An Attempt to Combine Large Eddy Simulation with the k-ε Model in a Channel-Flow Calculation

Large eddy simulation (LES) is combined with the Reynolds-averaged Navier–Stokes (RANS) equation in a turbulent channel-flow calculation. A one-equation subgrid-scale model is solved in a three-dimensional grid in the near-wall region whereas the standard k–ε model is solved in a one-dimensional grid in the outer region away from the wall. The two grid systems are overlapped to connect the two models smoothly. A turbulent channel flow is calculated at Reynolds numbers higher than typical LES and several statistical quantities are examined. The mean velocity profile is in good agreement with the logarithmic law. The profile of the turbulent kinetic energy in the near-wall region is smoothly connected with that of the turbulent energy for the k–ε model in the outer region. Turbulence statistics show that the solution in the near-wall region is as accurate as a usual LES. The present approach is different from wall modeling in LES that uses a RANS model near the wall. The former is not as efficient as the latter for calculating high-Reynolds-number flows. Nevertheless, the present method of combining the two models is expected to pave the way for constructing a unified turbulence model that is useful for many purposes including wall modeling. Received 11 June 1999 and accepted 15 December 2000     

A low Reynolds number variant of partially-averaged Navier-Stokes model for turbulence

A low Reynolds number (LRN) formulation based on the Partially Averaged Navier-Stokes (PANS) modelling method is presented, which incorporates improved asymptotic representation in near-wall turbulence modelling. The effect of near-wall viscous damping can thus be better accounted for in simulations of wall-bounded turbulent flows. The proposed LRN PANS model uses an LRN k-ε model as the base model and introduces directly its model functions into the PANS formulation. As a result, the inappropriate wall-limiting behavior inherent in the original PANS model is corrected. An interesting feature of the PANS model is that the turbulent Prandtl numbers in the k and ε equations are modified compared to the base model. It is found that this modification has a significant effect on the modelled turbulence. The proposed LRN PANS model is scrutinized in computations of decaying grid turbulence, turbulent channel flow and periodic hill flow, of which the latter has been computed at two different Reynolds numbers of Re = 10,600 and 37,000. In comparison with available DNS, LES or experimental data, the LRN PANS model produces improved predictions over the standard PANS model, particularly in the near-wall region and for resolved turbulence statistics. Furthermore, the LRN PANS model gives similar or better results - at a reduced CPU time - as compared to the Dynamic Smagorinsky model.     

A general one-equation turbulence model for free shear and wall-bounded flows

The purpose of thiswork is to introduce a complete and general one-equation model capable of correctly predicting a wide class of fundamental turbulent flows like boundary layer, wake, jet, and vortical flows. The starting point is the mature and validated two-equation k−ω turbulence model of Wilcox. The newly derived one-equation model has several advantages and yields better predictions than the Spalart-Allmaras model for jet and vortical flows while retaining the same efficiency and quality of the results for near-wall turbulent flows without using a wall distance. The derivation and validation of the new model using findings computed by the Spalart-Allmaras and the k−ω models are presented and discussed for several free shear and wall-bounded flows.     

A new low-Reynolds-number k--fμ model for predictions involving multiple surfaces

A new k--f_μ turbulence model is proposed, in which the near-wall effect without reference to distance and the non-equilibrium effect are incorporated. In this model, the non-local near-wall effect in a general coordinate system is taken into account by the f_μ equation, and the local anisotropy in strongly strained turbulent flows in introduced in the equation. The near-wall characteristics of the present model are ascertained from the DNS data. Also, the validation is explored to separate and reattaching flows. The backward-facing step flow is selected for benchmark test of the present model and the response of the model to flows involving complex surfaces is assessed. The predictions of the present model are checked with the existing measurements. The model performance is shown to be generally satisfactory.     

Exploring k and ? with R–Equation Model Using Elliptic Relaxation Function

An extended version of the isotropic R?Cequation model accompanied by an elliptic relaxation approach to account for the distinct effects of low-Reynolds number (LRN) and wall proximity is proposed. The turbulent kinetic energy k and the dissipation rate ? are evaluated using the R ( $=k^2/\tilde{\epsilon}$ ) transport equation together with some empirical relations. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and the Schwarz?? inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence in the sense that they are sensitized to rotational and nonequilibrium flows. The model is validated against a few well-documented flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Comparisons indicate that the present model offers some improvement over the Spalart?CAllmaras one?Cequation model and competitiveness with the SST k?C?? model.     

On Near-Wall Treatment in (U)RANS-Based Closure Models

The present work is concerned with computational evaluation of a recently formulated near-wall relationship providing the value of the dissipation rate ε of the kinetic energy of turbulence k through its exact dependence on the Taylor microscale λ: ε = 10νk/λ ², (Jakirli? and Jovanovi?, J. Fluid Mech. 656:530–539, 2010). Dissipation rate determination benefits from the asymptotic behavior of the Taylor microscale resulting in its linear variation in terms of the wall distance (λ?∝?y) being valid throughout entire viscous sublayer. Accordingly, it can be applied as a unified near-wall treatment in all computational frameworks relying on a RANS-based model of turbulence (including also hybrid LES/RANS schemes) independent of modeling level—both main modeling concepts eddy-viscosity and Reynolds stress models can be employed. Presently, the feasibility of the proposed formulation was demonstrated by applying a conventional near-wall second-moment closure model based on the homogeneous dissipation rate ε _h ( ${\varepsilon_h =\varepsilon -0.5\partial \left( {{\nu \partial k}/ {\partial x_j }} \right)} / {\partial x_j }$ ; Jakirli? and Hanjali?, J. Fluid Mech. 539:139–166, 2002) and its instability-sensitive version, modeled in terms of the inverse turbulent time scale ω _h (ω _h ?=?ε _h /k; Maduta and Jakirli?, 2011), to a fully-developed channel flow with both flat walls and periodic hill-shaped constrictions mounted on the bottom wall in a Reynolds number range. The latter configuration is subjected to boundary layer separation from a continuous curved wall. The influence of the near-wall resolution lowering with respect to the location of the wall-closest computational node, coarsened even up to the viscous sublayer edge situated at $y_P^+ \approx 5$ in equilibrium flows, is analyzed. The results obtained follow closely those pertinent to the conventional near-wall integration with the wall-next node positioned at $y_P^+ \le 0.5$ .     

A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow

This paper presents a manufactured solution (MS), resembling a two-dimensional, steady, wall-bounded, incompressible, turbulent flow for RANS codes verification. The specified flow field satisfies mass conservation, but requires additional source terms in the momentum equations. To also allow verification of the correct implementation of the turbulence models transport equations, the proposed MS exhibits most features of a true near-wall turbulent flow. The model is suited for testing six eddy-viscosity turbulence models: the one-equation models of Spalart and Allmaras and Menter; the standard two-equation k–ε model and the low-Reynolds version proposed by Chien; the TNT and BSL versions of the k–ω model.     

Turbulent flow and convective heat transfer in a wavy wall channel

This paper reports the numerical modeling of turbulent flow and convective heat transfer over a wavy wall using a two equations eddy viscosity turbulence model. The wall boundary conditions were applied by using a new zonal modeling strategy based on DNS data and combining the standard k– turbulence model in the outer core flow with a one equation model to resolve the near-wall region.It was found that the two-layer model is successful in capturing most of the important physical features of a turbulent flow over a wavy wall with reasonable amount of memory storage and computer time. The predicted results show the shortcomings of the standard law of the wall for predicting such type of flows and consequently suggest that direct integrations to the wall must be used instead. Moreover, Comparison of the predicted results of a wavy wall with that of a straight channel, indicates that the averaged Nusselt number increases until a critical value is reached where the amplitude wave is increased. However, this heat transfer enhancement is accompanied by an increase in the pressure drop.     

Finite element computation of a turbulent flow over a two-dimensional backward-facing step

The present paper is devoted to the computation of turbulent flows by a Galerkin finite element method. Effects of turbulence on the mean field are taken into account by means of a (k-ε) turbulence model. The wall region is treated through wall laws and, more specifically, Reichardt's law. An inlet profile for ε is proposed as a numerical treatment for physically meaningless values of k and ε. Results obtained for a recirculating flow in a two-dimensional channel with a sudden expansion in width are presented and compared with experimental values.     

Some characteristics of low-speed streaks under sheared air-water interfaces

The characteristics of low-speed fluid streaks occurring under sheared air-water interfaces were examined by means of hydrogen bubble visualization technique. A critical shear condition under which the streaky structure first appears was determined to beu _τ≈0.19 cm/s. The mean spanwise streak spacing increases with distance from the water surface owing to merging and bursting processes, and a linear relationship describing variation of non-dimensional spacing versusy ⁺ was found essentially independent of shear stress on the interface. Values of , however, are remarkably smaller than their counterparts in the near-wall region of turbulent boundary layers. Though low-speed streaks occur randomly in time and space, the streak spacing exhibits a lognormal probability distribution behavior. A tentative explanation concerning the formation of streaky structure is suggested, and the fact that takes rather smaller values than that in wall turbulence is briefly discussed. The project supported by the National Natural Science Foundation of China (19672070)     

RAS one-equation turbulence model with non-singular eddy-viscosity coefficient

A simplified consistency formulation for P_k/ε (production to dissipation ratio) is devised to obtain a non-singular C_μ (coefficient of eddy-viscosity) in the explicit algebraic Reynolds stress model of Gatski and Speziale. The coefficient C_μ depends non-linearly on both rotational/irrotational strains and is used in the framework of an improved RAS (Rahman–Agarwal–Siikonen) one-equation turbulence model to calculate a few well-documented turbulent flows, yielding predictions in good agreement with the direct numerical simulation and experimental data. The strain-dependent C_μ assists the RAS model in constructing the coefficients and functions such as to benefit complex flows with non-equilibrium turbulence. Comparisons with the Spalart–Allmaras one-equation model and the shear stress transport k-ω model demonstrate that C_μ improves the response of RAS model to non-equilibrium effects.     

Comparison of near-wall treatment methods for high Reynolds number backward-facing step flow

A comparison of near-wall treatment methods using different turbulence models for flow over a backward-facing step is presented. A Reynolds number (Re) of about 38,000 (U _∞ = 44.2 m/s), based on the step height and the mean stream velocity, was considered. An appropriate near-wall treatment method is critical to the choice of turbulence model used to predict wall-bounded flow. Predictions were obtained by applying standard wall functions, non-equilibrium wall functions and a two-layer model with six different turbulence models. These results were compared with data by Driver and Seegmiller (“Backward-facing step with inclined opposite wall—experiments by driver and seegmiller”, 1985a, http://cfd.me.umist.ac.uk/ercoftac [2003, Jan 31]). Non-equilibrium wall functions with modified k ? ? models predicted the closest reattachment length. However, the two-layer model gave results more representative of the entire flow pattern. The predictions show that a proper combination of turbulence models and near-wall treatment methods give reliable results.     

Predicting Turbulent Spectra in Drag-reduced Flows

In the present work we describe how turbulent skin-friction drag reduction obtained through near-wall turbulence manipulation modifies the spectral content of turbulent fluctuations and Reynolds shear stress with focus on the largest scales. Direct Numerical Simulations (DNS) of turbulent channels up to Re _τ = 1000 are performed in which drag reduction is achieved either via artificially removing wall-normal turbulent fluctuations in the vicinity of the wall or via streamwise-travelling waves of spanwise wall velocity. This near-wall turbulence manipulation is shown to modify turbulent spectra in a broad range of scales throughout the whole channel. Above the buffer layer, the observed changes can be predicted, exploiting the vertical shift of the logarithmic portion of the mean streamwise velocity profile, which is a classic performance measure for wall roughness or drag-reducing riblets. A simple model is developed for predicting the large-scale contribution to turbulent fluctuation and Reynolds shear stress spectra in drag-reduced turbulent channels in which a flow control acts at the wall. Any drag-reducing control that successfully interacts with large scales should deviate from the predictions of the present model, making it a useful benchmark for assessing the capability of a control to affect large scales directly.     

KOLMOGOROV BEHAVIOR OF NEAR-WALL TURBULENCE AND ITS APPLICATION IN TURBULENCE MODELING

The near-wall behavior of turbulence is re-examined in a way different from that proposed by Hanjalic and Launder¹ and followers^2,3,4,5. It is shown that at a certain distance from the wall, all energetic large eddies will reduce to Kolmogorov eddies (the smallest eddies in turbulence). All the important wall parameters, such as friction velocity, viscous length scale, and mean strain rate at the wall, are characterised by Kolmogorov microscales. According t o this Kolmogorov behavior of near-wall turbulence, the turbulence quantities, such as turbulent kinetic energy, dissipation rate, etc. at the location where the large eddies become “Kolmogorov” eddies, can be estimated by using both direct numerical simulation (DNS) data and asymptotic analysis of near-wall turbulence. This information will provide useful boundary conditions for the turbulent transport equations. As a n example, the concept is incorporated in the standard κ - εmodel which is then applied t o channel and boundary layer flows. Using appropriate boundary conditions (based on Kolmogorov behaviour of near-wall turbulence), there is no need for any wall-modification to the κ - ε equations (including model constants). Results compare very well with the DNS and experimental data.     

Prediction of turbulent wall shear flows directly from wall

The fully elliptic Reynolds-averaged Navier–Stokes equations have been used together with Lam and Bremhorst's low-Reynolds-number model, Chen and Patel's two-layer model and a two-point wall function method incorporated into the standard k-? model to predict channel flows and a backward-facig step flow. These flows enable the evaluation of the performance of different near-wall treatments in flows involving streamwise and normal pressure gradients, flows with separation and flows with non-equilibrium turbulence characteristics. Direct numerical simulation (DNS) of a channel flow with Re =3200 further provides the detailed budgets of each modelling term of the k and ?-transport equations. Comparison of model results with DNS data to evaluate the performance of each modelling term is also made in the present study. It is concluded that the low-Reynolds-number model has wider applicability and performs better than the two-layer model and wall function approaches. Comparison with DNS data further shows that large discrepancies exist between the DNS budgets and the modelled production and destruction terms of the ? equation. However, for simple channel flow the discrepancies are similar in magnitude but opposite in sign, so they are cancelled by each other. This may explain why, even when employing such an inaccurately modelled ?-equation, one can still predict satisfactorily some simple turbulent flows.     

A turbulent boundary layer over a two-dimensional rough wall

Particle image velocimetry (PIV) measurements and planar laser induced fluorescence (PLIF) visualizations have been made in a turbulent boundary layer over a rough wall. The wall roughness consisted of square bars placed transversely to the flow at a pitch to height ratio of λ/k = 11 for the PLIF experiments and λ/k = 8 and 16 for the PIV measurements. The ratio between the boundary layer thickness and the roughness height k/δ was about 20 for the PLIF and 38 for the PIV. Both the PLIF and PIV data showed that the near-wall region of the flow was populated by unstable quasi-coherent structures which could be associated to shear layers originating at the trailing edge of the roughness elements. The streamwise mean velocity profile presented a downward shift which varied marginally between the two cases of λ/k, in agreement with previous measurements and DNS results. The data indicated that the Reynolds stresses normalized by the wall units are higher for the case λ/k = 16 than those for λ/k = 8 in the outer region of the flow, suggesting that the roughness density effects could be felt well beyond the near-wall region of the flow. As expected the roughness disturbed dramatically the sublayer which in turn altered the turbulence production mechanism. The turbulence production is maximum at a distance of about 0.5k above the roughness elements. When normalized by the wall units, the turbulence production is found to be smaller than that of a smooth wall. It is argued that the production of turbulence is correlated with the form drag.     

Finite element computation of a turbulent flow over a two-dimensional backward-facing step

This paper is devoted to the computation of turbulent flows by a Galerkin finite element method. Effects of turbulence on the mean field are taken into account by means of a k-? turbulence model. The wall region is treated through wall laws and, more specifically, Reichardt's law. An inlet profile for ? is proposed as a numerical treatment for physically meaningless values of k and ?. Results obtained for a recirculating flow in a two-dimensional channel with a sudden expansion in width are presented and compared with experimental values.     

An eddy viscosity model with near‐wall modifications

An extended version of the isotropic k–ε model is proposed that accounts for the distinct effects of low‐Reynolds number (LRN) and wall proximity. It incorporates a near‐wall correction term to amplify the level of dissipation in nonequilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and the Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Comparisons indicate that the present model is a significant improvement over the standard eddy viscosity formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Direct numerical simulation of turbulent flows through concentric annulus with circumferential oscillation of inner wall

Periodic wall oscillations in the spanwise or circumferential direction can greatly reduce the friction drag in turbulent channel and pipe flows. In a concentric annulus, the constant rotation of the inner cylinder can intensify turbulence fluctuations and enhance skin friction due to centrifugal instabilities. In the present study, the effects of the periodic oscillation of the inner wall on turbulent flows through concentric annulus are investigated by the direct numerical simulation (DNS). The radius ratio of the inner to the outer cylinders is 0.1, and the Reynolds number is 2 225 based on the bulk mean velocity U_m and the half annulus gap H. The influence of oscillation period is considered. It is found that for short-period oscillations, the Stokes layer formed by the circumferential wall movement can effectively inhibit the near-wall coherent motions and lead to skin friction reduction, while for long-period oscillations, the centrifugal instability has enough time to develop and generate new vortices, resulting in the enhancement of turbulence intensity and skin friction.