Asymptotic description of incipient separation bubble bursting

The appearance of short laminar separation bubbles in high Reynolds number (Re) wall bounded flows due to appropriate adverse pressure gradient conditions is usually associated with minor effects on global flow properties (e.g. lift force). However, localized reverse flow regions are known to react very sensitively to perturbations and in further consequence may trigger the laminar-turbulent transition process or even cause global separation. The present investigation of marginally separated boundary layer flows is based on an asymptotic approach Re → ∞. Special emphasis is placed on solutions of the corresponding model equations which blow up within finite time indicating the ejection of a vortical structure and the emergence of shorter spatio-temporal scales reminiscent of the early transition scenario (‘ bubble bursting’ ). Within the framework of marginal separation theory, an alternative adjoint operator method is used to formulate evolution equations governing the viscous-inviscid interaction process in leading and higher order correction required for the study of later stages of the flow development. Their blow up structure specifies the initial condition of and the match to the subsequent triple deck stage. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On higher order effects in marginally separated flows

If the angle of attack α of a slender airfoil reaches a critical value α_s flow separation is known to occur at the upper s surface. Further increase of α initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field – a phenomenon known as marginal separation – but then rather rapidly causes a severe change of the flow behaviour, leading to leading edge stall. According to the asymptotic theory of marginal separation holding in the limit of large Reynolds numbers Re, the flow in the neighbourhood of the separation bubble is governed by an integro-differential equation. This so-called interaction equation contains a single controlling parameter which relates the angle of attack to the Reynolds number, with a value Γ_s corresponding to α_s. Some recent results concerning higher order s s corrections to this theory and their effect on the stability of steady solutions will be presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subsonic triple deck flow past a flat plate with an elastic stretch

The steady laminar subsonic flow past a flat plate having a stretch of an elastic membrane, the pressure on the other side of which is adjustable, is studied within the framework of the triple deck theory. The resulting lower deck problem is supplemented with a membrane equation relating the pressure difference across the membrane to its curvature. By pressurizing or depressurizing the membrane, it assumes the form of a hump or a dent that alters the flow characteristics. Numerical solutions obtained, in either case, give plausible account of the interaction between the membrane and the flow.     

Existence of the stationary solution of a Rayleigh-type equation

A fluid flow along a semi-infinite plate with small periodic irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure: a thin boundary layer (“lower deck”) and a classical Prandtl boundary layer (“upper deck”). The aim of this paper is to prove the existence and uniqueness of the stationary solution of a Rayleigh-type equation, which describes oscillations of the vertical velocity component in the classical boundary layer.     

High performance computation for DNS/LES

The paper is focused on high-order compact schemes for direct numerical simulation (DNS) and large eddy simulation (LES) for flow separation, transition, tip vortex, and flow control. A discussion is given for several fundamental issues such as high quality grid generation, high-order schemes for curvilinear coordinates, the CFL condition for complex geometry, and high-order weighted compact schemes for shock capturing and shock–vortex interaction. The computation examples include DNS for K-type and H-type transition, DNS for flow separation and transition around an airfoil with attack angle, control of flow separation by using pulsed jets, and LES simulation for a tip vortex behind the juncture of a wing and flat plate. The computation also shows an almost linear growth in efficiency obtained by using multiple processors.     

Theory of the hypersonic viscous shock layer at high Reynolds numbers and intensive injection of foreign gases : PMM vol. 38, n 6, 1974, pp. 1015–1024

The hypersonic flow around smooth blunted bodies in the presence of intensive injection from the surface of these is considered. Using the method of external and internal expansions the asymptotics of the Navier-Stokes equations is constructed for high Reynolds numbers determined by parameters of the oncoming stream and of the injected gas. The flow in the shock layer falls into three characteristic regions. In regions adjacent to the body surface and the shock wave the effects associated with molecular transport are insignificant, while in the intermediate region they predominate. In the derivation of solution in the first two regions the surface of contact discontinuity is substituted for the region of molecular transport (external problem). An analytic solution of the external problem is obtained for small values of parameters ₁ = ρ_∞/ρs^* and δ = ρ_ω^*1/2ν_ω^*/ρ_∞1/2ν_∞, in the form of corresponding series expansions in these parameters. Asymptotic formulas are presented for velocity profiles, temperatures, and constituent concentration across the shock layer and, also, the shape of the contact discontinuity and of shock wave separation. The derived solution is compared with numerical solutions obtained by other authors. The flow in the region of molecular transport is defined by equations of the boundary layer with asymptotic conditions at plus and minus infinity, determined by the external solution (internal problem). A numerical solution of the internal problem is obtained taking into consideration multicomponent diffusion and heat exchange. The problem of multicomponent gas flow in the shock layer close to the stagnation line was previously considered in [1] with the use of simplified Navier-6tokes equations.The supersonic flow of a homogeneous inviscid and non-heat-conducting gas around blunted bodies in the presence of subsonic injection was considered in [2–7] using Euler's equations. An analytic solution, based on the classic solution obtained by Hill for a spherical vortex, was derived in [2] for a sphere on the assumption of constant but different densities in the layers between the shock wave and the contact discontinuity and between the latter and the body. Certain results of a numerical solution of the problem of intensive injection at the surface of axisymmetric bodies of various forms, obtained by Godunov's method [3], are presented. Telenin's method was used in [4] for numerical investigation of flow around a sphere; the problem was solved in two formulations: in the first, flow parameters were determined for the whole of the shock layer, while in the second this was done for the sutface of contact discontinuity, which was not known prior to the solution of the problem, with the pressure specified by Newton's formula and flow parameters determined only in the layer of injected gases. The flow with injection over blunted cones was numerically investigated in [5] by the approximate method proposed by Maslen. The flow in the shock layer in the neighborhood of the stagnation line was considered in [6, 8], and intensive injection was investigated by methods of the boundary layer theory in [8–12].     

The generalized faro shuffle

The standard faro shuffle, an idealized riffle shuffle, divides the deck into two equal portions, and perfectly interlaces them. The simple cut takes one card from the top to the bottom of the deck. It is known that for decks of even size, the faro shuffle and simple cut generate all possible permutations, while if the deck is of odu size, only a small fraction are available. This paper considers a generalized faro shuffle wherein the deck is divided into n rather than 2, portions and these portions are “interlaced” together. It is shown that the generalized faro shuffle and the simple cut generate either the symmetric group of the deck, the alternating group of the deck, or in one special case, only a small fraction of the possible permutations. Whether the symmetric group or alternating group is generated depends on the parity of the simple cut and the generalized faro shuffle as group operations.     

A difference scheme with anti-diffusion terms to improve the computation of contact discontinuities and the study of weak discontinuities in compressible flow

The investigation of Mach reflection formed after the impingement of a weak plane shock wave on a wedge with shock Mach number M_s near 1, is still an open problem[12]. It's difficult for shock tube experiments with interferometer to detect contact discontinuities if it is too weak; also difficult to catch with due accuracy the transition condition between Mach reflection and regular reflection. The interest to this phenomenon is continuing, especially for weak shocks, because there was systematic discrepancy between simplified three shock theory of von Neumann [8] and shock tube results [15] which was named by G. Birkhoff as “von Neumann Paradox on three shock theory” [18].In 1972, K.O.Friedrichs called for more computational efforts on this problem. Recently it is known that for weak impinging shocks it's still difficult to get contact discontinuities and curved Mach stem with satisfactory accuracy. Recent numerical computation sometimes even fails to show reflected shock wave[6]. These explain why von Neumann paradox of the three shock theory in case of weak discontinuities is still a problem of interesting [9,12,14]. In this paper, on one hand, we investigate the numerical methods for Euler's equation for compressible inviscid flow, aiming at improving the computation of contact discontinuities, on the other hand, a methodology is suggested to correctly plot flow data from the massive information in storage. On this basis, all the reflected shock wave , contact discontinuities and the curved Mach stem are determined. We get Mach reflection under the condition when over-simplified shock theory predicts no such configuration[5].     

On the initial phase of the triple deck stage in marginally separated flows

The method of matched asymptotic expansions is used to investigate marginally separated boundary layer flows (laminar or alternatively transitional separation bubbles) at high Reynolds numbers. Typical examples include, among others, the flow past slender airfoils at small to moderate angels of attack and channel flows with suction. As is well-known, classical (hierarchical) boundary layer computations usually break down under the action of an adverse pressure gradient on the flow, a scenario associated with the appearance of the Goldstein separation singularity. If, however, the parameter controlling the strength of the pressure gradient (the angle of attack or the relative suction rate in the examples mentioned above) is adjusted accordingly, the application of a local viscous-inviscid interaction strategy is capable of describing localized boundary layer separation. Moreover, taking into account unsteady effects and flow control devices allows the investigation of the conditions leading to forced or self-sustained vortex generation and the subsequent evolution process culminating in bubble bursting. Within the asymptotic formulation of this stage bubble bursting is associated with the formation of finite time singularities in the solution of the underlying equations and a corresponding break down. The distinct blow-up structure gives rise to a fully non-linear triple deck interaction stage featuring shorter spatio-temporal scales characteristic of the successive vortex evolution process. The paper will focus on the numerical treatment of the initial phase of the latter stage. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

MHD analysis in hydromagnetic casting process of clad steel slabs

A novel continuous casting process for clad steel slabs has been developed by suppressing the mixing of molten steels in the mold pool of continuous casting strand with a level DC magnetic field (LMF) installed in the mold. In this process, two molten steels of different chemical composition are discharged by two nozzles into the upper and the lower pools respectively to solidify in the outer and the inner layers as a clad steel slabs. The mechanism of separation into two layers has been elucidated by using a three dimensional MHD analysis. The numerical prediction employing Maxwell's equation, Ohm's law, and the turbulent flow model shows that the mixing of the different type of steels is suppressed by the electromagnetic dividing of the upper and the lower recirculating flows. The principle of the new process has also been verified by steel casting trials of the stainless-steel clad steel slabs with an 8-ton scale pilot continuous casting machine.     

Wave propagation in a mixed convection boundary‐layer over a horizontal surface

The propagation of disturbances in a mixed convection boundary‐layer flow over a horizontal plate is described by a triple deck problem in the case of the buoyancy parameter being small. The pressure correction in the lower deck consists of two parts: One due to the buoyancy effects in the main deck and one due to the displacement of the outer flow field. The response of the boundary layer flow to an oscillator of frequency ω will be computed and upstream travelling waves will be identified.     

Sediment Transport and Deposition from a Two-layer Fluid Model of Gravity Currents on Sloping Bottoms

This article reports on a theoretical and numerical study of noneroding turbulent gravity currents moving down mildly inclined surfaces while depositing sediment. These flows are modeled by means of two-layer fluid systems appropriately modified to account for the presence of a sloping bottom and suspended sediment in the lower layer. A detailed scaling argument shows that when the density of the interstitial fluid is slightly greater than that of the ambient and the suspension is such that its volume fraction is of the order of the aspect ratio squared, for low aspect ratio flows a two-layer shallow-water theory is applicable. In this theory there is a decoupling of particle and flow dynamics. In contrast, however, when the densities of interstitial and ambient fluids are equal, so that it is the presence of the particles alone that drives the flow, we find that a consistent shallow-water theory is impossible no matter how small the aspect ratio or the initial volume fraction occupied by the particles. Our two-layer shallow-water formulation is employed to investigate the downstream evolution of flow and depositional characteristics for sloping bottoms. This investigation uncovers a new phenomenon in the formation of a rear compressive zone giving rise to shock formation in the post-end-wall-separation phase of the particle-bearing gravity flow. This separation of flow from the end wall in these fixed volume releases differs from what has been observed on horizontal surfaces where the flow always remains in contact with the end wall.     

Stoßwellenausbreitung in Rohrverzweigungen

Summary The interaction of a shock wave with the junction of a straight duct having a side branch is considered. The paper checks the validity of three hypotheses. The first and second one calculate the shock waves after the junctions only by a geometrical consideration. A comparison with experimental values shows only an unsatisfactory result for weak shock waves. The third theory assumes that the pressure drop across the junction, in the quasi-steady flow which takes place after the incident shock wave, is the same as in steady flow tests. The pressure drops were measured for different junctions and the results of the calculation were compared with experimental values from a shock tube, both for circular and rectangular channels. A good agreement was obtained.     

Generalized viscous shock layer equations with slip conditions on a surface in a flow field and on a head shock

The plane and axisymmetric problems of super- and hypersonic flow of a homogeneous viscous heat-conducting perfect gas over a blunt body are considered. Generalized viscous shock layer equations that take into account all the second-order effects of boundary-layer theory, i.e., the terms O(Re^?1/2), are derived from the Navier–Stokes equations by the asymptotic method, and all the out-of-order third-order terms O(Re^?1) and higher-order terms are also retained, except terms with second derivations in the marching coordinate (Re is Reynolds number, determined from the free-stream density and velocity the linear dimension, which is equal to the nose radius of the blunt Body, and the free-stream shear viscosity at the stagnation temperature). Thus, only the presence of terms with second derivatives in the marching coordinate, which specify the elliptical properties of the complete system of Navier–Stokes equations, distinguish it from the generalized viscous shock layer equations, which do not contain these terms. Slip and a temperature jump conditions on a body surface are presented with the same degree of accuracy, and generalized Rankine–Hugoniot conditions on a head shock, which take into account the effects of the viscosity and heat conduction, including their influence on the determination of the pressure, are derived. The incorrect and unfounded approximations used in preceding studies and the efficiency of iterative marching techniques for solving the generalized viscous shock layer equations, as well as the ability of the latter to provide a correct solution for the drag and heat-transfer coefficients in the transitional flow regime if the solution is constructed taking the slip and temperature jump on a surface and on a head shock into account, are noted.     

Continuum models in problems of the hypersonic flow of a rarefied gas around blunt bodiest

All possible continuum (hydrodynamic) models in the case of two-dimensional problems of supersonic and hypersonic flows around blunt bodies in the two-layer model (a viscous shock layer and shock-wave structure) over the whole range of Reynolds numbers, Re, from low values (free molecular and transitional flow conditions) up to high values (flow conditions with a thin leading shock wave, a boundary layer and an external inviscid flow in the shock layer) are obtained from the Navier-Stokes equations using an asymptotic analysis. In the case of low Reynolds numbers, the shock layer is considered but the structure of the shock wave is ignored. Together with the well-known models (a boundary layer, a viscous shock layer, a thin viscous shock layer, parabolized Navier-Stokes equations (the single-layer model) for high, moderate and low Re numbers, respectively), a new hydrodynamic model, which follows from the Navier-Stokes equations and reduces to the solution of the simplified (“local”) Stokes equations in a shock layer with vanishing inertial and pressure forces and boundary conditions on the unspecified free boundary (the shock wave) is found at Reynolds numbers, and a density ratio, k, up to and immediately after the leading shock wave, which tend to zero subject to the condition that (k/Re)^1/2 → 0. Unlike in all the models which have been mentioned above, the solution of the problem of the flow around a body in this model gives the free molecular limit for the coefficients of friction, heat transfer and pressure. In particular, the Newtonian limit for the drag is thereby rigorously obtained from the Navier-Stokes equations. At the same time, the Knudsen number, which is governed by the thickness of the shock layer, which vanishes in this model, tends to zero, that is, the conditions for a continuum treatment are satisfied. The structure of the shock wave can be determined both using continuum as well as kinetic models after obtaining the solution in the viscous shock layer for the weak physicochemical processes in the shock wave structure itself. Otherwise, the problem of the shock wave structure and the equations of the viscous shock layer must be jointly solved. The equations for all the continuum models are written in Dorodnitsyn--Lees boundary layer variables, which enables one, prior to solving the problem, to obtain an approximate estimate of second-order effects in boundary-layer theory as a function of Re and the parameter k and to represent all the aerodynamic and thermal characteristic; in the form of a single dependence on Re over the whole range of its variation from zero to infinity.

An efficient numerical method of global iterations, previously developed for solving viscous shock-layer equations, can be used to solve problems of supersonic and hypersonic flows around the windward side of blunt bodies using a single hydrodynamic model of a viscous shock layer for all Re numbers, subject to the condition that the limit (k/Re)^1/2 → 0 is satisfied in the case of small Re numbers. An aerodynamic and thermal calculation using different hydrodynamic models, corresponding to different ranges of variation Re (different types of flow) can thereby, in fact, be replaced by a single calculation using one model for the whole of the trajectory for the descent (entry) of space vehicles and natural cosmic bodies (meteoroids) into the atmosphere.     

Stability of a Mach configuration

We prove the stability of a Mach configuration, which occurs in shock reflection off an obstacle or shock interaction in compressible flow. The compressible flow is described by a full, steady Euler system of gas dynamics. The unperturbed Mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity. Among four regions divided by these four lines in the neighborhood of the intersection, two are supersonic regions, and other two are subsonic regions. We prove that if the constant states in the supersonic regions are slightly perturbed, then the structure of the whole configuration holds, while the other two shock fronts and the slip line, as well as the flow field in the subsonic regions, are also slightly perturbed. Such a conclusion asserts the existence and stability of the general Mach configuration in shock theory. In order to prove the result, we reduce the problem to a free boundary value problem, where two unknown shock fronts are free boundaries, while the slip line is transformed to a fixed line by a Lagrange transformation. In the region where the solution is to be determined, we have to deal with an elliptic‐hyperbolic composed system. By decoupling this system and combining the technique for both hyperbolic equations and elliptic equations, we establish the required estimates, which are crucial in the proof of the existence of a solution to the free boundary value problem. © 2005 Wiley Periodicals, Inc.     

Stream-function Co-ordinates in Rotational Flow and an Analysis of the Flow in a Shock Layer: I. Two-dimensional flow

For the analysis of two-dimensional rotational flow of inviscidfluids a new set of co-ordinates are proposed. The co-ordinatesare composed of a pair of functions, the stream function anda function to be constant on each trajectory perpendicular tothe stream-lines in the flow field. The theory is then appliedto the analysis of the flow in a shock layer around a blunt-nosedbody in a hypersonic stream. The problem treated in this paperis the inverse problem to be solved with a prescribed shockshape. Some numerical computations are carried out for a parabolicshape of shock, by making use of an electronic computer.