On the pressure and stress singularities induced by steady flows of a pair of nonmiscible, incompressible, viscous fluids in contact with a wall

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, only of late has there been much in the way of corresponding identifications of flow-induced stress singularities in fluid mechanics. These recent asymptotic identifications are for a single incompressible viscous fluid： Here the asymptotic approach is extended to apply to a configuration entailing two such fluids, For this configuration, various specifications leading to power or log singularities are determined. These results demonstrate that flow-induced stress singularities can occur in a structural container at a location where no singularities are identified within solid mechanics alone.     

Numerical simulation of expansion/compression effect on shock induced turbulent flow

The interaction of homogeneous and isotropic turbulence with a shock wave is observed by solving the Reynolds-averaged Navier–Stokes equations with the k–? turbulence model. All turbulent fluctuations are measured at the period of expansion in the turbulent field and during compression by the reflected shock on turbulent field, and it is observed that the longitudinal turbulent velocity fluctuation is enhanced more at the period of expansion due to incident shock wave movement far from the turbulent field. The amplification of the turbulent kinetic energy (TKE) level in the shock/turbulence interaction depends on the shock wave strength and the longitudinal velocity difference across the shock wave. On decreasing the longitudinal velocity difference across the shock, the turbulent kinetic energy (TKE) level is less amplified. The TKE level is amplified by the factor of 1.5–1.8 in the shock/turbulence interaction where the dissipation rate of TKE decreases in all cases of shock/turbulence interaction. After the shock/turbulence interaction, the turbulent dissipative-length scale is amplified slightly and the amplification of the length scales decreases when increasing the shock strength. To cite this article: M.A. Jinnah, K. Takayama, C. R. Mecanique 333 (2005).     

Numerical analysis on flutter of Busemann-type supersonic biplane airfoil

The Busemann-type supersonic biplane can effectively reduce the wave drag through shock interference effect between airfoils. However, considering the elastic property of the wing structure, the vibration of the wings can cause the shock oscillation between the biplane, which may result in relative aeroelastic problems of the wing. In this research, fluid–structure interaction characteristics of the Busemann-type supersonic biplane at its design condition have been studied. A theoretical two-dimensional structure model has been established to consider the main elastic characteristics of the wing structure. Coupled with unsteady Navier–Stokes equations, the fluid–structure dynamic system of the supersonic biplane is studied through the two-way computational fluid dynamics/computational structural dynamics (CFD/CSD) coupling method. The biplane system has been simulated at its design Mach number with different nondimensional velocities. Different initial disturbance has been applied to excite the system and the effects of the position of the mass center on the system’s aeroelastic stability is also discussed. The results reveal that the stability of the airfoil in supersonic biplane system is decreased compared with that of the airfoil isolated in supersonic flow and such stability reduction effect should be given due attention in practical design.     

Simulation of flow around rigid vegetation stems with a fast method of high accuracy

This paper develops the virtual cylinder model (VCM) to simulate flows through vegetation canopies at low to medium high Reynolds numbers in wetlands. Using a combined Eulerian and Lagrangian approach, the VCM is capable of simulating the flow field around each vegetation stem (modeled as a cylinder) as well as a bulk of numerous stems in a fluid. Numerical results of flows through regularly and randomly arranged cylinders have been compared with direct numerical simulations. Good agreement has been achieved. This model maintains an excellent balance between accuracy and efficiency in modeling interactions between the fluid and vegetation stems. Simulation results demonstrate that this model is promising for investigating the hydrodynamics and vegetation resilience in wetlands that serve as a first line of defense against the sea level rise, storm impacts and coastal erosion.     

A parallel Navier–Stokes solver using spectral discretisation in time

We investigate the performance of a multi-harmonic space–time approach to solve time-periodic flow problems. It employs a Fourier spectral discretisation of the time domain. The resulting large system of nonlinear equations is solved iteratively and in parallel. For illustration, a three-dimensional channel flow disturbed by an oscillating force is simulated.     

A Fully Coupled Solver for Incompressible Navier–Stokes Equations using Operator Splitting

The steady incompressible Navier–Stokes equations are coupled by a Poisson equation for the pressure from which the continuity equation is subtracted. The equivalence to the original N–S problem is proved. Fictitious time is added and vectorial operator-splitting is employed leaving the system coupled at each fractional-time step which allows satisfaction of the boundary conditions without introducing artificial conditions for the pressure. Conservative second-order approximations for the convective terms are employed on a staggered grid. The splitting algorithm for the 3D case is verified through an analytic solution test. The stability of the method at high values of Reynolds number is illustrated by accurate numerical solutions for the flow in a lid-driven rectangular cavity with aspect ratio 1 and 2, as well as for the flow after a back-facing step.     

Resonance-like phenomena in a submerged cylindrical shell subjected to two consecutive shock waves: The effect of the inner fluid

A submerged fluid-filled cylindrical shell subjected to a sequence of two shock waves originated at the same source is considered. It is demonstrated that, unlike in the case of a submerged evacuated shell, there exists a certain critical range of the values of the delay between the incident wavefronts where both the peak compressive and the peak tensile stress observed in the structure are significantly (60% or more) higher than the respective stresses in the same system subjected to a single-front loading. It is further demonstrated that the highest and the lowest hydrodynamic pressure attained in the system is also dramatically affected for certain values of the delay between the incident wavefronts, with the maximum double-front pressure being more than 30% higher than its single-front counterpart. The practical relevance of the findings is discussed in the context of the pre-design analysis of industrial systems subjected to shock loading.     

Three-dimensional hydrodynamic modelling of tidal flows interacting with aquaculture fish cages

Three-dimensional, non-linear, non-hydrostatic simulations of rotating tidal flows interacting with aquaculture cages (represented as drag elements) at the geophysical scale are performed using an adaptive, finite volume fluid code “Gerris”. Exploiting the Gerris grid structure, sub-metre scale resolution can be obtained even for the farm scale experiments, enabling examination of the impact of the cage on the imposed tidal flows. Passive tracers are used to try to quantify these cage impacts, representing either feed or faecal matter (with specified fall speeds), or other biogeochemical markers such as dissolved oxygen. Using a relatively simple drag formulation, we show that the model is able to reproduce laboratory observations. The farm scale simulations can also be “tuned” in a similar fashion, for example by comparison with observations of total drag force on such structures, or with field measurements of flow retardation by cages. Single and multi-scale cage experiments are then examined to explore the potential impacts of perturbed horizontal and vertical flows on material redistribution through and within the cages. Even with the relatively smooth forcing and drag formulation the experiments reveal a surprising level of complexity in terms of the perturbed flows and their impact on transporting and diffusing passive material.     

Investigations on the porous resistance coefficients for fishing net structures

The porous media model has been successfully applied to numerical simulation of current and wave interaction with traditional permeable coastal structures such as breakwaters. Recently this model was employed to simulate flow through and around fishing net structures, where the unknown porous resistance coefficients were adjusted by fitting the available experimental data. In the present paper, a new approach was proposed to calculate the porous resistance coefficients based on the transformation of Morison type load model. The transformation follows the principle that the total forces acting on a net panel from Morison type load model should be equal to the forces obtained from the porous media model. In order to account for the interaction effects in-between the twines, two coefficients were introduced, and they were calibrated by minimizing the least square error function. Extensive validation cases were carried out to examine the performance of the numerical model. This includes steady current flow through plane net panels and circular fish cages, and wave interaction with plane net panels. A variety of fishing nets with different solidity ratios were used in the validation cases, from which it was seen that the overall agreement between the numerical and experimental results is fair.     

A stabilised characteristic finite element method for transient Navier–Stokes equations

In this work, we consider a stabilised characteristic finite element method for the time-dependent Navier–Stokes equations based on the lowest equal-order finite element pairs. The diffusion term in these equations is discretised by using finite element method, the temporal differentiation and advection terms are treated by characteristic schemes. Unconditionally stable results and error estimates of optimal order for the velocity and pressure are established. Finally, some numerical results are provided to verify the performance of this method.     

A fictitious domain finite element method for simulations of fluid–structure interactions: The Navier–Stokes equations coupled with a moving solid

The paper extends a stabilized fictitious domain finite element method initially developed for the Stokes problem to the incompressible Navier–Stokes equations coupled with a moving solid. This method presents the advantage to predict an optimal approximation of the normal stress tensor at the interface. The dynamics of the solid is governed by Newton׳s laws and the interface between the fluid and the structure is materialized by a level-set which cuts the elements of the mesh. An algorithm is proposed in order to treat the time evolution of the geometry and numerical results are presented on a classical benchmark of the motion of a disk falling in a channel.     

A survey of weakly-nonlinear acoustic models: 1910–2009

A chronological survey of the weakly-nonlinear acoustic equations put forth between the years 1910–2009 is presented. The derivation, properties, and historical development of these models, along with the approximation scheme on which they are based, are discussed. Connections between the models are also discussed and errors/omissions in the literature are noted and, in some cases, corrected. Brief chronologies of the weakly-nonlinear models that describe propagation in relaxing fluids, bubbly liquids, and fluids that saturate porous solids are also presented.     

Semi-meshless stencil selection for anisotropic point distributions

Meshless methods are attractive for simulating moving body problems. The selection of the stencils over the domain for the meshless solver is crucial for the method to be competitive with established computational fluid dynamics techniques. Stencil selection is relatively straightforward if the point distributions are isotropic in nature, however, this is rarely the case in computations that solve the Navier–Stokes equations. In this paper, a fully automatic method of selecting the stencils from anisotropic point distributions, which are obtained from overlapping structured grids, is outlined. The original connectivity and the concept of a resolving direction are used to help construct good quality stencils with limited user input.     

Modelling turbulence around and inside porous media based on the second moment closure

To predict turbulence in porous media, a new approach is discussed. By double (both volume and Reynolds) averaging Navier–Stokes equations, there appear three unknown covariant terms in the momentum equation. They are namely the dispersive covariance, the macro-scale and the micro-scale Reynolds stresses, in the present study. For the macro-scale Reynolds stress, the TCL (two-component-limit) second moment closure is applied whereas the eddy viscosity models are applied to the other covariant terms: the Smagorinsky model and the one-equation eddy viscosity model, respectively for the dispersive covariance and the micro-scale Reynolds stress. The presently proposed model is evaluated in square rib array flows and porous wall channel flows with reasonable accuracy though further development is required.     

A case of strong nonlinearity: Intermittency in highly turbulent flows

It has long been suspected that flows of incompressible fluids at large or infinite Reynolds number (namely at small or zero viscosity) may present finite time singularities. We review briefly the theoretical situation on this point. We discuss the effect of a small viscosity on the self-similar solution to the Euler equations for inviscid fluids. Then we show that single-point records of velocity fluctuations in the Modane wind tunnel display correlations between large velocities and large accelerations in full agreement with scaling laws derived from Leray's equations (1934) for self-similar singular solutions to the fluid equations. Conversely, those experimental velocity–acceleration correlations are contradictory to the Kolmogorov scaling laws.     

Accurate iteration-free mixed-stabilised formulation for laminar incompressible Navier–Stokes: Applications to fluid–structure interaction

Stabilised mixed velocity–pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier–Stokes. In these formulations, the Newton–Raphson scheme is employed to solve the nonlinearity in the convection term. One fundamental issue with this approach is the computational cost incurred in the Newton–Raphson iterations at every load/time step. In this paper, we present an iteration-free mixed finite element formulation for incompressible Navier–Stokes that preserves second-order temporal accuracy of the generalised-alpha and related schemes for both velocity and pressure fields. First, we demonstrate the second-order temporal accuracy using numerical convergence studies for an example with a manufactured solution. Later, we assess the accuracy and the computational benefits of the proposed scheme by studying the benchmark example of flow past a fixed circular cylinder. Towards showcasing the applicability of the proposed technique in a wider context, the inf–sup stable P2–P1 pair for the formulation without stabilisation is also considered. Finally, the resulting benefits of using the proposed scheme for fluid–structure interaction problems are illustrated using two benchmark examples in fluid-flexible structure interaction.     

Error estimation using the error transport equation for finite-volume methods and arbitrary meshes

We present a single grid error estimation technique based on the derivation of a continuous equation for the discretization error. It is developed in the context of finite-volume methods for arbitrary meshes. The key issue of the evaluation of the source term is addressed through the use of a reconstruction operator. Using a higher order accurate evaluation of this term and solving the error equation with the same numerical methods and on the same computational grid as the primal problem leads to a higher order accurate error prediction. The methodology is presented in detail and its properties of asymptotic exactness and superconvergence are illustrated on several cases, including an application of practical engineering complexity. Also presented is the derivation of a powerful criterium for driving any adaptive procedure.     

A numerical simulation of boundary layer effects in a shock tube

A numerical scheme is used to investigate boundary layer effects in a shock tube. The method consists of a mixture of Roe's approximate Riemann solver and central differences for the convective fluxes and central differences for the viscous fluxes and is implicit in one space dimension. Comparisons are made with experimental data and with solutions obtained via boundary layer equations. Examination of the calculated flow field explains the observed behaviour and highlights the approximate nature of boundary layer solutions.     

A new immersed boundary method for compressible Navier–Stokes equations

This paper presents an immersed boundary method for compressible Navier–Stokes equations in irregular domains, based on a local radial basis function approximation. This approach allows one to define a reconstruction of the radial basis functions on each irregular interface cell to treat both the Dirichlet and Neumann boundary conditions accurately on the immersed interfaces. Several numerical examples, including problems with available analytical solutions and the well-documented flow past an airfoil, are presented to test the proposed method. The numerical results demonstrate that the proposed method provides accurate solutions for viscous compressible flows.     

Submerged circular cylindrical shell subjected to two consecutive shock waves: Resonance-like phenomena

A submerged evacuated circular cylindrical shell subjected to a sequence of two external shock waves generated at the same source is considered. A semi-analytical model combining the classical methods of mathematical physics with the finite-difference methodology is developed and employed to simulate the interaction. Both the hydrodynamic and structural aspects of the problem are considered, and it is demonstrated that varying the delay between the first and second wavefronts has a very significant effect on the stress–strain state of the structure. In particular, it is shown that for certain values of the delay, the constructive superposition of the elastic waves travelling around the shell results in a ‘resonance-like’ increase of the structural stress in certain regions. The respective stress can be so high that it sometimes exceeds the overall maximum stress observed in the same structure but subjected to a single-front shock wave with the same parameters, in some cases by as much as 50%. A detailed parametric analysis of the observed phenomenon is carried out, and an easy-to-use diagram summarizing the finding is proposed to aim the pre-design analysis of engineering structures.