Algebraic Eddy Viscosity Model Research Articles

Drag-related wave–current interaction inside a dense submerged aquatic canopy

The seabed in coastal regions is often covered by a dense submerged canopy formed by benthic organisms. The in-canopy flow is of great interest to coastal researchers. In this study, a semi-analytical model is proposed for the in-canopy current profile under the influence of coexisting waves. The current is driven by a mean shear stress at the canopy top and a mean streamwise pressure gradient. Wave–current interaction arises from the mean canopy drag, which is described by the conventional quadratic law. An algebraic eddy viscosity model, which accounts for the turbulence generated by a canopy-scale vortex and stem-scale wake, is proposed. The model is calibrated against unidirectional flume experiments, and is subsequently validated against a wave–current flume experiment. It is found that the mean canopy drag experienced by the current can be significantly amplified by a co-existing wave-driven oscillatory flow. Under the same current driving force, the current in wave–current flow is reduced from that in a pure-current flow. The reduction increases with canopy density and wave strength. For a wave and current at an angle, veering of the current velocity vector is suggested by the model, which makes the mean canopy drag align with the driving force of the current.

Reynolds-stress-constrained large-eddy simulation of wall-bounded turbulent flows

AbstractIn the traditional hybrid RANS/LES approaches for the simulation of wall-bounded fluid turbulence, such as detached-eddy simulation (DES), the whole flow domain is divided into an inner layer and an outer layer. Typically the Reynolds-averaged Navier–Stokes (RANS) equations are used for the inner layer, while large-eddy simulation (LES) is used for the outer layer. The transition from the inner-layer solution to the outer-layer solution is often problematic due to the lack of small-scale dynamics in the RANS region. In this paper, we propose to simulate the whole flow domain by large-eddy simulation while enforcing a Reynolds-stress constraint on the subgrid-scale (SGS) stress model in the inner layer. Both the algebraic eddy-viscosity model and the one-equation Spalart–Allmaras (SA) model have been used to constrain the Reynolds stress in the inner layer. In this way, we improve the LES methodology by allowing the mean flow of the inner layer to satisfy the RANS solution while small-scale dynamics is included. We validate the Reynolds-stress-constrained large-eddy simulation (RSC-LES) model by simulating three-dimensional turbulent channel flow and flow past a circular cylinder. Our model is able to predict mean velocity, turbulent stress and skin-friction coefficients more accurately in turbulent channel flow and to estimate the pressure coefficient after separation more precisely in flow past a circular cylinder compared with the pure dynamic Smagorinsky model (DSM) and DES using the same grid resolution. Furthermore, the computational cost of the RSC-LES is almost the same as that of DES.

Numerical simulation of transient roll-waves in two-phase pipe flow

This paper presents a transient roll-wave simulator based on a one-dimensional incompressible two-fluid model. Using an efficient numerical method and a grid cell length of one pipe diameter, the simulator is able to accurately predict experimental data in the roll-wave regime. Essential to obtaining the stable roll-wave solutions is the use of a modified version of the Biberg friction model. Based on an algebraic eddy viscosity model, the Biberg model yields mechanistic expressions for the wall and interfacial shear stresses in stratified two-phase flow. Input to the model is closure relations for the turbulence levels at either side of the interface. In order to model the increased dissipation of energy in breaking wave fronts, we propose to modify the original closures by adding a term proportional to the negative gradient of the liquid height. As the waves grow and come close to breaking, this term redistributes the shear stresses such that the waves are stabilized. The numerical method used in the simulator is an extension of a previously presented pseudospectral Fourier method. In particular, spectral vanishing viscosity terms are introduced to stabilize short wavelengths inconsistent with the long wavelength approximation of the two-fluid model. The result is a simulator that provides reliable and convergent numerical solutions for all stratified flow conditions. In the last part of the paper, the simulator is tested against roll-wave experiments with horizontal and upward inclined flows of water and a dense gas in a 10 cm pipe. Wave heights and speeds as well as mean pressure drops and holdups are predicted within 20% accuracy when compared with 474 experiments. Furthermore, the transient simulations enable us to study the dynamics and interactions between waves. In agreement with experimental observations, two different wave growth mechanisms are identified, and their influence on the evolution of the roll-wave flow regime is analyzed.

Large-eddy simulation of separated flow over a swept wing with approximate near-wall modelling

AbstractThe paper investigates, by means of a simulation methodology, the flow separating from a 40 degrees backward-swept wing at 9 degrees incidence and Reynolds number of 210,000, based on the wing-root chord length. The Simulation corresponds to LDA, PIV and suction-side-topology measurements for the same geometry, conducted by other investigators specifically to provide validation data. The finest block-structured mesh contains 23·6 million nodes and is organised in 256 blocks to maximise mesh quality and facilitate parallel solution on multi-processor machines. The near-wall layer is resolved, to a thickness of about y+ = 20, by means of parabolised URANS equations that include an algebraic eddy-viscosity model and from which the wall-shear stress is extracted to provide an unsteady boundary condition for the simulation. The numerical solution is in good agreement with the experimental behaviour over the 50-70% inboard portion of the span, but the simulation fails to resolve some complex features close to the wing tip, due to a premature leading-edge vortex breakdown and loss in vortex coherence. The comparisons and their discussion provide useful insight into various physical characteristics of this complex separated wing flow.

A time-varying diffusivity model for shear dispersion in oscillatory channel flow

An analysis is presented of the shear dispersion in turbulent channel flow forced by a purely oscillatory pressure gradient. The flow is described by an algebraic eddy viscosity model in which the instantaneous flow and viscosity are inter-dependent. Both eddy viscosity and eddy diffusivity contain a second-harmonic time-varying component. The effective transport equation is derived using a two-time-scale perturbation analysis, which also yields a closed-form expression for the dispersion coefficient. Numerical results demonstrate that, in this problem, the time-varying component of the eddy diffusivity can have a finite effect on the dispersion coefficient, which can be as much as a 30% difference from that obtained from a time-invariant eddy viscosity model.

Comparison of DES, RANS and LES for the separated flow around a flat plate at high incidence

AbstractThe separated turbulent flow past an inclined flat plate with sharp leading and trailing edges was computed based on three different simulation approaches for a reynolds number Rec= 20000 and a high angle of attack α=18°. The simulation techniques applied were the Reynolds‐averaged Navier–Stokes (RANS) equations combined with a one‐equation Spalart–Allmaras turbulence model, the large‐eddy simulation (LES) based on an algebraic eddy‐viscosity model, and a hybrid approach known as detached‐eddy simulation (DES) applying a slightly modified Spalart–Allmaras model in the entire integration domain. DES is a non‐zonal coupling technique of RANS and LES developed in the hope of reducing the large computational resources required for LES computations of turbulent flows with practical relevance. However, the objective of the present study was not to compare the resources (CPU time, memory) required for all three techniques but to investigate and evaluate the quality of the predicted results for RANS, DES and LES. For this purpose, a test case which is favourable to the basic concept of DES and which places special emphasis on the LES part of the DES concept was chosen. This last issue was important since the modified Spalart–Allmaras model applied as a subgrid scale model in the LES part is not as well validated as other models usually applied in LES. For this purpose, an LES prediction on a very fine grid served as a reference case for the evaluation. For all three techniques, predictions with different grid resolutions were carried out and compared with each other based on important integral parameters (e.g. Strouhal number, mean drag and lift coefficients and their standard deviations), the instantaneous and time‐averaged flow structures, and higher‐order statistics. As expected, the pure RANS calculation, although applied as unsteady RANS, failed to predict the unsteady characteristics of the separated flow. In contrast, the DES approach yielded reasonably the shedding phenomenon and some integral parameters. However, analysing the results in more detail led to remarkable deviations between the DES and LES predictions also when the same grid resolution was applied. Especially the free shear layer originating from the leading edge of the plate was not well reproduced by DES, showing strong deficiencies of the model applied as a subgrid scale model. The reasons for this behaviour of the model were analysed in detail. Two basic causes were identified; the first is given by some near‐wall corrections in the finite Reynolds number version of the model which are not working properly in the LES mode of DES. The second is a modified definition of the filter width typically applied for DES, leading to strongly increased values of the eddy viscosity. A revised version of the S–A model taking both issues seriously into account was used as a subgrid scale model in the LES mode. As a direct consequence, much better agreement with the reference LES solution was found for the DES prediction on the coarse grid, eliminating the deficiencies of the original formulation. Copyright© 2003 John Wiley & Sons, Ltd.

Numerical Study on Relaminarization in Fish-Like Locomotion

Unsteady viscous flow field around a fish-like advancing NACA wing is numerically studied. The main objective is to study the abilities of a flexible fish-like body to produce thrust, achieve higher propulsive efficiency as well as to catch the basic flow characteristics in viscous flow field. The Reynolds numbers based on the oncoming velocity and the body length range from 10 6 to 3x10 6 . The flow is simulated by solving two-dimensional unsteady Reynolds averaged Navier-Stokes equation in a primitive value formulation, while the turbulence transportation is described by updated algebraic eddy-viscosity model and transitional zone is computed by empirical formula. The numerical scheme is based on the MAC method with a body fitted coordinate system. The simulation shows that the high propulsive efficiency of fish-like locomotion comes from relaminarization and effective vorticity control. Relaminarization reduces the frictional force at higher Reynolds number. Fish-like locomotion can recapture energy contained in the eddies by skillfully controlling vorticity generation, travel and shed through the body undulation.

Editorial

This volume contains eight papers on turbomachinery covering CFD applications, detailed flow measurements on unsteady flow, new design methods and control devices, and conventional empirical approaches. Shi and Tsukamoto succeeded in predicting pressure fluctuations caused by the interaction between a centrifugal impeller and diffuser vanes, using a Navier-Stokes code with standard k−ε turbulence model. Using a commercial code, Gu et al. succeeded in predicting the performance of a single-stage centrifugal compressor and studied the volute/diffuser flow in detail at design and off design conditions. By combining a scheme called the Single-Sweep Method with an algebraic eddy viscosity model, Kochevsky made a parametric study of the energy loss in annular diffusers with a rotating hub and inlet flow swirl, with the purpose of minimizing the loss in annular diffusers. These papers suggest that CFD is now becoming capable of predicting not only steady flows under design flow conditions, but also unsteady flows and off-design flows. Sinha, Pinarbasi and Katz, using PIV and pressure fluctuation measurements, made detailed observations of unsteady flow process during the onset and developed stages of rotating stall within a vaned diffuser of a centrifugal pump. This work is useful for refining CFDs so that they can predict local flow process, as well as for obtaining a physical understanding of the stalling process. The combination of those efforts in CFD and detailed flow measurements will lead CFDs to more useful design tools that require less experience and fewer empirical factors, and also to methods for clarifying the flow physics in turbomachinery. New production technologies can expand the applications of turbomachinery. Liu, Nishi and Yoshida have shown that a high efficiency mini turbopump is possible by employing a larger outlet blade angle and a larger number of blades with a smaller outlet/inlet area ratio, combined with high precision manufacturing technology. Two papers propose new control methods. Saha et al. propose the application of radial grooves on the casing walls between impeller exit and vaned diffuser throat, to control vaned diffuser rotating stall in radial centrifugal impellers. In many cases the suppression was realized at the cost of head decrease, but it was shown that shallow grooves can decrease the rotating stall onset flow rate with acceptable head decrease. A pitch-flap coupling mechanism to control the output power of a horizontal axis wind turbine was studied by Shimizu and Kamada to determine the optimum pitch angle to flap angle ratio and also the flow physics of control. Despite the developments of CFD as mentioned before, we still need to depend largely on empirical relations in various aspects of turbomachinery design. Lazzarotto et al. studied the influence of Reynolds number on the performance of cross flow fans with various casing geometry. Engin and Gur studied the effects of solid particles on the performance of open type centrifugal pump impellers with various tip clearance. Such empirical studies will remain a requirement for the practical design of various types of turbomachinery.

Correlation of Numerical and Experimental Heat Transfer Data at the Turbine Blade Surface.

Transitional two-dimensional boundary-layer flows are considered. The algebraic eddy-viscosity model for a simulation of nonturbulent (laminar) boundary layers is proposed. It takes into account, in a simple form, an intensity and a length scale of the free stream turbulence (FST) and a value of the favorable pressure gradient. Transition is modeled by employing the intermittency factor. The Baldwin-Lomax model is used to calculate turbulent portions of the boundary layers. The boundary layer (BL) and heat transfer measurements for flows over a flat plate were employed to verify the proposed nonturbulence model and for an evaluation of the numerical code abilities to predict heat transfer for transitional flows. The code was applied for the calculation of heat transfer coefficients of two different gas turbine vanes, and the results were compared with available experimental data at different free-stream turbulence and Reynolds number conditions

Numerical prediction of three-dimensional mixing in a compound open channel

A numerical model has been developed to simulate time-dependent turbulent flows in open channels, including the transport of dissolved materials. The turbulence closure is provided by two algebraic eddy viscosity models. This paper's main focus is on the assessment of those two models' predicting capabilities for an open channel with compound cross section. A three-dimensional turbulent flow field and the transport of a neutrally buoyant solute are modeled. The results of the simulation are compared with experimental data and with similar calculations obtained using more advanced turbulence models. Considerations are made regarding the choice of the appropriate Schmidt number for the modeling of turbulent mass transport, and the use of anisotropic eddy diffusivity to account for different mixing rates along the vertical and transverse directions. Conclusions are drawn regarding the accuracy of the two models and their advantages and drawbacks when compared with more advanced turbulence models.

A Reexamination of the Core Helium Flash

The possibility of a dynamical core helium flash has been studied with two-dimensional hydrodynamic simulations using three separate turbulence models for the treatment of time-dependent convection. Two of these are simple algebraic eddy viscosity models. The third model involves a separate equation for the turbulent kinetic energy density, which then feeds into an expression for the eddy viscosity. One of the simple eddy viscosity models is that used by Cole & Deupree in studies of the core helium flash about fifteen years ago. I find that this model can be made to produce a more quiescent flash than Cole & Deupree found by substantially reducing the eddy viscosity from their value. When this is done, all three turbulence models produce the same primary result--the flash does not produce a hydrody-namic event. However, the traditional assumption of adiabatic convection through the flash evolution is found in all cases to be inadequate, primarily through the extensive heating of the region interior to the convective region.

An algebraic-Q 4 turbulence model for attached and separated airfoil flows

An algebraic eddy viscosity model, based on a new length scale has been developed. The model proposes the eddy viscosity as a solution of a quartic (Q 4) equation. The turbulent length scale for attached and separated flows is defined by employing a vorticity functionF =yΩD introduced in the Baldwin-Lomax model. The algebraic-Q 4 eddy viscosity model was incorporated into Navier-Stokes code and tested for complex transonic airfoil flows with separation. The results are compared with the experimental data.

Calculated effect of freestream turbulence on aerodynamic characteristics of a delta wing

An algebraic eddy viscosity model for the effect of freestream turbulence is developped to model the results of Pedisius et al. The purpose is to show that freestream turbulence is one possible source of discrepancy in tunnel data.

Navier-Stokes analysis of a circulation control airfoil

The two-dimensional, compressible, mass-averaged Navier-Stokes equations are used to investigate flows about a typical circulation control airfoil. The governing equations are solved using the implicit approximate-factorization algorithm of Beam-Warming with a modified algebraic eddy viscosity model. Results are compared with experimental data, and excellent agreement is obtained. The effects of different jet momentum coefficients and angles of attack on the flow are studied. The mechanism of genenating large lift by circulation control is discussed.

Effects of spatial order of accuracy on the computation of vortical flowfields

The objective of the present investigation is to assess the effect of the spatial order of accuracy used for the evaluation of the inviscid fluxes on the resolution of higher order quantitites, such as velocity gradients. The viscous terms are computed as second-order accurate with central difference formulas, even though for the explicit part of the algorithm higher order approximations may be used. A viscous/inviscid method is used, and the outer part of the flowfield is computed with the inviscid flow equations. The viscous boundary-layer type flow region close to the body surface is computed with an algebraic eddy viscosity model. Results obtained with the conservative and nonconservative formulations and the viscous/inviscid approach are compared with available experimental data. The effect of grid refinement on the accuracy of the solution is also presented.

Evaluation of algebraic eddy-viscosity models in three-dimensional turbulent boundary-layer flows

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Transonic shockwave/turbulent boundary layer interactions on a porous surface

AbstractTransonic shockwave/turbulent boundary layer interactions on a porous surface above a closed plenum chamber have been studied experimentally in the choked flow of a windtunnel test-section. The equivalent freestream Mach number is 0.76 and results were obtained for three shock strengths. Without the porous surface the Mach numbers ahead of the shock were 1.13, 1.18 and 1.26. The respective shock Mach numbers with the porous surface were 1.10, 1.11 and 1.19. Laser holographic interferometry results are used to measure the density flowfield and examine the nature of the interaction. These results show that the interaction on the porous surface is modified by a thin shear layer adjacent to the surface and the weakening of the Shockwave is attributed to this. The interaction was also studied by solving the two-dimensional Reynolds-averaged Navier-Stokes equations together with the two-layer algebraic eddy-viscosity model of Baldwin-Lomax modified with appropriate corrections for surface transpiration. The computed results show excellent agreement with the experimental data. The examination of these numerical results shows that the surface transpiration occurs at a low subsonic velocity and suggests that the effect of the transpiration through the porous surface on the interaction may be optimised.

An algebraic-Q4 turbulent eddy viscosity model: Boundary layer flow over a flat plate and flow in a pipe

An algebraic turbulent eddy viscosity model is proposed based on a length scale model coupled with the turbulent viscosity expression of the renormalization group theory of turbulence. The eddy viscosity is presented as a solution of a quartic equation. The new length scale model is based on boundary layer characteristics (displacement thickness, shape factor). The model is applied to transitional boundary layer flow over a flat plate and to flow in a smooth pipe. Predictions for the laminar-turbulent transition, and integral characteristics, such as the total skin friction coefficient, mean velocity profile across the boundary layer, and the friction coefficient in a pipe, are found to be in good agreement with experimental data.

Periodic solution of turbulent oscillating channel flows

AbstractThe time‐dependent turbulent Navier–Stokes equations are solved numerically by a finite element method with an algebraic eddy viscosity model (Baldwin–Lomax formulation) for oscillating turbulent channel flows. The method of averaging is used to analyse the resulting periodic motion of the fluid. Numerical results are obtained for various Strouhal numbers and relative amplitudes. A comparison is made between the numerical and published experimental results. It appears that for low relative amplitudes in a certain range of frequencies the agreement is satisfactory.

Modelling of turbulent transonic flow around aerofoils and wings

AbstractTurbulent flows around aerofoils and wings are simulated by solving the two‐and three‐dimensional Navier‐Stokes equations in quadrilaterals and hexahedrons, respectively. Turbulent flow is described by an algebraic eddy viscosity model. Boundary‐conforming O‐and O‐Otype meshes are generated by the transfinite interpolation method. The finite‐volume technique is employed for spatial discretization using two cells in each computational co‐ordinate direction more than the conventional approach to evaluate the viscous fluxes. An explicit three‐stage Runge–Kutta scheme is used for time integration. The method is applied to compute turbulent transonic flow over the NACA 0012 aerofoil and the ONERA M6 wing. The results are compared with experimental and computational data.