Analysis and Modeling of the Turbulent Diffusion of Turbulent Kinetic Energy in Natural Convection

Buoyant flows often contain regions with unstable and stable thermal stratification from which counter gradient turbulent fluxes are resulting, e.g. fluxes of heat or of any turbulence quantity. Basing on investigations in meteorology an improvement in the standard gradient-diffusion model for turbulent diffusion of turbulent kinetic energy is discussed. The two closure terms of the turbulent diffusion, the velocity-fluctuation triple correlation and the velocity-pressure fluctuation correlation, are investigated based on Direct Numerical Simulation (DNS) data for an internally heated fluid layer and for Rayleigh–Bénard convection. As a result it is decided to extend the standard gradient-diffusion model for the turbulent energy diffusion by modeling its closure terms separately. Coupling of two models leads to an extended RANS model for the turbulent energy diffusion. The involved closure term, the turbulent diffusion of heat flux, is studied based on its transport equation. This results in a buoyancy-extended version of the Daly and Harlow model. The models for all closure terms and for the turbulent energy diffusion are validated with the help of DNS data for internally heated fluid layers with Prandtl number Pr = 7 and for Rayleigh–Bénard convection with Pr = 0.71. It is found that the buoyancy-extended diffusion model which involves also a transport equation for the variance of the vertical velocity fluctuation gives improved turbulent energy diffusion data for the combined case with local stable and unstable stratification and that it allows for the required counter gradient energy flux.     

Removing of vortices in incompressible flows by distributed controls

We consider an optimal control problem of fluids flow. The fluid motion is governed by the incompressible time-dependent Navier-Stokes equations. A new optimal control formulation for the reduction and possibly extinction of vortices is proposed. A cost functional based on a local dynamical systems characterization of vortices is investigated. The resulting functional is a non-convex function of the velocity gradient tensor. The optimality system describing first order necessary optimality conditions is derived. The gradient and the second derivative of the cost functional with respect to the distributed control are established.     

Spectral methods for compressible reactive flows

High order simulations are necessary in order to capture fine details in resolving supersonic reactive flows. However, high Mach number compressible flows feature sharp gradients and discontinuities, which present a challenge to successful simulations using high order methods. Spectral methods have proven a powerful tool in simulation of incompressible turbulent flows, and recent advances allow the application of spectral methods to compressible reactive flows. We review the recent advances in the theory and application of spectral methods which allow stable computations of discontinuous phenomena, and the recovery of high order information via postprocessing, and present applications of high Mach number reactive flows. To cite this article: D. Gottlieb, S. Gottlieb, C. R. Mecanique 333 (2005).     

Experiment and modelling of birefringent flows using commercial CFD code

It is well-known that certain fluids are birefringent and when flows are viewed in polarised light interference fringes are observed. The fringes are caused by a phase shift in the light passing through the fluid and are proportional to the integral of the maximum shear strains in the fluid. In order to understand what is happening within the three dimensional flow and overcome the difficulties due to this integration, additional computational or experimental information is needed.

In this work, a commercially available computer code (Fluent) is used for the first time to model the flows. The flow data are then exported to a spreadsheet where the shear rates are integrated across the field and then banded for graphical output. The results from this are then compared to results generated from birefringent flow experiments and the agreement is found to be good since the modelled fringes show the same patterns as those in the experiment. This novel use of computational and experimental techniques together will allow quantitative analysis of three-dimensional flows in the future.

Currently, there are still a lot of empirical variables involved in fitting the computational fringes to the experiment, but the results of this preliminary study show that this is a promising approach to this type of problem.     

Orienting properties of discotic nematic liquid crystals in Jeffrey-Hamel flows

The orienting properties of incompressible discotic nematic liquid crystals for creeping flows between converging and diverging planar walls (Jeffrey-Hamel) are analyzed using the Leslie-Ericksen theory. The dependence of director orientation on the reactive parameter and the flow kinematics is presented. Closed form stationary solutions for the director orientation are found when elastic effects are neglected. Stationary numerical solutions for the velocity and director fields using the full Leslie-Ericksen theory are presented. The director field in converging flow is characterized by azimuthal (radial) centerline orientation, by being asymmetric with respect to the azimuthal (radial) direction, and by having an allowed orientation range that spans two half-quadrants (full quadrants). In the limiting case of perfectly flat disk ( –) the flow-induced director orientation in converging flow is the azimuthal direction, while in diverging flow the director rotates by a full n radians. By reducing the vertex angle between the walls to vanishingly small values, converging flow solutions properly reduce to those of flow between parallel plates, but diverging flows are expected to lead to a new instability.     

Transient flows of a second grade fluid

Exact analytical solutions for a class of unsteady unidirectional flows of an incompressible second-order fluid are constructed. The flows are generated impulsively from rest by motion of a plate or two plates or by sudden application of a pressure gradient. Expressions for velocity, flux and skin friction are obtained for both large and small times. It is found that large and small times solutions are dependent on the coefficient of viscoelasticity. The solutions corresponding to Newtonian fluids can be easily obtained from those for fluids of second order by letting the viscoelastic parameter to be zero.     

Quelques modèles diffusifs capillaires de type KortewegSome diffusive capillary models of Korteweg type.

The purpose of this Note is to propose new diffusive capillary models of Korteweg type and discuss their mathematical properties. More precisely, we introduce viscous models which provide some additional information on the behavior of the density close to vacuum. We actually prove that if some compatibility conditions between diffusion and capillarity are satisfied, some extra regularity information on a quantity involving the density is available. We obtain a non-trivial equality deduced from the special structure of the momentum equation. This Note generalizes to some extent the authors' previous works on the Korteweg model (with constant capillary coefficient) and on the shallow water equation. To cite this article: D. Bresch, B. Desjardins, C. R. Mecanique 332 (2004).     

Numerical Analysis of the Influence of Thermal Stresses on the Nonlinear Thermomolecular Pressure Difference Effect

On the basis of a numerical analysis of the non-Navier-Stokes gas-dynamic equations for slow non-isothermal gas flows, the nonlinear thermomolecular pressure difference effect due to a large temperature gradient along the lateral surface of a capillary is investigated. It is shown that the magnitude of the effect is substantially different from the values calculated using the Navier-Stokes equations. For two models of molecular interaction (Maxwell molecules and hard spheres), the possibility of a quasi-one-dimensional interpretation of the effect for experimental estimation purposes is demonstrated. The solutions of the relaxation kinetic equation for flow in a plane capillary at small Knudsen numbers and the gas-dynamic equations for slow non-isothermal flows are compared and the range of their applicability is estimated.     

Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier–Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier–Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.     

A high order moving boundary treatment for compressible inviscid flows

We develop a high order numerical boundary condition for compressible inviscid flows involving complex moving geometries. It is based on finite difference methods on fixed Cartesian meshes which pose a challenge that the moving boundaries intersect the grid lines in an arbitrary fashion. Our method is an extension of the so-called inverse Lax–Wendroff procedure proposed in [17] for conservation laws in static geometries. This procedure helps us obtain normal spatial derivatives at inflow boundaries from Lagrangian time derivatives and tangential derivatives by repeated use of the Euler equations. Together with high order extrapolation at outflow boundaries, we can impose accurate values of ghost points near the boundaries by a Taylor expansion. To maintain high order accuracy in time, we need some special time matching technique at the two intermediate Runge–Kutta stages. Numerical examples in one and two dimensions show that our boundary treatment is high order accurate for problems with smooth solutions. Our method also performs well for problems involving interactions between shocks and moving rigid bodies.