unify the treatment of these two cases by introducing a new wave-tracking method based on “instantaneous projection”, giving improved estimates even in the Lax case. Another important feature connected with the introduction of dispersion is the treatment of a non-sectorial operator. An immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all L ^p norms, p≥ 1 for the Lax case and p > 1 for the undercompressive case. Our approach extends immediately to the case of certain scalar equations of higher order, and would also appear suitable for extension to systems. Accepted May 29, 2000?Published online November 16, 2000     

Acceleration covariance in isotropic hydromagnetic turbulence

The present paper deals with the turbulent flow of an incompressible, viscous and conducting fluid which is isotropic, spatially homogeneous. The expression for acceleration covariance is derived. The obtained result shows that the defining scalars α(r, t) and β(r, t) of the acceleration covariance in MHD turbulence depend on the defining scalars of Q _ij , H _ij , Π _ij and S _{ik, j} .     

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.     

Boundary conditions for flows with chemical reactions occurring on a surface

With the use of a solution of a model Boltzmann equation for a binary mixture in the Knudsen layer, we obtain the boundary conditions for the equations of gas dynamics when the reactionl _iA_il _jA_j (l _i molecules of A_i change intol _j molecules of A_j, and vice versa) is occurring on a surface. The boundary condition that we obtain differs from those that are usually applicable by the presence of terms of the same order. This confirms the conclusion arrived at by the authors in [1], where it was shown that if the Knudsen layer is left out of account, which is precisely what is usually done, it is impossible to obtain correct boundary conditions.Moscow. Translated from Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–138, January–February, 1972.     

One-Dimensional Stability of Viscous Strong Detonation Waves

Building on Evans-function techniques developed to study the stability of viscous shocks, we examine the stability of strong-detonation-wave solutions of the Navier-Stokes equations for reacting gas. The primary result, following [1, 17], is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the Zeldovich-von Neumann-Döring limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided that the underlying shock is stable. Finally, for completeness, we include the calculation of the stability index for a viscous shock solution of the Navier-Stokes equations for a nonreacting gas.     

Product inhibition in packed-bed immobilized enzyme reactors for complex processes: Modelling of two-substrate processes

An analytical model was developed for describing the performance of packed-bed enzymic reactors operating with two cosubstrates, and when one of the reaction products is inhibitory to the enzyme. To this aim, the compartmental analysis technique was used. The relevant equations obtained were solved numerically, and the effect of the main operational parameters on the reactor characteristics were studied.Notation C _infa,i ^sup* local concentration of products in the pores of stage i - C _j,i concentration of substrate j in the pores of stage i - D _infa ^sup* internal (pore) diffusion coefficient for the reaction product a - D _j internal (pore) diffusion coefficient of substrate j - J _infa,i ^sup* net flux of product a, taking place from the pores of stage i into the corresponding bulk phase - J _j,i net flux of substrate j, taking place from the bulk phase of stage i into the corresponding pores - K _b inhibition constant - K _m,1, K _m,2 Michaelis constants for substrate 1 and 2, respectively - K _q inhibition constant - n total number of elementary stages in the reactor - Q volumetric flow rate throughout the reactor - R _j,i, R _infa,i ^sup* local reaction rates in pores of stage i, in terms of concentration of substrate j and product a respectively - S _infa,i ^sup* , S _infa,i-1 ^sup* bulk concentration of the reaction product a, in the stages i and i — 1, respectively - S _j,0 concentration of substrate j in the reactor feed - S _j,i-1, S _j,i concentration of substrate j in the bulk phase leaving stages i — 1 and i, respectively - V total volume of the reactor - V _m maximal reaction rate in terms of volumetric units - y axial coordinate of the pores - y ₀ depth of the pores - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ^* dimensionless parameter, defined in Equation (22) - volumetric packing density of catalytic particles (dimensionless) - porosity of the catalytic particles (dimensionless) - V _infi ^sup* dimensionless concentration of reaction product in pores of stage i, defined in Equation (17) - j,i dimensionless concentration of substrate j in pores of stage i; defined in Equation (6) - _j,i-1, _j.i dimensionless concentration of substrate j in the bulk phase of stage i; defined in Equation (6) - dimensionless position along the pore; defined in Equation (6)     

Instability of rarefaction shocks in systems of conservation laws

It is well-known that rarefaction shocks are unstable solutions of nonlinear hyperbolic conservation laws. Indeed, for scalar equations rarefaction shocks are unstable in the class of smooth solutions, but for systems one can only say in general that rarefaction shocks are unstable in the larger class of weak solutions. (Here unstable refers to a lack of continuous dependence upon perturbations of the initial data.) Since stability in the class of weak solutions is not well understood, ([T, TE]), entropy considerations have played a leading role in ruling out shocks that violate the laws of physics. However, for non-strictly hyperbolic systems the analogy with the equations of gas dynamics breaks down, and general entropy or admissibility criteria for the variety of shocks which appear, (see, e.g., [IMPT]), are not known. In this paper we address the question of when the instability of a shock can be demonstrated within the class of smooth solutions alone. We show by elementary constructions that this occurs whenever there exists an alternative solution to the Riemann problem with the same shock data which consists entirely of rarefaction waves and contact discontinuities with at least one non-zero rarefaction wave. We show that for 2×2 strictly hyperbolic, genuinely nonlinear systems the condition is both necessary and sufficient. We show too that for the full 3×3 (Euler) equations of gas dynamics with polytropic equations of state, rarefaction shocks of moderate strength are unstable in the class of smooth solutions if and only if the adiabatic gas constant satisfies 1 < < 5/3 (see Theorem 8). More precisely, there is a constant y _*, 0 < y _* < 1, depending only on , such that if y _* p _lp _rp _l for 1-shocks, and if y _* p _rP _lp _r for 3-shocks (where p _r and p _l denote the pressures on both sides of the rarefaction shock), then the shock is unstable if and only if 1 < < 5/3. Thus for such shocks, the theory of the Riemann problem for polytropic gases in the range 1 < < 5/3 can be rigorously developed with a knowledge of the smooth solutions alone by using stability under smoothing as an admissibility criterion, rather than by using the classical entropy inequalities.     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

A Boltzmann based model for open channel flows

A finite volume, Boltzmann Bhatnagar–Gross–Krook (BGK) numerical model for one‐ and two‐dimensional unsteady open channel flows is formulated and applied. The BGK scheme satisfies the entropy condition and thus prevents unphysical shocks. In addition, the van Leer limiter and the collision term ensure that the BGK scheme admits oscillation‐free solutions only. The accuracy and efficiency of the BGK scheme are demonstrated through the following examples: (i) strong shock waves, (ii) extreme expansion waves, (iii) a combination of strong shock waves and extreme expansion waves, and (iv) one‐ and two‐dimensional dam break problems. These test cases are performed for a variety of Courant numbers (C_r), with the only condition being C_r≤1. All the computational results are free of spurious oscillations and unphysical shocks (i.e., expansion shocks). In addition, comparisons of numerical tests with measured data from dam break laboratory experiments show good agreement for C_r≤0.6. This reduction in the stability domain is due to the explicit integration of the friction term. Furthermore, BGK schemes are easily extended to multidimensional problems and do not require characteristic decomposition. The proposed scheme is second‐order in both space and time when the external forces are zero and second‐order in space but first‐order in time when the external forces are non‐zero. However, since all the test cases presented are either for zero or small values of external forces, the results tend to maintain second‐order accuracy. In problems where the external forces become significant, it is possible to improve the order of accuracy of the scheme in time by, for example, applying the Runge–Kutta method in the integration of the external forces. Copyright © 2001 John Wiley & Sons, Ltd.     

Concentration-dependent diffusion in a composite slab with and without chemical reactions

Concentration-dependent diffusion of solute in a composite slab is investigated. The complex diffusion problem can be described by a set of nonlinear diffusion equations which is coupled to each other through the nonlinear interfacial boundary conditions. A two-layer diffusion is illustrated and the coupled nonlinear diffusion equations are conveniently solved by the orthogonal collocation method. Numerical simulation of the example reveals many interesting diffusion characteristics which are quite different from those in a single slab diffusion system.Nomenclature a _j expansion coefficient - A _i,j element of collocation matrix - B _i,j element of collocation matrix - C _a , C _b surface concentration - C _i concentration in the ith layer - D _i diffusion coefficient in the ith layer - D _i0 diffusion coefficient at very low concentration - k _i reaction rate in the ith layer - K _i dimensionless reaction rate, k _i l _i ² c _a ^m–1 /D ₁₀ - l _i thickness of the ith layer - m order of chemical reaction - n order of the orthogonal polynomial approximation - P _j–1(x _i ) orthogonal polynomial of order j - t time - x _i coordinate of the ith layer - X _i dimensionless coordinate of the ith layer, x _i/l _i - ratio of diffusion coefficient at low concentration, D ₂₀/D ₁₀ - ratio of thicknesses of layer, l ₁/l ₂ - _i dimensionless parameter in the concentration-dependent function of the ith layer - ratio of surface concentration, C _b /C _a - dimensionless time, tD ₁₀/l ₁ ² - _i dimensionless concentration in the ith layer, C _i /C _a      

Experimental study and direct numerical simulation of the evolution of disturbances in a viscous shock layer on a flat plate

The evolution of disturbances in a hypersonic viscous shock layer on a flat plate excited by slow-mode acoustic waves is considered numerically and experimentally. The parameters measured in the experiments performed with a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵ are the transverse profiles of the mean density and Mach number, the spectra of density fluctuations, and growth rates of natural disturbances. Direct numerical simulation of propagation of disturbances is performed by solving the Navier-Stokes equations with a high-order shock-capturing scheme. The numerical and experimental data characterizing the mean flow field, intensity of density fluctuations, and their growth rates are found to be in good agreement. Possible mechanisms of disturbance generation and evolution in the shock layer at hypersonic velocities are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 3–15, September–October, 2006.     

Effect of hydrodynamic forces on the pressure-difference equation

Because of the influence of hydrodynamic forces, the difference in macroscopic pressure which exists, at static equilibrium, between two immiscible phases located in a porous medium may be different from that which pertains during flow. In this paper, the concept of relative pressure difference, together with a new pressure-difference equation, is used to investigate the impact that the hydrodynamic forces have on the difference in macroscopic pressure which pertains when two immiscible fluids flow simultaneously through a homogeneous, water-wet porous medium. This investigation reveals that, in general, the equation defining the difference in pressure between two flowing phases must include a term which takes proper account of the hydrodynamic effects. Moreover, it is pointed out that, while neglect of the hydrodynamic effects introduces only a small amount of error when the two fluids are flowing cocurrently, such neglect is not permissible during steady-state, countercurrent flow. This is because failure to include the impact of the hydrodynamic effects in the latter case makes it impossible to explain the pressure behaviour observed in steady-state, countercurrent flow. Finally, the results of this investigation are used as a basis for arguing that, during steady-state, countercurrent flow, saturation is uniform, as is the case of steady-state, cocurrent flow.Roman Letters a parameter in Equation (18) - k absolute permeability, m² - k _i effective permeability to phasei;i=1, 2, m² - k _ij generalized effective permeability for phasei;i, j=1, 2, m² - p _d p ₂–p ₁=difference in macroscopic pressure between two flowing phases, N/m² - p _i pressure for phasei;i=1, 2, N/m² - p _h hydrodynamic contribution to difference in macroscopic pressure which exists during flow, N/m² - P _c macroscopic static capillary pressure, N/m² - R ₁₂ function defined by Equation (18) - S _i saturation of phasei;i=1, 2 - S _n normalized saturation of phase 1 - t time, s - u _i flux of phasei;i=1, 2m³/m²/s - x distance in direction of flow, m Greek Letters _R relative pressure difference - _i k _i / _i =mobility of phasei;i=1, 2m²/Pa·s - _ij k _ij / _j =generalized mobility of phasei;i, j=1, 2m²/Pa·s - _i viscosity of phasei;i=1, 2, Pa·s - porosity     

Optimization of a shock-wave system

The problem of a system consisting of n+m shock waves which realizes the maximum dynamic pressure is solved for given Mach numbers ahead of the first and the closing shocks provided that the sum of the flow turning angles in the last m waves is equal to the sum of the turning angles in the initial n waves minus the angle of attack. The corresponding necessary conditions of optimality of this shock-wave system, which constitutes a system of nonlinear algebraic equations, are obtained. An efficient iteration method of solving this system of equations, which makes it possible to solve the problem with high accuracy, is developed. The results of solving the problem make it possible to select the optimum supersonic air intake configuration in the preliminary design stage and in the case of a large number of shocks to estimate the limiting air intake parameters and the processes taking place in the air intake.     

An experimental investigation of the sonic criterion for transition from regular to Mach reflection of weak shock waves

The signal speed, namely the local sound speed plus the flow velocity, behind the reflected shocks produced by the interaction of weak shock waves (M _i < 1.4) with rigid inclined surfaces has been measured for several shock strengths close to the point of transition from regular to Mach reflection. The signal speed was measured using piezo-electric transducers, and with a multiple schlieren system to photograph acoustic signals created by a spark discharge behind a small aperture in the reflecting surfaces. Both methods yielded results with equal values within experimental error. The theoretical signal speeds behind regularly reflected shocks were calculated using a non-stationary model, and these agreed with the measured results at large angles of incidence. As the angle of incidence was reduced, for the same incident shock Mach number, so as to approach the point of transition from regular to Mach reflection, the measured values of the signal speed deviated significantly from the theoretical predictions. It was found, within experimental uncertainty, that transition from regular to Mach reflection occurred at the experimentally observed sonic point, namely, when the signal speed was equal to the speed of the reflection point along the reflecting surface. This sonic condition did not coincide with the theoretical value.     

Equilibrium states in homogeneous turbulence with baroclinic instability

The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ?U _i /?x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise ${(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)}The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ∂U _i /∂x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise (?B/?x₂ = N_h² = -2WS){(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)} and vertical (?B/?x₃ = N_v²){(\partial B{/}\partial x_3\,{=}\,N_v^2)} directions. Computations based on the rapid distortion theory (RDT) are performed for several values of the rotation number R = 2Ω/S and the Richardson number R_i = N_v²/S² < 1{R_i\,{=}\,N_v^2/S^2 <1 }. It is shown that, during an initial phase, the energies and the buoyancy fluxes are sensitive to the effects of pressure and viscosity. At large time, the ratios of energies, as well as the normalized fluxes, evolve to an asymptotically constant value, while the pressure–strain correlation scaled with the product of the turbulent kinetic energy by the shear rate approaches zero. Accordingly, an analytical parametric study based on the “pressure-less” approach (PLA) is also presented. The analytical study indicates that, when R _i  < 1, there is an exponential instability and equilibrium states of turbulence, in agreement with RDT. The energies and the buoyancy fluxes grow exponentially for large times with the same rate (γ in St units). The asymptotic value of the ratios of energies yielded by RDT is well described by its PLA counterpart derived analytically. At R _i  = 0, the asymptotic value of γ increases with increasing R approaching 2 for high rotation rates. At low rotation rates, an important contribution to the kinetic energy comes from the streamwise kinetic energy, whereas, at high rotation rates, the contribution of the vertical kinetic energy is dominant. When 0 < R _i  < 1 and R 1 0{R\ne 0}, the asymptotic value of γ decreases as R _i increases so as it becomes zero at R _i  = 1.     

Determination of a system consisting of 2<Emphasis Type="Italic">n</Emphasis> shock pairs realizing the total pressure maximum

The problem of a symmetric system consisting of 2n pairs of intersecting shock waves in a plane breaking duct which realizes the maximum total pressure is solved for given Mach numbers upstream of the leading shocks and downstream of the closing shocks provided that in each pair consisting of impinging and reflected waves the flow turning angles are equal in absolute values and have opposite directions. The corresponding necessary conditions of optimality of this shock-wave system, which constitutes a system of nonlinear algebraic equations, are obtained. An efficient iteration method of solving this system of equations, which makes it possible to solve the above-mentioned problem with high accuracy, is developed. An approximate analytic solution is obtained for large n. The results of solving the problem make it possible to select the optimum configuration of the plane internal-compression duct.     

Nonstationary pandom vibpation analysis of linear elastic structures with finite element method

At present, the finite element method is an efficient method for analyzing structural dynamic problems. When the physical quantities such as displacements and stresses are resolved in the spectra and the dynamic matrices are obtained in spectral resolving form, the relative equations cannot be solved by the vibration mode resolving method as usual. For solving such problems, a general method is put forward in this paper. The excitations considered with respect to nonstationary processes are as follows: P(t)={P_i(t)},P_i(t)=a_i(t)P_i(t), a_i(t) is a time function already known. We make Fourier transformation for the discretized equations obtained by finite element method, and by utilizing the behaviour of orthogonal increment of spectral quantities in random process^[1], some formulas of relations about the spectra of excitation and response are derived. The cross power spectral denisty matrices of responses can be found by these formulas, then the structrual safety analysis can be made. When a_i(t)=l (i= 1,2,…n), the. method stated in this paper will be reduced to that which is used in the special case of stationary process.     

Estimation of three-phase relative permeabilities

Starting from the statistical structural model of Alemánet al. (1988), we have developed an alternative to Stone's (1970, 1973; Aziz and Settari, 1979) methods for estimating steady-state, three-phase relative permeabilities from two sets of steady-state, two-phase relative permeabilities. Our result reduces to Stone's (1970; Aziz and Settari, 1979) first method, when the steady-state, two-phase relative permeability of the intermediate-wetting phase with respect to either the wetting phase or the nonwetting phase is a linear function of the saturation of the intermediate-wetting phase. As the curvature of either of these relative permeability functions increases, the deviation of our result from Stone's (1970; Aziz and Settari, 1979) first method increases. Currently, there are no data available that are sufficiently complete to form the basis of a comparison between our result and either of the methods of Stone (1970, 1973; Aziz and Settari, 1979).Notation a free parameter in Equation (19) - B(m, n) Beta function defined by Equation (17) - F ^(w), F^(nw) defined by Equations (31) and (27), respectively - G ⁽ⁱ⁾ defined by Equations (37) and (39) - H ⁽ⁱ⁾ defined by Equations (38) and (40) - k ⁽ⁱ⁾ three-phase relative permeability fo phasei - k ^(i)* defined by Equations (34) through (36) - k ^(i,j) relative permeability to phasei during a two-phase flow with phasej, possibly in the presence of an immobile phase - k ^(i,j)* defined by analogy with Equations (41) and (42) - k ^(i,j)** defined by Equations (49), (50), (53), and (54) - k _max ⁽ⁱ⁾ defined by Equation (11) - k ₁₉₇₀ ^(iw) defined by Equation (10) - k ₁₉₇₃ ^(iw)* defined by Equation (58) - k ₁₉₇₃ ^(iw) defined by Equation (13) - L length and diameter of cylindrical averaging surfaceS - L _t length of an individual capillary tube enclosed byS - L _t ^* defined by Equation (19) - L _t,min length of pore whose radius isR _max - N total number of pores contained within the averaging surfaceS - p ₁ ⁽ⁱ⁾ ,p ₂ ⁽ⁱ⁾ pressure of phasei at entrance and exit of averaging surfaceS, respectively - p defined by Equation (21) - p _c ^(i,j) capillary pressure function - p _c ^(i,j)* defined by Equations (23), (29), and (32) - p ⁽ⁱ⁾ intrinsic average of pressure within phasei defined by Alemánet al. (1988) - R pore radius - R ^* defined by Equation (18) - R _max maximum pore radius that occurs withinS - s ⁽ⁱ⁾ local saturation of phasei - s ^(i)* defined by Equation (7) - s _min ⁽ⁱ⁾ minimum or immobile saturation of phasei - S averaging surface introduced in local volume averaging - V ⁽ⁱ⁾ volume of phasei occupying the pore space enclosed byS Greek Letters , parameters in the Beta distribution defined by Equation (16) - ^(w), ^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (6) - ^(i,j) interfacial tension between phasesi andj - (x) Gamma function - defined by Equation (57) - , spherical coordinates in system centered upon the axis of the averaging surfaceS - _max maximum value of , 45 °, in view of assumption (9) - ^(i,j) contact angle between phasesi andj measured through the displacing phase - ^(w),^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (12) Other gradient operator Amoco Production Company, PO Box 591 Tulsa, OK 74102, U.S.A.     

Viscous Profiles for Shock Waves in Isothermal Magnetohydrodynamics

We prove existence of viscous profiles for slow and fast shock waves in isothermal magnetohydrodynamics with a general pressure law both using the Conley index and connection matrices.