Numerical analysis of transpiration cooling in carbon dioxide atmosphere at hypersonic Mach numbers

The numerical solutions are obtained for skin friction, heat transfer to the wall and growth of boundary layer along the flat plate by employing two dimensional Navier-Stokes equations governing the hypersonic flow coupled with species continuity equations. Flow fields have been computed along the flat plate in CO2 atmosphere in the presence of transpiration cooling using air and carbon dioxide.

Arbitrary Lagrangian-Eulerian finite-element method for computation of two-phase flows with soluble surfactants

A finite-element scheme based on a coupled arbitrary Lagrangian-Eulerian and Lagrangian approach is developed for the computation of interface flows with soluble surfactants. The numerical scheme is designed to solve the time-dependent Navier-Stokes equations and an evolution equation for the surfactant concentration in the bulk phase, and simultaneously, an evolution equation for the surfactant concentration on the interface. Second-order isoparametric finite elements on moving meshes and second-order isoparametric surface finite elements are used to solve these equations. The interface-resolved moving meshes allow the accurate incorporation of surface forces, Marangoni forces and jumps in the material parameters. The lower-dimensional finite-element meshes for solving the surface evolution equation are part of the interface-resolved moving meshes. The numerical scheme is validated for problems with known analytical solutions. A number of computations to study the influence of the surfactants in 3D-axisymmetric rising bubbles have been performed. The proposed scheme shows excellent conservation of fluid mass and of the total mass of the surfactant. (C) 2012 Elsevier Inc. All rights reserved.

Stability of developing film flow down an inclined surface

Film flows on inclined surfaces are often assumed to be of constant thickness, which ensures that the velocity profile is half-Poiseuille. It is shown here that by shallow water theory, only flows in a portion of Reynolds number-Froude number (Re-Fr) plane can asymptotically attain constant film thickness. In another portion on the plane, the constant thickness solution appears as an unstable fixed point, while in other regions the film thickness seems to asymptote to a positive slope. Our simulations of the Navier-Stokes equations confirm the predictions of shallow water theory at higher Froude numbers, but disagree with them at lower Froude numbers. We show that different regimes of film flow show completely different stability behaviour from that predicted earlier. Supercritical decelerating flows are shown to be always unstable, whereas accelerating flows become unstable below a certain Reynolds number for a given Froude number. Subcritical flows on the other hand are shown to be unstable above a certain Reynolds number. In some range of parameters, two solutions for the base flowexist, and the attached profile is found to be more stable. All flows except those with separation become more stable as they proceed downstream. (C) 2012 American Institute of Physics. http://dx.doi.org/10.1063/1.4758299]

Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics

This paper attempts to unravel any relations that may exist between turbulent shear flows and statistical mechanics through a detailed numerical investigation in the simplest case where both can be well defined. The flow considered for the purpose is the two-dimensional (2D) temporal free shear layer with a velocity difference Delta U across it, statistically homogeneous in the streamwise direction (x) and evolving from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain L x +/-infinity. Extensive computer simulations of the flow are carried out through appropriate initial-value problems for a ``vortex gas'' comprising N point vortices of the same strength (gamma = L Delta U/N) and sign. Such a vortex gas is known to provide weak solutions of the Euler equation. More than ten different initial-condition classes are investigated using simulations involving up to 32 000 vortices, with ensemble averages evaluated over up to 10(3) realizations and integration over 10(4)L/Delta U. The temporal evolution of such a system is found to exhibit three distinct regimes. In Regime I the evolution is strongly influenced by the initial condition, sometimes lasting a significant fraction of L/Delta U. Regime III is a long-time domain-dependent evolution towards a statistically stationary state, via ``violent'' and ``slow'' relaxations P.-H. Chavanis, Physica A 391, 3657 (2012)], over flow time scales of order 10(2) and 10(4)L/Delta U, respectively (for N = 400). The final state involves a single structure that stochastically samples the domain, possibly constituting a ``relative equilibrium.'' The vortex distribution within the structure follows a nonisotropic truncated form of the Lundgren-Pointin (L-P) equilibrium distribution (with negatively high temperatures; L-P parameter lambda close to -1). The central finding is that, in the intermediate Regime II, the spreading rate of the layer is universal over the wide range of cases considered here. The value (in terms of momentum thickness) is 0.0166 +/- 0.0002 times Delta U. Regime II, extensively studied in the turbulent shear flow literature as a self-similar ``equilibrium'' state, is, however, a part of the rapid nonequilibrium evolution of the vortex-gas system, which we term ``explosive'' as it lasts less than one L/Delta U. Regime II also exhibits significant values of N-independent two-vortex correlations, indicating that current kinetic theories that neglect correlations or consider them as O(1/N) cannot describe this regime. The evolution of the layer thickness in present simulations in Regimes I and II agree with the experimental observations of spatially evolving (3D Navier-Stokes) shear layers. Further, the vorticity-stream-function relations in Regime III are close to those computed in 2D Navier-Stokes temporal shear layers J. Sommeria, C. Staquet, and R. Robert, J. Fluid Mech. 233, 661 (1991)]. These findings suggest the dominance of what may be called the Kelvin-Biot-Savart mechanism in determining the growth of the free shear layer through large-scale momentum and vorticity dispersal.

Numerical modeling of the non-isothermal liquid droplet impact on a hot solid substrate

The heat transfer from a solid phase to an impinging non-isothermal liquid droplet is studied numerically. A new approach based on an arbitrary Lagrangian-Eulerian (ALE) finite element method for solving the incompressible Navier Stokes equations in the liquid and the energy equation within the solid and the liquid is presented. The novelty of the method consists in using the ALE-formulation also in the solid phase to guarantee matching grids along the liquid solid interface. Moreover, a new technique is developed to compute the heat flux without differentiating the numerical solution. The free surface and the liquid solid interface of the droplet are represented by a moving mesh which can handle jumps in the material parameter and a temperature dependent surface tension. Further, the application of the Laplace-Beltrami operator technique for the curvature approximation allows a natural inclusion of the contact angle. Numerical simulation for varying Reynold, Weber, Peclet and Biot numbers are performed to demonstrate the capabilities of the new approach. (C) 2014 Elsevier Ltd. All rights reserved.

Simulations of impinging droplets with surfactant-dependent dynamic contact angle

An arbitrary Lagrangian-Eulerian (ALE) finite element scheme for computations of soluble surfactant droplet impingement on a horizontal surface is presented. The numerical scheme solves the time-dependent Navier-Stokes equations for the fluid flow, scalar convection-diffusion equation for the surfactant transport in the bulk phase, and simultaneously, surface evolution equations for the surfactants on the free surface and on the liquid-solid interface. The effects of surfactants on the flow dynamics are included into the model through the surface tension and surfactant-dependent dynamic contact angle. In particular, the dynamic contact angle (theta(d)) of the droplet is defined as a function of the surfactant concentration at the contact line and the equilibrium contact angle (theta(0)(e)) of the clean surface using the nonlinear equation of state for surface tension. Further, the surface forces are included into the model as surface divergence of the surface stress tensor that allows to incorporate the Marangoni effects without calculating the surface gradient of the surfactant concentration on the free surface. In addition to a mesh convergence study and validation of the numerical results with experiments, the effects of adsorption and desorption surfactant coefficients on the flow dynamics in wetting, partially wetting and non-wetting droplets are studied in detail. It is observed that the effects of surfactants are more in wetting droplets than in the non-wetting droplets. Further, the presence of surfactants at the contact line reduces the equilibrium contact angle further when theta(0)(e) is less than 90 degrees, and increases it further when theta(0)(e) is greater than 90 degrees. Nevertheless, the presence of surfactants has no effect on the contact angle when theta(0)(e) = 90 degrees. The numerical study clearly demonstrates that the surfactant-dependent contact angle has to be considered, in addition to the Marangoni effect, in order to study the flow dynamics and the equilibrium states of surfactant droplet impingement accurately. The proposed numerical scheme guarantees the conservation of fluid mass and of the surfactant mass accurately. (C) 2015 Elsevier Inc. All rights reserved.

Modeling of the non-isothermal liquid droplet impact on a heated solid substrate with heterogeneous wettability

A comprehensive numerical investigation on the impingement and spreading of a non-isothermal liquid droplet on a solid substrate with heterogeneous wettability is presented in this work. The time-dependent incompressible Navier-Stokes equations are used to describe the fluid flow in the liquid droplet, whereas the heat transfer in the moving droplet and in the solid substrate is described by the energy equation. The arbitrary Lagrangian-Eulerian (ALE) formulation with finite elements is used to solve the time-dependent incompressible Navier-Stokes equation and the energy equation in the time-dependent moving domain. Moreover, the Marangoni convection is included in the variational form of the Navier-Stokes equations without calculating the partial derivatives of the temperature on the free surface. The heterogeneous wettability is incorporated into the numerical model by defining a space-dependent contact angle. An array of simulations for droplet impingement on a heated solid substrate with circular patterned heterogeneous wettability are presented. The numerical study includes the influence of wettability contrast, pattern diameter, Reynolds number and Weber number on the confinement of the spreading droplet within the inner region, which is more wettable than the outer region. Also, the influence of these parameters on the total heat transfer from the solid substrate to the liquid droplet is examined. We observe that the equilibrium position depends on the wettability contrast and the diameter of the inner surface. Consequently. the heat transfer is more when the wettability contrast is small and/or the diameter of inner region is large. The influence of the Weber number on the total heat transfer is more compared to the Reynolds number, and the total heat transfer increases when the Weber number increases. (C) 2015 Elsevier Ltd. All rights reserved.

Implicit and multigrid procedures for steady-state computations with upwind algorithms

A computational study for the convergence acceleration of Euler and Navier-Stokes computations with upwind schemes has been conducted in a unified framework. It involves the flux-vector splitting algorithms due to Steger-Warming and Van Leer, the flux-difference splitting algorithms due to Roe and Osher and the hybrid algorithms, AUSM (Advection Upstream Splitting Method) and HUS (Hybrid Upwind Splitting). Implicit time integration with line Gauss-Seidel relaxation and multigrid are among the procedures which have been systematically investigated on an individual as well as cumulative basis. The upwind schemes have been tested in various implicit-explicit operator combinations such that the optimal among them can be determined based on extensive computations for two-dimensional flows in subsonic, transonic, supersonic and hypersonic flow regimes. In this study, the performance of these implicit time-integration procedures has been systematically compared with those corresponding to a multigrid accelerated explicit Runge-Kutta method. It has been demonstrated that a multigrid method employed in conjunction with an implicit time-integration scheme yields distinctly superior convergence as compared to those associated with either of the acceleration procedures provided that effective smoothers, which have been identified in this investigation, are prescribed in the implicit operator.

Statistically steady turbulence in thin films: direct numerical simulations with Ekman friction

We present a detailed direct numerical simulation (DNS) of the two-dimensional Navier-Stokes equation with the incompressibility constraint and air-drag-induced Ekman friction; our DNS has been designed to investigate the combined effects of walls and such a friction on turbulence in forced thin films. We concentrate on the forward-cascade regime and show how to extract the isotropic parts of velocity and vorticity structure functions and hence the ratios of multiscaling exponents. We find that velocity structure functions display simple scaling, whereas their vorticity counterparts show multiscaling, and the probability distribution function of the Weiss parameter 3, which distinguishes between regions with centers and saddles, is in quantitative agreement with experiments.

Simulation of laminar flow in a three-dimensional lid-driven cavity by lattice Boltzmann method

Purpose - The purpose of this paper is to apply lattice Boltzmann equation method (LBM) with multiple relaxation time (MRT) model, to investigate lid-driven flow in a three-dimensional (3D), rectangular cavity, and compare the results with flow in an equivalent two-dimensional (2D) cavity. Design/methodology/approach - The second-order MRT model is implemented in a 3D LBM code. The flow structure in cavities of different aspect ratios (0.25-4) and Reynolds numbers (0.01-1000) is investigated. The LBM simulation results are compared with those from numerical solution of Navier-Stokes (NS) equations and with available experimental data. Findings - The 3D simulations demonstrate that 2D models may predict the flow structure reasonably well at low Reynolds numbers, but significant differences with experimental data appear at high Reynolds numbers. Such discrepancy between 2D and 3D results are attributed to the effect of boundary layers near the side-walls in transverse direction (in 3D), due to which the vorticity in the core-region is weakened in general. Secondly, owing to the vortex stretching effect present in 3D flow, the vorticity in the transverse plane intensifies whereas that in the lateral plane decays, with increase in Reynolds number. However, on the symmetry-plane, the flow structure variation with respect to cavity aspect ratio is found to be qualitatively consistent with results of 2D simulations. Secondary flow vortices whose axis is in the direction of the lid-motion are observed; these are weak at low. Reynolds numbers, but become quite strong at high Reynolds numbers. Originality/value - The findings will be useful in the study of variety of enclosed fluid flows.

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier-Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier-Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (eta, tau). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.

Nonaxisymmetric unsteady motion over a rotating disk in the presence of free convection and magnetic field

The nonaxisymmetric unsteady motion produced by a buoyancy-induced cross-flow of an electrically conducting fluid over an infinite rotating disk in a vertical plane and in the presence of an applied magnetic field normal to the disk has been studied. Both constant wall and constant heat flux conditions have been considered. It has been found that if the angular velocity of the disk and the applied magnetic field squared vary inversely as a linear function of time (i.e. as (1??t*)?1, the governing Navier-Stokes equation and the energy equation admit a locally self-similar solution. The resulting set of ordinary differential equations has been solved using a shooting method with a generalized Newton's correction procedure for guessed boundary conditions. It is observed that in a certain region near the disk the buoyancy induced cross-flow dominates the primary von Karman flow. The shear stresses induced by the cross-flow are found to be more than these of the primary flow and they increase with magnetic parameter or the parameter ? characterizing the unsteadiness. The velocity profiles in the x- and y-directions for the primary flow at any two values of the unsteady parameter ? cross each other towards the edge of the boundary layer. The heat transfer increases with the Prandtl number but reduces with the magnetic parameter.

Turbulent flow computations using upwind RANS technique for a flat plate and an airfoil

Reynolds Averaged Navier Stokes (RANS) equations are solved using third order upwind biased Roe's scheme for the inviscid fluxes and second order central difference scheme for the viscous fluxes. The Baldwin & Lomax turbulence model is employed for Reynolds stresses. The governing equations are solved using finite-volume implicit scheme in body fitted curvilinear coordinate O-grid system. Computations axe reported for a flat plate apart from RAE 2822 and NACA 0012 airfoils. Results for the flat plate at M = 0.3, R-c = 4.0 x 10(6) compare favourably with the analytical solution. Results for the two airfoils are compared with experiment. There is a good agreement in C-p distribution between experiment and computation for both the airfoils. Comparison of C-f distribution with experiment for RAE 2822 airfoil is reasonable.

Cumulus clouds and convective boundary layers: a tropical perspective on two turbulent shear flows

This paper is a review prepared for the second Marseille Colloquium on the mechanics of turbulence, held in 2011, 50 years after the first. The review covers recent developments in our understanding of the large-scale dynamics of cumulus cloud flows and of the atmospheric boundary layer in the low-wind convective regime that is often encountered in the tropics. It has recently been shown that a variety of cumulus cloud forms and life cycles can be experimentally realized in the laboratory, with the transient diabatic plume taken as the flow model for a cumulus cloud. The plume is subjected to diabatic heating scaled to be dynamically similar to heat release from phase changes in clouds. The experiments are complemented by exact numerical solutions of the Navier-Stokes-Boussinesq equations for plumes with scaled off-source heating. The results show that the Taylor entrainment coefficient first increases with heating, reaches a positive maximum and then drops rapidly to zero or even negative values. This reduction in entrainment is a consequence of structural changes in the flow, smoothing out the convoluted boundaries in the non-diabatic plume, including the tongues engulfing the ambient flow. This is accompanied by a greater degree of mixedness in the core flow because of lower dilution by the ambient fluid. The cloud forms generated depend strongly on the history of the diabatic heating profile in the vertical direction. The striking effects of heating on the flow are attributable to the operation of the baroclinic torque due to the temperature field. The mean baroclinic torque is shown to peak around a quasi-cylindrical sheet situated midway between the axis of the flow and the edges. This torque is shear-enhancing and folds down the engulfment tongues. The increase in mixedness can be traced to an explosive growth in the enstrophy, triggered by a strong fluctuating baroclinic torque that acts as a source, especially at the higher wave numbers, thus enhancing the mixedness. In convective boundary layers field measurements show that, under conditions prevailing in the tropics, the eddy fluxes of momentum and energy do not follow the Monin-Obukhov similarity. Instead, the eddy momentum flux is found to be linear in the wind speed at low winds; and the eddy heat flux is, to a first approximation, governed by free convection laws, with wind acting as a small perturbation on a regime of free convection. A new boundary layer code, based on heat flux scaling rather than wall-stress scaling, shows promising improvements in predictive skills of a general circulation model.