Numerical study of the flow over shallow cavities

This paper reports on a series of numerical simulations of both laminar and turbulent flows over shallow cavities. For the turbulent case the influences of the following parameters were considered: (i) cavity aspect ratios, (ii) turbulence level of the oncoming flow, and (iii) Reynolds number. Several important results and conclusions are reported. We have found that for the turbulent case the external flow touches the floor of the cavity, and this depends on a specific value of each of these parameters. This condition has an important impact upon convective effects inside the cavity. The mathematical model corresponds to the incompressible, Reynolds-averaged, Navier-Stokes equations plus a high-Reynolds κ-ε model of turbulence, and the numerical computation is performed using the SIMPLER algorithm.     

A modified full multigrid algorithm for the Navier-Stokes equations

A modified full multigrid (FMG) method for the solution of the Navier-Stokes equations is presented. The method proposed is based on a V-cycle omitting the restriction procedure for dependent variables but retaining it for the residuals. This modification avoids possible mismatches between the mass fluxes and the restricted velocities as well as the turbulent viscosity and the turbulence quantities on the coarse grid. In addition, the pressure on the coarse grid can be constructed in the same way as the velocities. These features simplify the multigrid strategy and corresponding programming efforts. This algorithm is applied to accelerate the convergence of the solution of the Navier-Stokes equations for both laminar and high-Reynolds number turbulent flows. Numerical simulations of academic and practical engineering problems show that the modified algorithm is much more efficient than the FMG-FAS (Full Approximation Storage) method.     

A parallel, finite-volume algorithm for large-eddy simulation of turbulent flows

A parallel, finite-volume algorithm has been developed for large-eddy simulation (LES) of compressible turbulent flows. This algorithm includes piecewise linear least-square reconstruction, trilinear finite-element interpolation, Roe flux-difference splitting (FDS), and second-order MacCormack time marching. A systematic and consistent means of evaluating the surface and volume integrals of the control volume is described. Parallel implementation is done using the message-passing programming model. To validate the numerical method for turbulence simulation, LES of fully developed turbulent flow in a square duct is performed for a Reynolds number of 320 based on the average friction velocity and the hydraulic diameter of the duct. Direct numerical simulation (DNS) results are available for this test case, and the accuracy of this algorithm for turbulence simulations can be ascertained by comparing the LES solutions with the DNS results. For the first time, a finite volume method with Roe FDS was used for LES of turbulent flow in a square duct, and the effects of grid resolution, upwind numerical dissipation, and subgrid-scale dissipation on the accuracy of the LES are examined. Comparison with DNS results shows that the standard Roe FDS adversely affects the accuracy of the turbulence simulation. For accurate turbulence simulations, only 3–5% of the standard Roe FDS dissipation is needed.     

An implicit gas-kinetic scheme for turbulent flow on unstructured hybrid mesh

In this study, an implicit scheme for the gas-kinetic scheme (GKS) on the unstructured hybrid mesh is proposed. The Spalart–Allmaras (SA) one equation turbulence model is incorporated into the implicit gas-kinetic scheme (IGKS) to predict the effects of turbulence. The implicit macroscopic governing equations are constructed and solved by the matrix-free lower-upper symmetric-Gauss–Seidel (LU-SGS) method. To reduce the number of cells and computational cost, the hybrid mesh is applied. A modified non-manifold hybrid mesh data(NHMD) is used for both unstructured hybrid mesh and uniform grid. Numerical investigations are performed on different 2D laminar and turbulent flows. The convergence property and the computational efficiency of the present IGKS method are investigated. Much better performance is obtained compared with the standard explicit gas-kinetic scheme. Also, our numerical results are found to be in good agreement with experiment data and other numerical solutions, demonstrating the good applicability and high efficiency of the present IGKS for the simulations of laminar and turbulent flows.     

Unsteady turbulence in plane channel flow

Direct numerical simulations were conducted for oscillating flow with zero time mean (reciprocating flow) in a plane channel subject to a harmonic forcing term of varying amplitude and frequency. The results confirmed the existence of four flow regimes (laminar, “disturbed laminar”, intermittently turbulent, and fully turbulent) depending on the above parameters. The flow behaviour was found to depend on the complex interplay of mean and turbulence quantities, as described by the closed loop formed by the streamwise Reynolds-averaged momentum equation in conjunction with the exact transport equations for the turbulent (Reynolds) stresses. A crucial role in this loop appeared to be played by the different time response of the mean flow to the applied forcing at different cross-stream locations, due to the laminar and turbulent diffusion of momentum from the walls and causing characteristic distortions of the cross-stream mean velocity profiles at different phases (i.e. acceleration vs. deceleration). The intrinsic inertia of turbulence quantities themselves played only a minor role, as confirmed by the fact that, in a broad range of conditions, turbulent stresses were roughly in phase with the respective production and dissipation terms. The structure of turbulence was found to depend largely on the instantaneous mean velocity profile, as confirmed by “frozen velocity” simulations.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

Performance of a bend-diffuser system: Experimental and numerical studies

The turbulent flow inside a combined bend-diffuser configuration with a rectangular cross section is experimentally and numerically studied. The experimental study includes the outer and inner-wall-pressure measurements and the overall system/diffuser loss determination. Simulation is performed using the high-Reynolds number k-ε turbulence model improved by the low-Reynolds number k-ε turbulence model near the walls, because of its success to predict the flow with strong adverse pressure gradient. So the present paper provides a numerical procedure for the calculation of turbulent flow in a sequence curved, expanding passages, with emphasis on the bend-diffuser configuration system consisting of a 90° bend followed by a diffuser with different expanding angles ranges from 2θ = 6-30° at different inflow Reynolds numbers. Satisfied comparisons with reported experimental data in the literature as well as that carried out by the present authors at the heat engine laboratory of Menoufiya university show that the numerical method with the utilized closure turbulence model reproduces the essential features of upstream curved flow effects on the diffuser performance. The effect of spacer length (between the bend and diffuser) is also experimentally and numerically included. The results show that there is an optimum diffuser angle which depends on the inflow Reynolds number and produces the minimum pressure loss and hence good performance of such complex geometry is obtained.     

A fast numerical method for flow analysis and blade design in centrifugal pump impellers

A numerical methodology is developed to simulate the turbulent flow in a 2-dimensional centrifugal pump impeller and to compute the characteristic performance curves of the entire pump. The flow domain is discretized with a polar, Cartesian mesh and the Reynolds-averaged Navier-Stokes (RANS) equations are solved with the control volume approach and the k-ε turbulence model. Advanced numerical techniques for adaptive grid refinement and for treatment of grid cells that do not fit the irregular boundaries are implemented in order to achieve a fully automated grid construction for any impeller design, as well as to produce results of adequate precision and accuracy. After estimating the additional hydraulic losses in the casing and the inlet and outlet sections of the pump, the performance of the pump can be predicted using the numerical results from the impeller section only. The regulation of various energy loss coefficients involved in the model is carried out for a commercial pump, for which there are available measurements. The predicted overall efficiency curve of the pump was found to agree very well with the corresponding experimental data. Finally, a numerical optimization algorithm based on the unconstrained gradient approach is developed and combined with the evaluation software in order to find the impeller geometry that maximizes the pump efficiency, using as free design variables the blade angles at the leading and the trailing edge. The results verified that the optimization process can converge very fast and to reasonable optimal values.     

Synthetic Controllable Turbulence Using Robust Second Vorticity Confinement

Capturing fine details of turbulence on a coarse grid is one of the main tasks in real‐time fluid simulation. Existing methods for doing this have various limitations. In this paper, we propose a new turbulence method that uses a refined second vorticity confinement method, referred to as robust second vorticity confinement, and a synthesis scheme to create highly turbulent effects from coarse grid. The new technique is sufficiently stable to efficiently produce highly turbulent flows, while allowing intuitive control of vortical structures. Second vorticity confinement captures and defines the vortical features of turbulence on a coarse grid. However, due to the stability problem, it cannot be used to produce highly turbulent flows. In this work, we propose a robust formulation to improve the stability problem by making the positive diffusion term to vary with helicity adaptively. In addition, we also employ our new method to procedurally synthesize the high‐resolution flow fields. As shown in our results, this approach produces stable high‐resolution turbulence very efficiently.     

Using the dinus code for direct numerical simulation of hydrodynamic processes in VVER-440 fuel rod bundles

The paper presents results of the DINUS code verification against data from experiments conducted at a full-size test facility in Finland. The DINUS code is intended for direct numerical simulation of thermal-hydraulic processes in reactor fuel rod bundles. The experiments studied hydrodynamic processes in the VVER-440 fuel rod bundle at the Loviisa NPP and were carried out for the Reynolds number of 50000. A comparison is given for calculated and experiment data on the axial velocity and flow turbulence intensity distributions at the rod bundle cross sections at different distances from the spacer grid. The DINUS code was used to analyze the effect of the spacer grid on the intensity of inter-subchannel turbulent mixing and convective transfer.     

Numerical analysis of hydraulically-driven fractures

A general method-of-lines numerical approach for modeling hydraulically-driven fractures is developed and tested. The methodology employs several novel features: a straining coordinate system that elongates as the fracture grows, an evolutionary equation to describe growth of the fracture length, direct treatment of the fluid/elastic-solid coupling, and a control volume equation which governs fluid motion near the tip and thus circumvents local degeneracy of the differential equations. Spatial discretization of the governing equations leads to a nonsparse system of implicit, coupled ordinary differential equations that is solved for time derivatives that are then integrated with a Runge-Kutta algorithm. The numerical solutions agree very well with known similarity solutions for laminar and for turbulent flow. New solutions for nonsimilar flows are also presented and these converge to proper limits as the fracture becomes very long. Acceptable accuracy, in all cases, is obtained using a very few numerical grid points and with modest execution times.     

Multiscale modelling strategy using the lattice Boltzmann method for polymer dynamics in a turbulent flow

Polymer dynamics in a turbulent flow is a problem spanning several orders of magnitude in length and time scales. A microscopic simulation covering all those scales from the polymer segment to the inertial scale of turbulence remains improbable within the foreseeable future. We propose a multiscale simulation strategy to enhance the spatio-temporal resolution of the local Lagrangian turbulent flow by matching two different simulation techniques, i.e. direct numerical simulation for the flow as a whole, and the lattice Boltzmann method coupled to polymer dynamics at the Kolmogorov dissipation scale. Local turbulent flows sampled by Lagrangian tracer particles in the direct numerical simulation are reproduced in the lattice Boltzmann model with a finer resolution, by supplying the latter with both the correct initial condition as well as the correct time-dependent boundary condition, sampled from the former. When combined with a Molecular Dynamics simulation of a polymer chain in the lattice Boltzmann model, it provides a strategy to simulate the passive dynamics of a polymer chain in a turbulent flow covering all these scales. Although this approach allows for a fairly realistic model of the macromolecule, the back-coupling to the flow on the large scales is missing.     

Assessment of grid interface treatments for multi-block incompressible viscous flow computation

In the multi-block computation of the Navier-Stokes equations, the interface treatment is a key issue. In the present work, we investigate this issue in the context of a pressure-based method using a non-orthogonal grid. For the momentum equations, a straightforward bilinear interpolation seems satisfactory as the interface treatment; on the other hand, because the pressure field depends on the satisfaction of the mass continuity equation, a conservative interface treatment has been found necessary for the pressure-correction equation. Two alternative interface treatments for the pressure-correction equation, one employing the Neumann boundary condition in both grid blocks, based on explicit local, cell-by-cell mass flux conservation, and the other utilizing Neumann-Dirichlet boundary conditions, allowing the interface condition in one block to be derived by interpolating the pressure field from the adjacent block, are assessed in the present work. To evaluate these interface schemes, the laminar flow inside a lid-driven cavity flow, and the turbulent flow around cascades of multiple airfoils have been investigated. For the case tested, both interface treatments give comparable accuracy. The finding that more than one type of interface treatment can work well allows one to devise a flexible multi-block strategy for complex flow computations.     

Procedural Synthesis using Vortex Particle Method for Fluid Simulation

We propose a fast and effective technique to improve sub‐grid visual details of the grid based fluid simulation. Our method procedurally synthesizes the flow fields coming from the incompressible Navier‐Stokes solver and the vorticity fields generated by vortex particle method for sub‐grid turbulence. We are able to efficiently animate smoke which is highly turbulent and swirling with small scale details. Since this technique does not solve the linear system in high‐resolution grids, it can perform fluid simulation more rapidly. We can easily estimate the influence of turbulent and swirling effect to the fluid flow.     

Finite volume method on the unstructured grid system

Applying the Voronoi diagram to the cell system for the finite volume method, a new method on the unstructured grid system is devised for the simulation of incompressible steady flow. In this method, the SIMPLE algorithm can be applied with little expansion. The turbulent flow around the two-dimensional vehicle model is simulated with the k-ε turbulence model by this method. Comparing the calculation result with another result obtained using the structured grid system and the experimental data, the new method is shown to be suitable for the simulation of complex flow fields.     

Low Reynolds number turbulent flows around a dynamically shaped airfoil

A computational investigation for flows surrounding a dynamically shaped airfoil, at a chord Reynolds number of 78,800, is conducted along with a parallel experimental effort. A piezo-actuated flap on the upper surface of a fixed airfoil is adopted for active control. The actuation frequency focused on is 500 Hz. The computational framework consists of a multi-block, moving grid technique, the eⁿ-based laminar-turbulent transition model, the two-equation turbulence closure, and a pressure-based flow solver. The moving grid technique, which handles the geometric variations in time, employs the transfinite interpolation scheme with a spring network approach. Comparing the experimental and computational results for pressure and velocity fields, implications of the detailed flap geometry, the flapping amplitude, turbulence modeling, and grid distributions on the flow structure are assessed. The effect of the flap movement on the separation location and vortex dynamics is also investigated.     

Numerical correlation for natural convective flows in isothermal heated, inclined and convergent channels, for high Rayleigh numbers

A numerical study of natural convective flows, mainly for high Rayleigh numbers, in a sloped converging channel, for different inclination and convergence angles has been carried out, taking into account the lacks of the literature on some aspects of this configuration. Two-dimensional, laminar, transitional and turbulent simulations were obtained by solving the fully elliptic governing equations using two different general-purpose codes: Fluent and Phoenics. The low-Reynolds k-ω turbulence model has been employed. Special emphasis is made on the systematic comparisons of computational results with experimental and numerical data taken from literature for turbulent regime, so as on the transitional conditions, studying the influence of Rayleigh number and channel aspect ratio. A generalized turbulent correlation for the average Nusselt number has been obtained from numerical results in a channel with isothermal heated plates, for symmetric heating conditions. This correlation is valid for wide and not yet covered ranges of Rayleigh number (based on length of the channel) varying from 10¹⁰ to 10¹⁶, aspect ratio between 0.03 and 0.25, the converging angle from 1° to 30° and sloping angle from 0° to 60°. Finally, an application of this numerical correlation in two experimental prototypes is presented.     

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.     

One- and two-equation turbulence models for the prediction of complex cascade flows using unstructured grids

One- and two-equation, low-Reynolds eddy-viscosity turbulence models are employed in the context of a primitive variable, finite volume, Navier-Stokes solver for unstructured grids. Through the study of the complex flow in a controlled-diffusion compressor cascade at off-design conditions, the ability of the models under consideration to predict the laminar separation bubble close to the leading edge and the boundary layer development is investigated. In order to control the unphysical growth of turbulent kinetic energy near the leading edge stagnation point, appropriate modifications to the conventional models are employed and tested. All of them improve the leading edge flow patterns and significantly affect the size of the predicted laminar separation bubble. The use of an adequately refined mesh around the airfoil, that is formed by triangles placed in a quasi-structured way, allows for the generation of grid elements of moderate aspect ratios. This helps to readily overcome any relevant problems of accuracy; a second-order upwind scheme without flux limiters or least squares approximations is successfully employed for the gradients. The test case includes quasi-3D effects by considering the streamtube thickness variation in the governing equations.     

An efficient and high-order sliding mesh method for computational aeroacoustics

This paper presents an efficient sliding mesh method to simulate the noise emission from aero-engines. The vortexes shed from the rotating blade and interact with the downstream stationary outlet guide vanes (OGVs), producing noise. The challenges in simulating the problem are the accurate modelling of the wake turbulence, and the capabilities to capture the acoustic waves, the energy of which is several orders lower than the turbulent components. To model the relative motion between the rotors and OGVs, a sliding mesh method is developed to account for the rotation of the rotor blades and wakes, which can lead to efficiency and accuracy challenges. In this work, an advanced treatment is developed for efficient and high-accuracy interpolation by combining both patch and sliding interfaces. The grid along the sliding interface is uniformly distributed taking advantage of the patch interface, providing huge benefits to the overall performance by reusing data and omitting repeated calculation. The algorithm using message passing interface is well designed for maintaining ideal performance of the code. The fan–OGV geometry is represented as unwrapped two-dimensional cascades with isotropic and anisotropic turbulence synthesised and injected to simulate the fan-wake. The numerical results are compared to analytical solutions for accuracy validation. The simulations numerically reveal the effect of turbulence intensity, length scale and anisotropy in the fan wake on the noise emission due to the turbulence-OGV interaction. Also the blockage effect of rotating blades on the noise propagation and its impact on the hearing of observers are discussed. Moreover, it is shown that the new method is able to maintain a high accuracy for acoustic computation and an ideal performance is obtained from the numerical code using a parallel computing algorithm.