Computational study of the transonic flow around a rocket-shaped body

Transonic flows around a rocket were computed using the second-order TVD scheme proposed by Harten for solving the two- and three-dimensional unsteady, compressible Navier-Stokes equations in the conservation-law form. LU-ADI and DD-ADI schemes were employed to the implicit part. The influences of the Reynolds number on the shock-wave/boundary-layer and the shock-wave/vortex interaction were clarified by the two-dimensional analysis. The three-dimensional computations show that there is no stationary shock-wave except near the nose cone. In the wake region, a pair of asymmetrical vortices separates periodically, and the shear layer oscillates. The computed pressure distribution on the surface of the body was compared with that of the three-dimensional experiment. The qualitative agreement of the general profile was good.     

Mathematical simulation of turbulent separated transonic flows around the bodies of revolution

The properties of the turbulent separated flows around boat-tailed objects, especially in transonic regimes, are very complex and have still not been fully understood. With a variation of the free-stream Mach number, the flow structure, the size and location of separation areas, internal supersonic regions, and the position and intensity of internal shocks vary significantly. These flow properties determine the complexity of the numerical modeling problem and high demands on the algorithms used. The paper presents a comparison of the numerical results obtained on the basis of different mathematical models with the experimental data. The investigations into fundamental properties of transonic flow transformation are presented as well.     

Numerical simulation of the flow around rows of cylinders

Two-dimensional, laminar, unsteady, water flow around cylinder arrays of unequal sizes was simulated using FLUENT™ at Reynolds numbers below 150 (based on the free-stream velocity and first row cylinder diameter). The flow pattern through two rows of inline cylinders showed incomplete vortex shedding behind the first row at a separation distance less than 2d. Karman vortices were not formed and a near-stagnant separated flow region appeared between the aligned cylinders. Cylinders in staggered arrangements shed Karman vortices regardless of the separation between the two rows. This research has shed light on the detailed flow through paper machine forming fabrics.     

Numerical simulation of interaction between turbulent flow and a vibrating airfoil

The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil. A solid elastically supported airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the stabilized finite element treatment of the Reynolds averaged Navier–Stokes (RANS) approach, the use of turbulence models and the solution of the system of ordinary differential equations describing the airfoil motion. The time dependent computational domain and a moving grid are taken into account with the aid of the Arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier–Stokes equations. High Reynolds numbers up to 10⁶ require to use a suitable stabilization of the finite element discretization and the application of a turbulence model. We apply the algebraic turbulence model, which was designed by Baldwin and Lomax and modified by Rostand. The developed technique was tested by the simulation of flow past a flat rigid plate and the computation of pressure distribution around a rotating airfoil with prescribed motion. Finally, the method was applied to the simulation of flow induced airfoil vibrations. This research was supported under the Grant No. IAA200760613 of the Grant Agency of Academy of Sciences of the Czech Republic. The research of M. Feistauer was partly supported by the research project MSM 0021620839 financed by the Ministry of Education of the Czech Republic and the research of L. Dubcová was partly supported by the grant No. 48607 of the Grant Agency of the Charles University. The authors acknowledge the support of these institutions.     

Numerical simulation of turbulent flow in the inlet region of a smooth pipe

An algebraic-Q4 turbulent eddy viscosity model expresses the eddy viscosity as a solution of a quartic (Q4) equation. The model is applied to numerical simulation of developing turbulent flow in the inlet region of a smooth pipe. Predictions of the flow characteristics, such as velocity profiles accross and along a pipe, pressure drop along a pipe are found in good agreement with experimental data.     

Numerical modeling of two-phase transonic flow

Our work is aimed at the development of numerical method for the modeling of transonic flow of wet steam including condensation/evaporation phase change. We solve a system of PDE’s consisting of Euler or Navier-Stokes equations for the mixture of vapor and liquid droplets and transport equations for the integral parameters describing the droplet size spectra. Numerical method is based on a fractional step technique due to the stiff character of source terms, i.e. we solve separately the set of homogenous PDE’s by the finite volume method and the remaining set of ODE’s either by explicit Runge-Kutta or implicit Euler method. The finite volume method is based on the Lax-Wendroff scheme with conservative artificial dissipation terms for structured grid. We also note result achieved by recently developed finite volume method with VFFC scheme. We discuss numerical results of steady and unsteady two-phase transonic flow in 2D nozzle, 2D and 3D turbine cascade and 2D turbine stage with moving rotor cascade.     

Vortex simulation of active control strategies for transitional backward-facing step flows

In this work a vortex method is used to simulate an incompressible two-dimensional transitional flow over a backward-facing step. The simulations are validated for two different Reynolds numbers comparing to previous studies. Then, two different control strategies are implemented to modify the shedding, the recirculation zone behind the step and the transport in the channel. The first technique consists in using a pulsing inlet velocity and the second one is based on local oscillating jets implemented on the step vertical wall. The influence of these controls on several characteristic functionals related to the flow is carefully investigated. Both, open-loop and closed-loop active control approaches are performed in order to choose the most efficient control methods.     

Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions

Numerical solutions of 2-D laminar flow over a backward-facing step at high Reynolds numbers are presented. The governing 2-D steady incompressible Navier-Stokes equations are solved with a very efficient finite difference numerical method which proved to be highly stable even at very high Reynolds numbers. Present solutions of the laminar flow over a backward-facing step are compared with experimental and numerical results found in the literature.     

Numerical simulation of turbulent flow in complex geometries used in power plants

Performance degradations or improvements of coal-fired power stations depend on effective functioning of pulveriser equipment and combustion efficiency of furnaces in boilers. The function of a pulveriser is to grind the lumped coal and transfer the fine coal to the furnace for efficient combustion. However, the presence of several solid objects inside the mill, flow of air and particles takes turn around from inlet to outlet. The flow simulation process involves the geometrical modelling, grid generation and particle trajectories for the given flow conditions, and has been investigated to understand the flow path in grinding chamber, separator and classifier. The behaviour of turbulent air flow motion with fly ash particle paths on Lagrangian scale in computational domain are obtained through CFD/CAD software packages. The understanding developed with reference to recirculation flows in the inlet duct, non-uniform flow over the height of bowl mill and unequal flow at exit, provides valuable insights to designers for optimisation of components for better efficiency.     

Numerical simulation of turbulent reactive flows

A direct numerical simulation of a turbulent homogeneous field is used study the decay of the concentration of a scalar quantity which is advected, diffused and undergoes the effect of a sink term which models the effect of a chemical reaction. The reaction rate and the one-species formulation used herein are oriented towards the simulation of the combustion of a premixed gas in order to study various quantities useful for turbulent combustion models. Computations yield results depending on the “chemical time” under the form of various probability density functions (PDF) calculated from the realizations of the reactive scalar fields.     

An investigation of transonic flow over axisymmetric rigid body

The aim of this study is to investigate transonic flow over the axisymmetric rigid body of revolutions using matched asymptotic expansions of high Reynolds number flow. For this purpose the triple-deck model is employed. It allows to study the flow separation near a junction line where a circular cylinder is connected to a divergent conical body. It is found that in the axisymmetric transonic flow the interaction region is governed by the viscous-inviscid interaction process, where the axisymmetric Karman-Guderley equation in the inviscid part of the flow should be coupled with Prandtl’s boundary layer equations for the viscous sublayer. The coupled governing equations of the interaction region is solved using a semi-direct numerical method considering proper boundary conditions. Numerical results imply that incipience of separation may appear over the axisymmetric rigid body subject to body shape and transonic axisymmetric nature makes the flow much less prone to separation as compared to the two-dimensional flow.     

Numerical analysis of confined turbulent flow

This work outlines a second order accurate, coupled, conservative new numerical scheme for solving a two dimensional incompressible turbulent flow filed. Mean vorticity, ω, and mean stream function, ψ, are used as the mean flow dependent variables. The turbulent kinetic energy $\text{k}$ and the turbulent energy decay rate, ?, are used to define the turbulent state. In the present computational scheme two systems of equations and variables are considered: the mean flow system, ψ-ω, and the turbulent state system, $\text{k-?}$. Every system is solved implicity in a coupled double loop manner, and all the flow equations are solved iteratively in the global sense. Since the turbulence boundary conditions have a non-regular variation near a solid wall, they are coupled to the equations implicitly in both systems. In this way the numerical instabilities due to the irregular form of the equations near the solid walls are suppressed. The rate of convergence of the new numerical scheme of the coupled systems ψ-ω and $\text{k-?}$ is twice that realized when solving these equations separately. The necessary conditions for convergence of the numerical equations are investigated as well as the rate of convergence features. The detailed stability conditions are derived. As an example, the axisymmetric mixing of two confined jets with an internal heat source is considered with this numerical scheme.     

Numerical predictions of backward-facing step flows in microchannels using extended Navier–Stokes equations

Flows in microchannels were successfully predicted, in the past, both analytically and numerically, employing the extended Navier–Stokes equations (ENSE). In ENSE, the self-diffusion transport of mass, together with the resulting momentum and heat transport, is taken into account properly and the same is omitted in the classical Navier–Stokes equations. The ENSE have been employed here to numerically predict backward-facing step flows in microchannels, and the predictions are summarized in this paper. The results obtained by employing ENSE are compared with the available literature data computed by both direct simulation Monte Carlo and slip-velocity-based simulations. The good agreement of the present results with those given in the literature evidently points out that the ENSE can be applied to gas flows through complex microchannel geometries.     

Numerical studies of specific features of the transonic flow reconstruction on a hammerhead model

The results of the numerical studies of transonic flow reconstruction occurring with an increase in the free stream Mach number on a hammerhead cone-cylinder body with a small break angle in the generatrix are presented. A turbulent flow regime is considered. The Reynolds equations with different turbulence models are used. The numerical results are compared to the experimental data and the results of the calculation using the Euler model.     

Three-dimensional compressible-incompressible turbulent flow simulation using a pressure-based algorithm

In this work, we extend a finite-volume pressure-based incompressible algorithm to solve three-dimensional compressible and incompressible turbulent flow regimes. To achieve a hybrid algorithm capable of solving either compressible or incompressible flows, the mass flux components instead of the primitive velocity components are chosen as the primary dependent variables in a SIMPLE-based algorithm. This choice warrants to reduce the nonlinearities arose in treating the system of conservative equations. The use of a new Favre-averaging like technique plays a key role to render this benefit. The developed formulations indicate that there is less demand to interpolate the fluxes at the cell faces, which is definitely a merit. To impose the hyperbolic behavior in compressible flow regimes, we introduce an artificial hyperbolicity in pressure correction equation. We choose k-ω turbulence model and incorporate the compressibility effect as a correction. It is shown that the above considerations grant to achieve a robust algorithm with great capabilities in solving both flow regimes with a reasonable range of Mach number applications. To evaluate the ability of the new pressure-based algorithm, three test cases are targeted. They are incompressible backward-facing step problem, compressible flow over a wide range of open to closed cavities, and compressible turbulent flow in a square duct. The current results indicate that there are reliable agreements with those of experiments and other numerical solutions in the entire range of investigation.     

Assessment of turbulence-transport models including non-linear rng eddy-viscosity formulation and second-moment closure for flow over a backward-facing step

A computational study is performed in which the predictive capabilities of a range of eddy-viscosity and second-moment-closure models are examined by reference to a separated flow behind a backward-facing step in an expanding channel. The models include three second-moment-closure variants, all being of the ‘Launder-Reece-Rodi’ type, two RNG k—ε forms, one combining the RNG approach with a non-linear eddy-viscosity formulation, and a low-Re k—ε model. The study demonstrates that to achieve a solution similar to that returned by second-moment closure, the RNG formulation needs to be implanted into a non-linear eddy-viscosity framework; neither returns, on its own, the correct behaviour, not even for mean-flow features. Moreover, relatively minor variations within second-moment closure—specifically, such relating to wall-induced effects on turbulence isotropisation and to stress diffusion—can significantly alter the overall performance of the closure. All models specifically designed to return realistic solutions for normal stresses seriously over-estimate anisotropy.     

Pseudospectral simulation of turbulent viscoelastic channel flow

The methodology and validation of direct numerical simulations of viscoelastic turbulent channel flow are presented here. Using differential constitutive models derived from kinetic and network theories, numerical simulations have demonstrated drag reduction for various values of the parameters, under conditions where there is a substantial increase in the extensional viscosity compared to the shear viscosity (Sureshkumar, Beris, Handler, Direct numerical simulation of turbulent channel flow of a polymer solution, Phys. Fluids 9 (1997) 743–755 and Dimitropoulos, Sureshkumar, Beris, Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological parameters, J. Non-Newtonian Fluid Mech. 79 (1998) 433–468). In this work, new results pertaining to the Reynolds stress and the pressure are presented, and the convergence of the pseudospectral algorithm utilized in the simulations, as well as its parallel implementation, are discussed in detail. It is shown that the lack of mesh refinement, or the use of a larger value for the artificial stress diffusivity used to stabilize the conformation tensor evolution equations, introduce small quantitative errors which qualitatively have the effect of lowering the drag reduction capability of the simulated fluid. However, an insufficient size of the periodic computational domain can also introduce errors in certain cases, which albeit usually small, can qualitatively alter various features of the solution.     

Large-eddy simulation of streamwise-rotating turbulent channel flow

In many engineering and industrial applications the investigation of rotating turbulent flow is of great interest. Whereas some research has been done concerning channel flows with a spanwise rotation axis, only few investigations have been performed on channel flows with a rotation about the streamwise axis. In the present study an LES of a turbulent streamwise-rotating channel flow at Re_τ = 180 is performed using a moving grid method. The three-dimensional structures and the details of the secondary flow distribution are analyzed and compared with experimental data. The numerical-experimental comparison shows a convincing agreement as to the overall flow features. The results confirm the development of a secondary flow in the spanwise direction, which has been found to be correlated to the rotational speed. Furthermore, the findings show the distortion of the main flow velocity profile, the slight decrease of the streamwise Reynolds stresses in the vicinity of the walls, and the pronounced increase of the spanwise Reynolds stresses at higher rotation rates near the walls and particularly in the symmetry region. As to the numerical set-up it is shown that periodic boundary conditions in the spanwise direction suffice if the spanwise extent of the computational domain is larger than 10 times the channel half width.     

A computer simulation of turbulent jet flow

The evolution of a round turbulent jet is simulated numerically under the constraint of axial symmetry. The vortex sheet shed from the orifice is represented by vortex ring elements, with a velocity field cut-off to control close encounters. Large-scale vortex clusters form in the model jet, similar to those inferred from laboratory flow visualization experiments. However, comparisons of statistical properties reveal significant differences between the axisymmetric model flow and real turbulent jets.     

Lattice Boltzmann simulation of turbulent flow laden with finite-size particles

The paper describes a particle-resolved simulation method for turbulent flow laden with finite size particles. The method is based on the multiple-relaxation-time lattice Boltzmann equation. The no-slip boundary condition on the moving particle boundaries is handled by a second-order interpolated bounce-back scheme. The populations at a newly converted fluid lattice node are constructed by the equilibrium distribution with non-equilibrium corrections. MPI implementation details are described and the resulting code is found to be computationally efficient with a good scalability. The method is first validated using unsteady sedimentation of a single particle and sedimentation of a random suspension. It is then applied to a decaying isotropic turbulence laden with particles of Kolmogorov to Taylor microscale sizes. At a given particle volume fraction, the dynamics of the particle-laden flow is found to depend mainly on the effective particle surface area and particle Stokes number. The presence of finite-size inertial particles enhances dissipation at small scales while reducing kinetic energy at large scales. This is in accordance with related studies. The normalized pivot wavenumber is found to not only depend on the particle size, but also on the ratio of particle size to flow scales and particle-to-fluid density ratio.