Nonreflecting boundary conditions based on nonlinear multidimensional characteristics

Nonlinear characteristic boundary conditions based on nonlinear multidimensional characteristics are proposed for 2‐ and 3‐D compressible Navier–Stokes equations with/without scalar transport equations. This approach is consistent with the flow physics and transport properties. Based on the theory of characteristics, which is a rigorous mathematical technique, multidimensional flows can be decomposed into acoustic, entropy, and vorticity waves. Nonreflecting boundary conditions are derived by setting corresponding characteristic variables of incoming waves to zero and by partially damping the source terms of the incoming acoustic waves. In order to obtain the resulting optimal damping coefficient, analysis is performed for problems of pure acoustic plane wave propagation and arbitrary flows. The proposed boundary conditions are tested on two benchmark problems: cylindrical acoustic wave propagation and the wake flow behind a cylinder with strong periodic vortex convected out of the computational domain. This new approach substantially minimizes the spurious wave reflections of pressure, density, temperature, and velocity as well as vorticity from the artificial boundaries, where strong multidimensional flow effects exist. The numerical simulations yield accurate results, confirm the optimal damping coefficient obtained from analysis, and verify that the method substantially improves the 1‐D characteristics‐based nonreflecting boundary conditions for complex multidimensional flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.     

三维扰动波的非平行边界层稳定性研究

导出了三维扰动波的原始变量形式的抛物化稳定性方程（PSE）,研究了三维空间模态TS波的非平行边界层稳定性问题.采用了法向四阶紧致格式,以提高计算精度.通过给出不会导致奇性的坐标变换、修改外边界条件以及克服平行流初始值的瞬态影响和推进步长的限制,保证了计算的数值稳定.用补全元素带状矩阵法求解块三对角矩阵,大大提高了速度.计算结果清楚地显示了三维扰动波的演化过程和非平行性对边界层稳定性的影响,特别是,观察到非平行性对三维扰动波的影响,有时会使其稳定性出现逆转的现象.还研究了逆压梯度的作用.算例的结果与其他结果符合良好.     

Spatial Simple Waves on a Shear Flow

The paper studies simple waves of the shallowwater equations describing threedimensional wave motions of a rotational liquid in a freeboundary layer. Simple wave equations are derived for the general case. The existence of unsteady or steady simple waves adjacent continuously to a given steady shear flow along a characteristic surface is proved. Exact solutions of the equations describing steady simple waves were found. These solutions can be treated as extension of Prandtl–Mayer waves for sheared flows. For shearless flows, a general solution of the system of equations describing unsteady spatial simple waves was found.     

Characteristics of unsteady type IV shock/shock interaction

Characteristics of the unsteady type IV shock/shock interaction of hypersonic blunt body flows are investigated by solving the Navier–Stokes equations with high-order numerical methods. The intrinsic relations of flow structures to shear, compression, and heating processes are studied and the physical mechanisms of the unsteady flow evolution are revealed. It is found that the instantaneous surface-heating peak is caused by the fluid in the “hot spot” generated by an oscillating and deforming jet bow shock (JBS) just ahead of the body surface. The features of local shock/boundary layer interaction and vortex/boundary layer interaction are clarified. Based on the analysis of flow evolution, it is identified that the upstream-propagating compression waves are associated with the interaction of the JBS and the shear layers formed by a supersonic impinging jet, and then the interaction of the freestream bow shocks and the compression waves results in entropy and vortical waves propagating to the body surface. Further, the feedback mechanism of the inherent unsteadiness of the flow field is revealed to be related to the impinging jet. A feedback model is proposed to reliably predict the dominant frequency of flow evolution. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to this complex flow.     

Boundary layer flows behind constant speed shock waves moving into a dusty gas

Laminar boundary layer flows behind constant speed shock waves moving into a dusty gas are analyzed numerically. The basic equations of two-phase flows are derived in shock fixed coordinates and solved by an implicit finite-difference method for the side wall boundary layer in a dusty gas shock tube. The development of the boundary layer and resulting velocity and temperature profiles, respectively, for the gas and particles are given from the shock front to far downstream. The effects of diaphragm pressure ratio, mass loading ratio of particles and particle size upon the flow properties are discussed in detail.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Multi-shock structure of roll wavesStructure multi-chocs d'ondes à rouleaux

The special class of periodic travelling waves which is known as roll waves is investigated for nonhomogeneous hyperbolic equations of gas dynamics type. In this Note these equations are applied to shallow water flows in inclined open channels, but the results obtained are more general and far-reaching. The necessary conditions for the existence of a roll wave are derived. It is shown that for a nonconvex pressure term, multi-shock configurations of roll waves of finite amplitude exist. A new type of periodic travelling wave, which corresponds to the slug flow regime in two-layer flows, is found. To cite this article: A. Boudlal, V.Yu. Liapidevskii, C. R. Mecanique 332 (2004).     

Simulation of non‐hydrostatic,density‐stratified flow in irregular domains

A numerical model has been developed for simulating density‐stratified flow in domains with irregular but simple topography. The model was designed for simulating strong interactions between internal gravity waves and topography, e.g. exchange flows in contracting channels, tidally or convectively driven flow over two‐dimensional sills or waves propagating onto a shoaling bed. The model is based on the non‐hydrostatic, Boussinesq equations of motion for a continuously stratified fluid in a rotating frame, subject to user‐configurable boundary conditions. An orthogonal boundary fitting co‐ordinate system is used for the numerical computations, which rely on a fourth‐order compact differentiation scheme, a third‐order explicit time stepping and a multi‐grid based pressure projection algorithm. The numerical techniques are described and a suite of validation studies are presented. The validation studies include a pointwise comparison of numerical simulations with both analytical solutions and laboratory measurements of non‐linear solitary wave propagation. Simulation results for flows lacking analytical or laboratory data are analysed a posteriori to demonstrate satisfaction of the potential energy balance. Computational results are compared with two‐layer hydraulic predictions in the case of exchange flow through a contracting channel. Finally, a simulation of circulation driven by spatially non‐uniform surface buoyancy flux in an irregular basin is discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

纳米示踪平面激光散射技术在激波复杂流场测量中的应用

在激波以及激波边界层相互作用这类含激波的复杂流场中,流场结构具有明显的三维特征.研究这类流场,采用纹影、阴影和干涉等传统流动显示技术空间分辨率较低,难以分辨流场的三维特性.基于纳米示踪的平面激光散射技术(nano-tracer planar laser scattering,NPLS),是作者近年来开发的一种新的研究超声速流场的测试与显示技术,可对超声速复杂三维流场进行高时空分辨率流动显示与测量.NPLS技术的特点使其成为测量激波复杂流场的有力手段.近年来,作者以NPLS技术为主要手段,对航空航天领域典型的激波复杂流场进行了试验研究,包括超声速弹头绕流、超声速混合层、超声速边界层,以及激波边界层相互作用流场,显示出NPLS技术在激波复杂流场精细测试与流动显示中优势.本文简要介绍NPLS技术在激波复杂流场测量中应用的研究进展.      

Interaction of viscous wakes with a free surface

The interaction of laminar wakes with.free-surface waves generated by a moving body beneath the surface of an incompressible viscous fluid of infinite depth was investigated analytically. The analysis was based on the steady Oseen equations for disturbed flows.The kinematic and dynamic boundary conditions were linearized for the small-amplitude free-surface waves. The effect of the moving body was mathematically modeled as an Oseenlet.The disturbed flow was regarded as the sum of an unbounded singular Oseen flow which represents the effect of the viscous wake and a bounded regular Oseen flow which represents the influence of the free surface. The exact solution for the free-surface waves was obtained by the method of integral transforms. The asymptotic representation with additive corrections for the free-surface waves was derived by means of Lighthill‘s two-stage scheme. The symmetric solution obtained shows that the amplitudes of the free-surface waves are exponentially damped by the presences of viscosity and submergence depth.     

Resonance in rarefied gases

Dispersion and damping of ultrasound waves are a standard test for mathematical models of rarefied gas flows. Normally, one considers waves in semi-infinite systems in relatively large distance of the source. For a more complete picture, ultrasound propagation in finite closed systems of length L is studied by means of several models for rarefied gas flows: the Navier-Stokes-Fourier equations, Grad’s 13 moment equations, the regularized 13 moment equations, and the Burnett equations. All systems of equations are considered in simple 1-D geometry with their appropriate jump and slip boundary conditions. Damping and resonance are studied in dependence of frequency and length. For small L, all wave modes contribute to the solution.     

Nonlinearwaves in a liquid film-gas flow system

The instability and regular nonlinear waves in the film of a heavy viscous liquid flowing along the wall of a round tube and interacting with a gas flow are investigated. The solutions for the wave film flows are numerically obtained in the regimes from free flow-down in a counter-current gas stream to cocurrent upward flow of the film and the gas at fairly large gas velocities. Continuous transition from the counter-current to the cocurrent flow via the state with a maximum amplitude of nonlinear waves and zero values of the liquid flow rate and the phase velocity is investigated. The Kapitsa-Shkadov method is used to reduce a boundary value problem to a system of evolutionary equations for the local values of the layer thickness and the liquid flow rate.     

Simultaneous solution of the Navier-Stokes and elastic membrane equations by a finite element method

Fluid flow through a significantly compressed elastic tube occurs in a variety of physiological situations. Laboratory experiments investigating such flows through finite lengths of tube mounted between rigid supports have demonstrated that the system is one of great dynamical complexity, displaying a rich variety of self-excited oscillations. The physical mechanisms responsible for the onset of such oscillations are not yet fully understood, but simplified models indicate that energy loss by flow separation, variation in longitudinal wall tension and propagation of fluid elastic pressure waves may all be important. Direct numerical solution of the highly non-linear equations governing even the most simplified two-dimensional models aimed at capturing these basic features requires that both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. To accomplish this, previous algorithms have decoupled the solid and fluid mechanics, solving for each separately and converging iteratively on a solution which satisfies both. This paper describes a finite element technique which solves the incompressible Navier-Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary. The elastic boundary position is parametized in terms of distances along spines in a manner similar to that which has been used successfully in studies of viscous free surface flows, but here the membrane curvature equation rather than the kinematic boundary condition of vanishing normal velocity is used to determine these diatances and the membrane tension varies with the shear stresses exerted on it by the fluid motions. Bothy the grid and the spine positions adjust in response to membrane deformation, and the coupled fluid and elastic equations are solved by a Newton-Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse. Solutions to the steady problem are discussed, along with an indication of how the time-dependent problem might be approached.     

Global volume conservation in unsteady free surface flows with energy absorbing far‐end boundaries

A wave absorption filter for the far‐end boundary of semi‐infinite large reservoirs is developed for numerical simulation of unsteady free surface flows. Mathematical model is based on finite volume solution of the Navier–Stokes equations and depth‐integrated continuity equation to track the free surface. The Sommerfeld boundary condition is applied at the far‐end of the truncated computational domain. A dissipation zone is formed by applying artificial pressure on water surface to dissipate the kinetic energy of the outgoing waves. The computational scheme is tested to verify the conservation of total fluid volume in the domain for long simulation durations. Combination of the Sommerfeld boundary and dissipation zone can effectively minimize reflections and prevent cumulative changes in total fluid volume in the domain. Solitary wave, nonlinear periodic waves and irregular waves are simulated to illustrate the numerical developments. Earthquake excited surface waves and nonlinear hydrodynamic pressures in a dam–reservoir are computed. Copyright © 2009 John Wiley & Sons, Ltd.     

Higher order stability effects in a natural convection boundary layer over a vertical heated wall

 The stability of a laminar boundary layer flow under natural convection on a vertical isothermally heated wall is studied analytically. The analysis is performed by using two different two-dimensional linear models: (1) The non-parallel flow model in which the steady mean flow as well as the disturbance amplitude functions can change in the streamwise direction; (2) The parallel flow model in which the effects of the mean flow and disturbance changes in the streamwise direction are neglected. The linear non-parallel stability analysis is based on the so-called parabolised stability equations (PSEs) which have been successfully applied to the stability analysis of forced convection boundary layers. In this study the PSE equations are applied to natural convection boundary layers in order to show the difference between parallel and non-parallel stability analysis. A second part of this study deals with the effects of variable properties, which are always present in natural convection flows. They are analysed by an extended version of the Orr–Sommerfeld equation (EOSE). Received on 31 May 2000     

Numerical simulation of high‐Reynolds number flow around circular cylinders by a three‐step FEM–BEM model

An innovative computational model, developed to simulate high‐Reynolds number flow past circular cylinders in two‐dimensional incompressible viscous flows in external flow fields is described in this paper. The model, based on transient Navier–Stokes equations, can solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the projection method. The pressure is assumed to be zero at infinite boundary and the external flow field is simulated using a direct boundary element method (BEM) by solving a pressure Poisson equation. A three‐step finite element method (FEM) is used to solve the momentum equations of the flow. The present model is applied to simulate high‐Reynolds number flow past a single circular cylinder and flow past two cylinders in which one acts as a control cylinder. The simulation results are compared with experimental data and other numerical models and are found to be feasible and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

3D free‐surface flow computation using a RANSE/Fourier–Kochin coupling

A coupling method for numerical calculations of steady free‐surface flows around a body is presented. The fluid domain in the neighbourhood of the hull is divided into two overlapping zones. Viscous effects are taken in account near the hull using Reynolds‐averaged Navier–Stokes equations (RANSE), whereas potential flow provides the flow away from the hull. In the internal domain, RANSE are solved by a fully coupled velocity, pressure and free‐surface elevation method. In the external domain, potential‐flow theory with linearized free‐surface condition is used to provide boundary conditions to the RANSE solver. The Fourier–Kochin method based on the Fourier–Kochin formulation, which defines the velocity field in a potential‐flow region in terms of the velocity distribution at a boundary surface, is used for that purpose. Moreover, the free‐surface Green function satisfying this linearized free‐surface condition is used. Calculations have been successfully performed for steady ship‐waves past a serie 60 and then have demonstrated abilities of the present coupling algorithm. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of 3D free surface flows,with multiple incompressible immiscible phases. Applications to impulse waves

A numerical method for the solution to the density‐dependent incompressible Navier–Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid phases together with an additional vacuum phase separated with a free surface. It is based on a volume‐of‐fluid approach involving N indicator functions (one per phase, identified by its density) that guarantees mass conservation within each phase. An additional indicator function for the whole liquid domain allows to treat boundary conditions at the interface between the liquid domain and a vacuum. The system of partial differential equations is solved by implicit operator splitting at each time step: first, transport equations are solved by a forward characteristics method on a fine Cartesian grid to predict the new location of each liquid phase; second, a generalized Stokes problem with a density‐dependent viscosity is solved with a FEM on a coarser mesh of the liquid domain. A novel algorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the N phases is validated on benchmark flows. Then, we focus on a novel application and compare the numerical and physical simulations of impulse waves, that is, waves generated at the free surface of a water basin initially at rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is to predict the effects of a landslide‐generated impulse wave in a reservoir. Copyright © 2014 John Wiley & Sons, Ltd.     

Large eddy simulation of boundary layer flow under cnoidal waves

Water waves in coastal areas are generally nonlinear, exhibiting asymmetric velocity profiles with different amplitudes of crest and trough. The behaviors of the boundary layer under asymmetric waves are of great significance for sediment transport in natural circumstances. While previous studies have mainly focused on linear or symmetric waves, asymmetric wave-induced flows remain unclear, particularly in the flow regime with high Reynolds numbers.Taking cnoidal wave as a typical example of asymmetric waves, we propose to use an infinite immersed plate oscillating cnoidally in its own plane in quiescent water to simulate asymmetric wave boundary layer. A large eddy simulation approach with Smagorinsky subgrid model is adopted to investigate the flow characteristics of the boundary layer. It is verified that the model well reproduces experimental and theoretical results. Then a series of numerical experiments are carried out to study the boundary layer beneath cnoidal waves from laminar to fully developed turbulent regimes at high Reynolds numbers, larger than ever studied before.Results of velocity profile, wall shear stress, friction coefficient, phase lead between velocity and wall shear stress, and the boundary layer thickness are obtained. The dependencies of these boundary layer properties on the asymmetric degree and Reynolds number are discussed in detail.     

Stationary simple waves in a plasma with anisotropic pressure

Stationary simple waves in a plasma with anisotropic pressure are investigated on the basis of the hydrodynamic equations of Chew, Goldberger, and Low. In Sec. 1, for the case where the vectors of the average flow velocity and the magnetic field intensity are parallel, the system of equations is reduced to two quasilinear equations for the velocity components. In Sec. 2 the equations for the characteristics are obtained, the system being assumed to be hyperbolic. For the special case of irrotational flow the character of simple waves in flows adjacent to various contours is studied. Section 3 contains a qualitative investigation of changes in the flow parameters in simple waves. In Sec. 4 the possibility of a transition to an unstable state of the plasma is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 12–19, March–April, 1971.The author thanks V. B. Baranov for the formulation of the problem and for his advice and constant attention to the work and also A. G. Kulikovskii for discussion of the results.