Solution of a hyperbolic system of turbulence-model equations by the “box” scheme

The momentum and continuity equations for a two-dimensional boundary layer, together with an empirical first-order partial differential equation for shear-stress transport, form a hyperbolic system of equations for u, v and τ. Here the Bradshaw-Ferriss-Atwell version of this turbulence model is solved by the Keller-Cebeci “box” scheme, which is particularly suited to systems of equations that are individually of first order. Computing time is about equal to that taken by a method-of-characteristics program if the same number of grid points are used across the layer and in the streamwise direction. However, the box scheme allows larger x-steps to be taken in the streamwise direction leading to smaller computing times.     

Large-eddy simulations of film cooling flows

Large-eddy simulations (LES) of a jet in a cross-flow (JICF) problem are carried out to investigate the turbulent flow structure and the vortex dynamics in gas turbine blade film cooling. A turbulent flat plate boundary layer at a Reynolds number of Re_∞ = 400,000 interacts with a jet issued from a pipe. To study the effect of the jet inclination angle α on the flow field, two angles are chosen, the perpendicular injection at 90° and the streamwise inclined injection at 30°. For the normal injection case a small blowing ratio of the jet velocity to the cross-stream velocity R = 0.1 is examined. For the streamwise inclined injection case two blowing ratios R = 0.1 and R = 0.48 are investigated to check the impact of the jet velocity on the cooling performance. The time-dependent turbulent inflow information for the cross-flow is provided by a simultaneously performed LES of a spatially developing turbulent boundary layer. Whereas in the perpendicular injection case a rather large separation region is found at the leading edge of the jet hole, in the streamwise inclined injection cases no separation is observed. Compared with the normal injection case at the same blowing ratio, the streamwise inclination weakens the jet-cross-flow interaction significantly. Thus, the first appearance of the counter-rotating vortex pair (CVP) is shifted downstream and its strength is reduced. The increase of the blowing ratio leads to a stronger penetration of the jet into the cross-flow, resulting in a more upstream located and more pronounced CVP. Downstream of the jet exit the streamwise vortices are so large that besides the jet fluid also the cross-stream is partially entrained into this zone, which yields the worst cooling performance.     

Simulations of mixing for a confined co-flowing planar jet

Simulations are performed for the evolution of a mixing layer in a channel with two splitter plates at the inlet. The effect of three different velocity ratios are studied for a channel flow configuration. Solutions are obtained from the time-dependent incompressible Navier–Stokes equations by means of an artificial compressibility formulation with dual-time stepping. The transport of a conserved passive scalar is examined to assess the mechanisms of entrainment and mixing within the shear flow. Scalar probability density functions are evaluated to trace the evolution of the mixing layer along the streamwise direction. It is seen that better mixing is obtained at higher velocity ratios.     

Analysis of the turbulent boundary layer on a rotating disk

 An order of magnitude analysis of the equations of motion governing the turbulent boundary layer on a rotating disk was performed in a stationary coordinate system located at the center of the disk. The governing equations in the wall layer were derived. Using the log law for the tangential velocity, a relation between the tangential friction velocity and the Reynolds number was derived. The forms of the three components of mean velocity in the viscous sublayer were derived. Analysis of the radial momentum equation indicated the existence of two velocity scales that are both dependent on the Reynolds number. This is the reason for the lack of inner scaling observed for the mean radial velocity. The far-wall behavior of the mean tangential and vertical velocity was documented. The analysis, along with measurements, indicated that the mean downward vertical velocity induced at the edge of the boundary layer is on the order of 15% of the tangential shear velocity. Received: 5 July 2001/Accepted: 17 October 2001     

Coherent structures produced by the interaction between synthetic jets and a laminar boundary layer and their surface shear stress patterns

In this paper, the results from 3D numerical simulations of circular synthetic jets issued into a zero-pressure-gradient laminar boundary layer developing along a flat plate are reported. The simulations are undertaken using FLUENT at a wide range of actuator operating conditions. The formation and development of the coherent structures produced as a result of the interaction between the synthetic jets and the boundary layer were examined using the Q-criterion. Non-dimensional parameter space maps were established to illustrate the variations in the appearance of these resultant structures and their shear stress footprints upon the changes in the operating conditions of synthetic jets. Finally, the parameter boundary separating the two distinct types of vortical structures and surface shear stress patterns is identified. It is found that the location of this boundary correlates closely with the jet-to-freestream velocity ratio of VR = 0.4 when the Strouhal number (Str) is less than 1, whereas for Str > 1 the boundary deviates from this trend, approaching the line of dimensionless stroke length of L = 1.6. In order to investigate the potential impact of the synthetic jets on the boundary layer, the increase in the space- and time-averaged skin friction coefficient relative to the baseline case without the synthetic jets is calculated. It appears that in order to maximise the impact on the near-wall flow while keeping the energy expenditure down, it is wise to maximise the accumulated effect of hairpin vortices by keeping the spacing between consecutive hairpin vortices similar to the local boundary layer thickness upstream of the separated flow instead of producing stronger individual structures.     

Direct numerical simulation of rarefied turbulent microchannel flow

Direct numerical simulation (DNS) has been carried out to investigate the effect of weak rarefaction on turbulent gas flow and heat transfer characteristics in microchannel. The Reynolds number based on the friction velocity and the channel half width is 150. Grid number is 64 × 128 × 64. Fractional time-step method is employed for the unsteady Navier–Stokes equations, and the governing equations are discretized with finite difference method. Statistical quantities such as turbulent intensity, Reynolds shear stress, turbulent heat flux and temperature variance are obtained under various Knudsen number from 0 to 0.05. The results show that rarefaction can influence the turbulent flow and heat transfer statistics. The streamwise mean velocity and temperature increase with increase of Kn number. In the near-wall-region rarefaction can increase the turbulent intensities and temperature variance. The effects of rarefaction on Reynolds shear stress and wall-normal heat flux are presented. The instantaneous velocity fluctuations in the vicinity of the wall are visualized and the influence of Kn number on the flow structure is discussed.     

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v_θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.     

Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study

A numerical simulation for mixed convective three-dimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented. The flow is caused by nonlinear stretching surface. Conservation of energy equation involves the radiation heat flux term. Applied transverse magnetic effect of variable kind is also incorporated. Suitable nonlinear similarity transformations are used to reduce the governing equations into a set of self-similar equations. The subsequent equations are solved numerically by using shooting method. The solutions for the velocity and temperature distributions are computed for several values of flow pertinent parameters. Further, the numerical values for skin-friction coefficients and Nusselt number in respect of different nanoparticles are tabulated. A comparison between our numerical and already existing results has also been made. It is found that the velocity and thermal slip boundary condition showed a significant effect on momentum and thermal boundary layer thickness at the wall. The presence of nanoparticles stabilizes the thermal boundary layer growth.

     

Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction

The boundary layer flow of a micro-polar fluid due to a linearly stretching sheet is investigated. The influence of various flow parameters like ‘suction and injection velocity through the porous surface’, ‘viscosity parameter causing the coupling of the micro-rotation field and the velocity field’ and ‘vortex viscosity parameter’ on ‘shear stress at the surface’, ‘fluid velocity’ and ‘micro-rotation’ are studied. The governing equations of the transformed boundary layer are solved analytically using homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly studied and a proper discussion is given for the obtained results. Comparison between the HAM and numerical solutions showed excellent agreement.     

On a method for direct numerical simulation of shear layer/compression wave interaction for aeroacoustic investigations

Direct numerical simulation (DNS) of a spatially developing mixing layer was performed. The compressible three-dimensional Navier-Stokes equations were solved for pressure, velocities and entropy for this flow using a compact finite-difference scheme of sixth-order accuracy in space, combined with Runge-Kutta three-step time advancement. On one of the transverse boundaries of the box-shaped domain, a compression wave profile was imposed in pressure and velocity components via a wave decomposition of the governing equations, in order to study the interaction of an isolated weak shock wave entering the domain with the mixed subsonic/supersonic shear layer. This flow situation is found along the shear layer of supersonic, imperfectly expanded jets containing a shock cell structure. In the present work, an isolated compression-expansion structure constitutes the model problem. The domain setup and the boundary conditions were chosen such as to allow analysis of the sound field generated by the turbulent flow and the shock-turbulence interaction. The numerical method used to impose the boundary conditions and solve the compressible Navier-Stokes equations, and the choice of numerical parameters, are described in detail. Some results on the two-dimensional and three-dimensional flow field computed are presented as well.     

MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation $\epsilon$ is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.     

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.     

An investigation of turbulent open channel flow with heat transfer by large eddy simulation

Large eddy simulation of fully developed turbulent open channel flow with heat transfer is performed. The three-dimensional filtered Navier-Stokes and energy equations are numerically solved using a fractional-step method. Dynamic subgrid-scale (SGS) models for the turbulent SGS stress and heat flux are employed to close the governing equations. Two typical temperature boundary conditions, i.e., constant temperature and constant heat flux being maintained at the free surface, respectively, are used. The objective of this study is to explore the behavior of heat transfer in the turbulent open channel flow for different temperature boundary conditions and to examine the reliability of the LES technique for predicting turbulent heat transfer at the free surface, in particular, for high Prandtl number. Calculated parameters are chosen as the Prandtl number (Pr) from 1 up to 100, the Reynolds number (Re_τ) 180 based on the wall friction velocity and the channel depth. Some typical quantities, including the mean velocity, temperature and their fluctuations, heat transfer coefficients, turbulent heat fluxes, and flow structures based on the velocity, vorticity and temperature fluctuations, are analyzed.     

Numerical consistency and spurious boundary layer in the projection method

In this work, the controversial issue regarding the boundary condition for the Laplace equation has been addressed. It turned out to be that the velocity field can be superimposed by a gradient of the Helmholtz potential part that satisfies the Neumann type boundary condition and a correction term. By this representation it follows that the physical variables (u,P) are independent of the flux imposed along the computational boundary at each fractional step. Provided that, the same flux distribution is imposed at each time step along the computational boundary in the decoupled case. Furthermore, for each fractional component, the Helmholtz term generates pseudo-boundary layers, in opposite sign, along the computational boundary, to accommodate the imposed flux. The summation of both component annihilates the pseudo-boundary layer. A failure to comply with the restriction that the same flux distribution should be imposed at each time step may result in inconsistency of the solution. For a flux that is slowly varying over time, this type of inconsistency may lower the order of the scheme accuracy, while for a rapid variation over time the solution is inconsistent. The rapid variation in time is related to the initial stage of the decoupled problem with a given flux along the computational boundary. For such a case an educated guess is required for the initial velocity-potential, to prevent an inconsistent solution at an early stage of the solution.     

The parallel implementation of the one-dimensional Fourier transformed Vlasov-Poisson system

A parallel implementation of an algorithm for solving the one-dimensional, Fourier transformed Vlasov-Poisson system of equations is documented, together with the code structure, file formats and settings to run the code. The properties of the Fourier transformed Vlasov-Poisson system is discussed in connection with the numerical solution of the system. The Fourier method in velocity space is used to treat numerical problems arising due the filamentation of the solution in velocity space. Outflow boundary conditions in the Fourier transformed velocity space removes the highest oscillations in velocity space. A fourth-order compact Padé scheme is used to calculate derivatives in the Fourier transformed velocity space, and spatial derivatives are calculated with a pseudo-spectral method. The parallel algorithms used are described in more detail, in particular the parallel solver of the tri-diagonal systems occurring in the Padé scheme.

Program summary

Title of program:vlasovCatalogue identifier:ADVQProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVQProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandOperating system under which the program has been tested: Sun Solaris; HP-UX; Read Hat LinuxProgramming language used: FORTRAN 90 with Message Passing Interface (MPI)Computers: Sun Ultra Sparc; HP 9000/785; HP IPF (Itanium Processor Family) ia64 Cluster; PCs clusterNumber of lines in distributed program, including test data, etc.:3737Number of bytes in distributed program, including test data, etc.:18 772Distribution format: tar.gzNature of physical problem: Kinetic simulations of collisionless electron-ion plasmas.Method of solution: A Fourier method in velocity space, a pseudo-spectral method in space and a fourth-order Runge-Kutta scheme in time.Memory required to execute with typical data: Uses typically of the order 10⁵-10⁶ double precision numbers.Restriction on the complexity of the problem: The program uses periodic boundary conditions in space.Typical running time: Depends strongly on the problem size, typically few hours if only electron dynamics is considered and longer if both ion and electron dynamics is important.Unusual features of the program: No     

Electric potential approximations for an eight node plate finite element

The aim of this work is to develop a computational tool for multilayered piezoelectric plates: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account the transverse shear stress effects and using only five unknown generalized displacements: C⁰ finite element approximation based on first-order shear deformation theories (FSDT) [Polit O, Touratier M, Lory P. A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 1994;37:387–411] and C¹ finite element approximations using a high order shear deformation theory (HSDT) [Polit O, Touratier M. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comput Meth Appl Mech Eng 2000;185:305–24]. In this article, we present the piezoelectric extension of the FSDT eight node plate finite element. The electric potential is approximated using the layerwise approach and an evaluation is proposed in order to assess the best compromise between minimum number of degrees of freedom and maximum efficiency. On one side, two kinds of finite element approximations for the electric potential with respect to the thickness coordinate are presented: a linear variation and a quadratic variation in each layer. On the other side, the in-plane variation can be quadratic or constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post-processing level, the transverse shear stresses are deduced using the equilibrium equations.Numerous tests are presented in order to evaluate the capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions. Finally, an adaptative composite plate is evaluated using the best compromise finite element.     

Eduction of near wall flow structures responsible for large deviations of the momentum-heat transfer analogy and fluctuations of wall transfer rates in turbulent channel flow

In this paper we analyze the flow structures responsible for large local instantaneous deviations of the conventional momentum-heat transfer analogy and large fluctuations of the wall shear stress and the wall heat flux in a forced convection turbulent channel flow at low-Reynolds numbers (Re = 4570, Pr = 0.7). The analysis was carried out using a database obtained from a direct numerical simulation performed with a second-order finite volume code. The ensemble averaged velocity and temperature profiles and profiles of the turbulence intensities and turbulent heat fluxes agree well with direct numerical simulations available in the literature. When the flow was statistically fully developed, we recorded the time evolution of the velocities and temperatures near one wall of the channel. The near wall structures responsible for the extreme values of the deviations were educed by a conditional sampling technique. Results show that extreme values of the wall shear stress and wall heat transfer rates, as well as departures from the conventional analogy between momentum and heat transfer, occur within the high-speed streaks on the wall and are associated with fluctuations of the streamwise pressure gradient. These large fluctuations on the wall are produced by the combined effect of two quasi-parallel counterrotating streamwise vortices.     

Thermophoresis and MHD mixed convection three-dimensional flow of viscoelastic fluid with Soret and Dufour effects

Heat and mass transfer effects in three-dimensional mixed convection flow of viscoelastic fluid over a stretching surface with convective boundary conditions are investigated. The fluid is electrically conducting in the presence of constant applied magnetic field. Conservation laws of energy and concentration are based upon the Soret and Dufour effects. First order chemical reaction effects are also taken into account. By using the similarity transformations, the governing boundary layer equations are reduced into the ordinary differential equations. The transformed boundary layer equations are computed for the series solutions. Dimensionless velocity, temperature, and concentration distributions are shown graphically for different values of involved parameters. Numerical values of local Nusselt and Sherwood numbers are computed and analyzed. It is found that the behaviors of viscoelastic, mixed convection, and concentration buoyancy parameters on the Nusselt and Sherwood numbers are similar. However, the Nusselt and Sherwood numbers have qualitative opposite effects for Biot number, thermophoretic parameter, and Soret-Dufour parameters.

     

The method of lines applied to crack problems including plasticity effect

To seek stress and strain distributions in flawed structures, the governing equations in elasto-plasticity are first derived in terms of displacement increments. The method of lines is then applied to derive the sets of ordinary differential equations with appropriate boundary conditions. Their solutions are sought along continuous lines of a discretized region. They are solved by a combination of power series and modal matrix method. A step by step integration is devised to determine the displacements at nodal points. Non-linear work-hardening is taken care of by the effective stress and strain approach. The cases of uncracked and cracked hollow cylinders of finite length under axisymmetrical loadings are studied in detail. The growth of plastic zone, crack-opening displacement and J-integral along various paths are illustrated.     

A Fourth Order Scheme for Incompressible Boussinesq Equations

A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.