Verification testing in computational fluid dynamics: an example using Reynolds‐averaged Navier–Stokes methods for two‐dimensional flow in the near wake of a circular cylinder

Verification testing was performed for various Reynolds‐averaged Navier–Stokes methods for uniform flow past a circular cylinder at Re= 5232. The standard and renormalized group (RNG) versions of the k–ε method were examined, along with the Boussinesq, Speziale and Launder constitutive relationships. Wind tunnel experiments for flow past a circular cylinder were also performed to obtain a comparative data set. Preliminary studies demonstrate poor convergence for the Speziale relationship. Verification testing with the standard and RNG k–ε models suggests that the simulations exhibit global monotonic convergence for the Boussinesq models. However, the global order of accuracy of the methods was much lower than the expected order of accuracy of 2. For this reason, pointwise convergence ratios and orders of accuracy were computed to show that not all sampling locations had converged (standard k–ε model: 19% failed to converge; RNG k–ε model: 14% failed to converge). When the non‐convergent points were removed from consideration, the average orders of accuracy are closer to the expected value (standard k–ε model: 1.41; RNG k–ε model: 1.27). Poor iterative and global grid convergence was found for the RNG k–ε/Launder model. The standard and RNG k–ε models with the Boussinesq relationship were compared with experimental data and yielded results significantly different from the experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

Measurements of mean and turbulence quantities in the curved wake of an airfoil

Measurements of mean and turbulence quantities are presented for a curved wake of an airfoil. The wake is generated by placing a NACA 0012 airfoil of 0.150 m chord length at one chord length upstream of a 90° bend. The bend has a square cross-section of 0.457 m × 0.457 m, a mean radius-to-height ratio of R/H=1.17, and concave and convex radii of curvature 0.764 and 0.307 m, respectively. In addition to streamwise curvature, the wake is subjected to varying streamwise and radial pressure. The measurements were carried out at mainstream air velocities of 10, 15 and 20 m/s. The results are presented for the mean streamwise velocity, five components of turbulence stresses, the calculated wake half-width and the maximum velocity defect. The results showed the formation of an asymmetric wake about the wake centreline, with a larger wake half-width on the inner side. The wake half-width on both inner side and outer side of the wake decrease with mainstream velocity, whereas the maximum velocity defect, turbulence stresses increase with mainstream velocity. The turbulence stresses are enhanced on the inner side but suppressed on the outer side.     

Prediction of developing turbulent flow in 90°‐curved ducts using linear and non‐linear low‐Re k–ε models

This paper reports the outcome of applying two different low‐Reynolds‐number eddy‐viscosity models to resolve the complex three‐dimensional motion that arises in turbulent flows in ducts with 90^° bends. For the modelling of turbulence, the Launder and Sharma low‐Re k–ε model and a recently produced variant of the cubic non‐linear low‐Re k–ε model have been employed. In this paper, developing turbulent flow through two different 90^° bends is examined: a square bend, and a rectangular bend with an aspect ratio of 6. The numerical results indicate that for the bend of square cross‐section the curvature induces a strong secondary flow, while for the rectangular cross‐section the secondary motion is confined to the corner regions. For both curved ducts, the secondary motion persists downstream of the bend and eventually slowly disappears. For the bend of square cross‐section, comparisons indicate that both turbulence models can produce reasonable predictions. For the bend of rectangular cross‐section, for which a wider range of data is available, while both turbulence models produce satisfactory predictions of the mean flow field, the non‐linear k–ε model returns superior predictions of the turbulence field and also of the pressure and friction coefficients. Copyright © 2006 John Wiley & Sons, Ltd.     

Two‐dimensional prediction of time dependent,turbulent flow around a square cylinder confined in a channel

This paper presents two‐dimensional and unsteady RANS computations of time dependent, periodic, turbulent flow around a square block. Two turbulence models are used: the Launder–Sharma low‐Reynolds number k–ε model and a non‐linear extension sensitive to the anisotropy of turbulence. The Reynolds number based on the free stream velocity and obstacle side is Re=2.2×10⁴. The present numerical results have been obtained using a finite volume code that solves the governing equations in a vertical plane, located at the lateral mid‐point of the channel. The pressure field is obtained with the SIMPLE algorithm. A bounded version of the third‐order QUICK scheme is used for the convective terms. Comparisons of the numerical results with the experimental data indicate that a preliminary steady solution of the governing equations using the linear k–ε does not lead to correct flow field predictions in the wake region downstream of the square cylinder. Consequently, the time derivatives of dependent variables are included in the transport equations and are discretized using the second‐order Crank–Nicolson scheme. The unsteady computations using the linear and non‐linear k–ε models significantly improve the velocity field predictions. However, the linear k–ε shows a number of predictive deficiencies, even in unsteady flow computations, especially in the prediction of the turbulence field. The introduction of a non‐linear k–ε model brings the two‐dimensional unsteady predictions of the time‐averaged velocity and turbulence fields and also the predicted values of the global parameters such as the Strouhal number and the drag coefficient to close agreement with the data. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

Assessment of two‐equation turbulence modelling for high Reynolds number hydrofoil flows

This paper presents an evaluation of the capability of turbulence models available in the commercial CFD code FLUENT 6.0 for their application to hydrofoil turbulent boundary layer separation flow at high Reynolds numbers. Four widely applied two‐equation RANS turbulence models were assessed through comparison with experimental data at Reynolds numbers of 8.284×106 and 1.657×107. They were the standard k–εmodel, the realizable k–εmodel, the standard k–ωmodel and the shear‐stress‐transport (SST) k–ωmodel. It has found that the realizable k–εturbulence model used with enhanced wall functions and near‐wall modelling techniques, consistently provides superior performance in predicting the flow characteristics around the hydrofoil. Copyright © 2004 John Wiley & Sons, Ltd.     

Computational study on turbulent flows around modern tanker hull forms

To enhance the applicability of CFD techniques to the hull form design of modern commercial ships, an efficient and robust numerical method for turbulent flow calculation is developed. The preprocessor is composed of hull form presentation, surface mesh generation, and field grid generation. The finite‐volume method is utilised to solve the Reynolds‐averaged Navier–Stokes equations. Three k–? turbulence models, i.e., the standard k–? model (SKE), the RNG‐based k–? model (RNG), and the realisable k–? model (RKE), are evaluated to investigate the difference caused by the turbulence model. The developed numerical method is applied to two practical VLCC hull forms with the same forebody and the slightly different afterbodies in order to find out whether CFD can capture the difference of stern flow due to hull form variation. It is found that RKE successfully predict the strength and location of bilge vortex, while SKE and RNG fail. However, all three models can provide the right information on the nominal wake difference between two hull forms. Copyright © 2002 John Wiley & Sons, Ltd.     

Turbulence modelling of problem aerospace flows

Unsteady Reynolds averaged Navier–Stokes (URANS) and detached eddy simulation (DES) related approaches are considered for high angle of attack NACA0012 airfoil, wing–flap, generic tilt‐rotor airfoil and double‐delta geometry flows. These are all found to be problem flows for URANS models. For DES fifth‐order upwinding is found too dissipative and the use of, for high speed flows, instability prone centred differencing essential. An existing hybrid ILES–RANS modelling approach, intended for flexible geometry, relatively high numerical dissipation codes is tested along with differential wall distance algorithms. The former gives promising results. The standard turbulence modelling approaches are found to give perhaps a surprising results variation. Results suggest that for the problem flows, the explicit algebraic stress and Menter shear stress transport (SST) URANS models are more accurate than the economical Spalart–Allmaras (SA). However, the explicit algebraic stress model (EASM) in its k–ε form is impractically expensive to converge. Here, SA predictions lack a rotation correction term and this is likely to improve these results. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluation of erosion in a two‐way coupled fluid–particle system

In this study, the effects of flow turbulence intensity, temperature, particle sizes and impinging velocity on erosion by particle impact are demonstrated numerically. Underlying turbulent flow on an Eulerian frame is described by the compressible Reynolds averaged Navier–Stokes equations with a RNG k–ε turbulence model. The particle trajectories and particle–wall interactions are evaluated by a Eulerian–Lagrangian approach in a two‐way coupling system. An erosion model considering material weight removal from surfaces is used to predict erosive wear. Computational validation against measured data is demonstrated satisfactorily. The analysis of erosion shows that the prevention of erosion is enhanced by increasing the effects of flow temperature and turbulence intensity and reducing particle inertial momentum. Copyright © 2001 John Wiley & Sons, Ltd.     

Time‐dependent turbulent cavitating flow computations with interfacial transport and filter‐based models

Turbulent cavitating flow computations need to address both cavitation and turbulence modelling issues. A recently developed interfacial dynamics‐based cavitation model (IDCM) incorporates the interfacial transport into the computational modelling of cavitation dynamics. For time‐dependent flows, it is known that the engineering turbulence closure such as the original k–ε model often over‐predicts the eddy viscosity values reducing the unsteadiness. A recently proposed filter‐based modification has shown that it can effectively modulate the eddy viscosity, rendering better simulation capabilities for time‐dependent flow computations in term of the unsteady characteristics. In the present study, the IDCM along with the filter‐based k–ε turbulence model is adopted to simulate 2‐D cavitating flows over the Clark‐Y airfoil. The chord Reynolds number is Re=7.0 × 10⁵. Two angles‐of‐attack of 5 and 8° associated with several cavitation numbers covering different flow regimes are conducted. The simulation results are assessed with the experimental data including lift, drag and velocity profiles. The interplay between cavitation and turbulence models reveals substantial differences in time‐dependent flow results even though the time‐averaged characteristics are similar. Copyright © 2005 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

Experimental and numerical study of turbulent flow and heat transfer inside hexagonal duct

Flow and heat transfer characteristics in transition and turbulent regions are studied experimentally and numerically in a horizontal smooth regular hexagonal duct under constant wall temperature boundary condition covering a range of Reynolds number from 2.3 × 10³ to 52 × 10³. Two types of k-omega (standard and shear stress transport (SST)) and three types of k-ε (standard, renormalization (RNG), and realizable) turbulence model are employed for transition and turbulent regions, respectively. Both average and fully developed Darcy friction factor and Nusselt number are presented as a function of Reynolds number. It is seen that k-omega SST and k-ε realizable turbulence models gave the best agreement with the experimental data in transition and turbulent regions, respectively. All the experimental results are correlated within an accuracy of ±13 % and ±7 % for Nusselt number and Darcy friction factor, respectively. Results obtained in this study are compared with circular duct results using hydraulic diameter.     

A fully coupled Navier–Stokes solver for calculation of turbulent incompressible free surface flow past a ship hull

This paper deals with the calculation of free surface flow of viscous incompressible fluid around the hull of a boat moving with rectilinear motion. An original method used to avoid a large part of the theoretical problems connected with free surface boundary conditions in three‐dimensional Navier–Stokes–Reynolds equations is proposed here. The linearised system of convective equations for velocities, pressure and free surface elevation unknowns is discretised by finite differences and two methods to solve the fully coupled resulting matrix are presented here. The non‐linear convergence of fully coupled algorithm is compared with the velocity–pressure weakly coupled algorithm SIMPLER. Turbulence is taken into account through Reynolds decomposition and k–ε or k–ω model to close the equations. These two models are implemented without wall function and numerical calculations are performed up to the viscous sub‐layer. Numerical results and comparisons with experiments are presented on the Series 60 CB=0.60 ship model for a Reynolds number Rn=4.5×10⁶ and a Froude number Fn=0.316. Copyright © 1999 John Wiley & Sons, Ltd.     

Effect of flexure on aerodynamic propulsive efficiency of flapping flexible airfoil

The aim of present study is to investigate the effect of chord-wise flexure amplitude on unsteady aerodynamic characteristics for a flapping airfoil with various combinations of Reynolds number and reduced frequency. Unsteady, viscous flows over a single flexible airfoil in plunge motion are computed using conformal hybrid meshes. The dynamic mesh technique is applied to illustrate the deformation modes of the flexible flapping airfoil. In order to investigate the influence of the flexure amplitude on the aerodynamic performance of the flapping airfoil, the present study considers eight different flexure amplitudes (a₀) ranging from 0 to 0.7 in intervals of 0.1 under conditions of Re=10⁴, reduced frequency k=2, and dimensionless plunge amplitude h₀=0.4. The computed unsteady flow fields clearly reveal the formation and evolution of a pair of leading edge vortices along the body of the flexible airfoil as it undergoes plunge motion. Thrust-indicative wake structures are generated when the flexure amplitude of the airfoil is less than 0.5 of the chord length. An enhancement in the propulsive efficiency is observed for a flapping airfoil with flexure amplitude of 0.3 of the chord length. This study also calculates the propulsive efficiency and thrust under various Reynolds numbers and reduced frequency conditions. The results indicate that the propulsive efficiency has a strong correlation with the reduced frequency. It is found that the flow conditions which yield the highest propulsive efficiency correspond to Strouhal number St of 0.255.     

Cubic eddy‐viscosity turbulence models for strongly swirling confined flows with variable density

An investigation on the predictive performance of cubic eddy‐viscosity turbulence models for strongly swirling confined flows with variable density is presented. Comparisons of the prediction with the experiments show some improvements of cubic models over the linear k–ε model. The linear k–ε model does not contain any mechanism to represent the interaction of swirl and density variation and as a consequence it performs poorly. With appropriate modelling, two‐equation cubic turbulence models can capture the subcritical nature of the flow, represent the azimuthal velocity profiles of combined forced‐free vortex motion, and predict the combined effects of swirl and density variation fairly well. However, the calibration of model coefficients is still a topic of investigation. Further amendments are also needed for the equations of k and ε to take into account the effects of swirl and density gradients correctly. Copyright © 2004 John Wiley & Sons, Ltd.     

Experimental validation of two depth‐averaged turbulence models

A finite volume turbulence model for the resolution of the two‐dimensional shallow water equations with turbulent term is presented. After making a finite volume discretization of the depth‐averaged k–ε equations in conservative form, the q–r equations, that give stability to the process, are obtained. Wall and inlet boundary conditions for the turbulent equations and wall conditions for the hydrodynamic equations are discussed. A comparison between the k–ε and q–r models and some experimental results is made. Copyright © 2008 John Wiley & Sons, Ltd.     

Stochastic estimation of flow near the trailing edge of a NACA0012 airfoil

A stochastic estimation technique has been applied to simultaneously acquired data of velocity and surface pressure as a tool to identify the sources of wall-pressure fluctuations. The measurements have been done on a NACA0012 airfoil at a Reynolds number of Re _c  = 2 × 10⁵, based on the chord of the airfoil, where a separated laminar boundary layer was present. By performing simultaneous measurements of the surface pressure fluctuations and of the velocity field in the boundary layer and wake of the airfoil, the wall-pressure sources near the trailing edge (TE) have been studied. The mechanisms and flow structures associated with the generation of the surface pressure have been investigated. The “quasi-instantaneous” velocity field resulting from the application of the technique has led to a picture of the evolution in time of the convecting surface pressure generating flow structures and revealed information about the sources of the wall-pressure fluctuations, their nature and variability. These sources are closely related to those of the radiated noise from the TE of an airfoil and to the vibration issues encountered in ship hulls for example. The NACA0012 airfoil had a 30 cm chord and aspect ratio of 1.     

Direct numerical simulation of a NACA0012 in full stall

This work aims at investigating the mechanisms of separation and the transition to turbulence in the separated shear-layer of aerodynamic profiles, while at the same time to gain insight into coherent structures formed in the separated zone at low-to-moderate Reynolds numbers. To do this, direct numerical simulations of the flow past a NACA0012 airfoil at Reynolds numbers Re = 50,000 (based on the free-stream velocity and the airfoil chord) and angles of attack AOA = 9.25° and AOA = 12° have been carried out. At low-to-moderate Reynolds numbers, NACA0012 exhibits a combination of leading-edge/trailing-edge stall which causes the massive separation of the flow on the suction side of the airfoil. The initially laminar shear layer undergoes transition to turbulence and vortices formed are shed forming a von Kármán like vortex street in the airfoil wake. The main characteristics of this flow together with its main features, including power spectra of a set of selected monitoring probes at different positions on the suction side and in the wake of the airfoil are provided and discussed in detail.     

Reynolds‐stress modelling of M = 2.25 shock‐wave/turbulent boundary‐layer interaction

M = 2.25 shock‐wave/turbulent‐boundary‐layer interactions over a compression ramp for several angles (8, 13 and 18°) at Reynolds‐number Re=7 × 10³ were simulated with three low‐Reynolds second‐moment closures and a linear low‐Reynolds standard k–ε model. A detailed assessment of the turbulence closures by comparison with both mean‐flow and turbulent experimental quantities is presented. The Reynolds‐stress model which is wall‐topology free and which uses an optimized redistribution closure, is in good agreement with experimental data both for wall‐pressure and mean‐velocity profiles. Detailed analysis of three components of the Reynolds‐stress tensor (comparison with measurements and transport‐equation budgets) provides a critical evaluation of full Reynolds‐stress models for the separated supersonic compression ramp. The discrepancy observed in the shock‐wave foot region, between computations and measurements for the Reynolds‐stresses profiles, could be explained by considering the experimental shock‐wave oscillation and directions for future modelling work are indicated. Copyright © 2007 John Wiley & Sons, Ltd.     

Turbulent separated flow over and downstream of a two-element airfoil

Flow characteristics in the vicinity of the flap of a single-slotted airfoil are presented and analysed. The flow remained attached over the model surfaces except in the vicinity of the flap trailing edge where a small region of boundary-layer separation extended over the aft 7% of flap chord. The airfoil configuration was tested at a Mach number of 0.09 and a chord Reynolds number of 1.8 × 10⁶ in the NASA Ames Research Center 7- by 10-Foot Wind Tunnel. The flow was complicated by the presence of a strong, initially inviscid, jet, emanating from the slot between airfoil and flap, and a gradual merging of the main airfoil wake and flap suction-side boundary layer.Research Engineer, NRC Research AssociateAerospace Engineer