Von Karman Length Scale Research Articles

Exact Eddy-Viscosity Equation for Turbulent Wall Flows—Implications for Computational Fluid Dynamics Models

For turbulent channel flow, pipe flow, and zero-pressure gradient boundary layer, Heinz yielded recently analytical formulas for the eddy viscosity as a product of a function of (the wall-normal distance scaled in inner units) and a function of (the same scaled in outer units). By calculating the eddy-viscosity turbulent diffusion term, an exact high-Reynolds-number equation with one production and two dissipation terms is constructed for those flows. One dissipation term is universal, peaks near the wall, and scales mainly with . The second, smaller one, is flow dependent, peaks in the wake, and scales mainly with . The production term is flow dependent, peaks in between, and scales similarly. The universal dissipation term implies a length scale analogous to the von Karman length scale used in the scale-adaptive simulation models of Menter. This length scale also appears in the production term. This confirms the relevance of these length scales. An asymptotic analysis of all terms in the budget in the limit of infinite Reynolds numbers is provided. This yields a test bench of existing Reynolds-averaged Navier–Stokes models with a similar equation. It is shown that some models, e.g., the one of Spalart and Allmaras, do not respect the flow physics: they display a production peak in the near-wall region. The most promising model, a scale-adaptive simulation model, is modified. As a step forward toward a solution to the wall damping problem, the equation of our model behaves much more correctly in the near-wall region.

Effect of von Karman length scale in scale adaptive simulation approach on the prediction of supersonic turbulent flow

Numerical investigations of the supersonic axisymmetric base flow at Mach number 2.46 and Reynolds number 2.858×106 are carried out using the scale-adaptive simulation (SAS) approach. To achieve SAS computation, three mathematical forms of von Karman length scale are utilized. The first one is calculated from the strain rate tensor and second derivative of the velocity field (this SAS can be marked as “SAS-1”). Computation of the second one is employed using the vorticity and its first derivative (this SAS can be denoted as “SAS-2”). The third one is computed from the vorticity and the second derivative of the velocity field (this SAS can be named as “SAS-3”). Their effects on the quantitative predictions are assessed. Two inflow conditions are also investigated, i.e., inflow condition with/without turbulent-boundary-layer profiles. Results show that the inflow conditions have the significant effects on the base-pressure and streamwise velocity distributions, and the predicted results at the inflow condition with profiles show better agreement with experimental data than that at the inflow condition without profiles. The inflow conditions and von Karman length scales only have the relatively slight effects on the radial velocity distributions. Similarly, at the inflow condition with profiles, the base-pressure distributions are also not sensitive to mathematical forms of von Karman length scale. More comparisons reveal that the best predictions can be obtained by SAS-2 and SAS-3. It may be associated with the vortex stretching effects explicitly considered in the von Karman length scales of SAS-2 and SAS-3. Although the mean flow field predicted by SAS-2 is slightly better than that obtained by SAS-3. However, SAS-3 can give the slightly better predictions for the Reynolds stresses than SAS-2.

A hybrid RANS/LES approach with scale-adaptive capabilities for highly separated flows

ABSTRACTA new hybrid RANS/LES approach with scale-adaptive capabilities is developed. The blending function in the SST model is adopted to prevent the invasion of the von Karman length scale to the RANS region, and the compressibility correction proposed by Wilcox is incorporated to produce a realistic shear layer development in compressible flows. The new model is validated for a subcritical flow past a circular cylinder and a supersonic base flow. Time-averaged turbulent statistics predicted by the new model show fairly good agreement with the experimental data, slight improvements over DES simulations, and are much better than SAS results. The main advantage of the new model over the DES method is that the distribution of the blending function reflects local vortex structures instead of grid spacing in the turbulent wake. The sequence of the effect intensity of the compressibility correction from strong to weak is SAS, the new model and DES.

Wall-modeled large eddy simulation of turbulent channel flow at high Reynolds number using the von Karman length scale

The von Karman length scale is able to reflect the size of the local turbulence structure. However, it is not suitable for the near wall region of wall-bounded flows, for its value is almost infinite there. In the present study, a simple and novel length scale combining the wall distance and the von Karman length scale is proposed by introducing a structural function. The new length scale becomes the von Karman length scale once local unsteady structures are detected. The proposed method is adopted in a series of turbulent channel flows at different Reynolds numbers. The results show that the proposed length scale with the structural function can precisely simulate turbulence at high Reynolds numbers, even with a coarse grid resolution.

A one-equation turbulence model for recirculating flows

A one-equation turbulence model which relies on the turbulent kinetic energy transport equation has been developed to predict the flow properties of the recirculating flows. The turbulent eddy-viscosity coefficient is computed from a recalibrated Bradshaw’s assumption that the constant a1 = 0.31 is recalibrated to a function based on a set of direct numerical simulation (DNS) data. The values of dissipation of turbulent kinetic energy consist of the near-wall part and isotropic part, and the isotropic part involves the von Karman length scale as the turbulent length scale. The performance of the new model is evaluated by the results from DNS for fully developed turbulence channel flow with a wide range of Reynolds numbers. However, the computed result of the recirculating flow at the separated bubble of NACA4412 demonstrates that an increase is needed on the turbulent dissipation, and this leads to an advanced tuning on the self-adjusted function. The improved model predicts better results in both the non-equilibrium and equilibrium flows, e.g. channel flows, backward-facing step flow and hump in a channel.

A Scale-Adaptive Turbulence Model Based on the k-Equation and Recalibrated Reynolds Stress Constitutive Relation

An algebraic relationship between turbulent dissipation rate and von Karman length are used to dismiss the transport equation of turbulent dissipation rate in standard k−ε (SKE) turbulence model. Meanwhile, a recalibrated Bradshaw's assumption is built based on the data from a boundary layer flow of turbulent flat plate simulated by direct numerical simulation (DNS). The JL model is reformed to a one-equation model which only depends on the turbulent energy, so the new model can also be called kinetic-energy dependent only (KDO) turbulence model. As the KDO model is using the von Karman length scale, it can automatically adjust to fit the resolved structures of the local flow. Results will be shown for the boundary layer flow on a turbulent flat plate, and the external flows of an NACA4412 airfoil, an ONERA-M6 wing, a three dimension delta wing, and an NACA0012 airfoil at deep stall.

Modeling the 3-d flow around a cylinder using the SAS hybrid model

<div> <p>Three-dimensional calculations were performed to simulate the flow around a cylindrical vegetation element using the Scale Adaptive Simulation (SAS) model; commonly, this is the first step of the modeling of the flow through multiple vegetation elements. SAS solves the Reynolds Averaged Navier-Stokes equations in stable flow regions, while in regions with unstable flow it goes unsteady producing a resolved turbulent spectrum after reducing eddy viscosity according to the locally resolved vortex size represented by the von Karman length scale. A finite volume numerical code was used for the spatial discretisation of the rectangular computational domain with stream-wise, cross-flow and vertical dimensions equal to 30D, 11D and 1D, respectively, which was resolved with unstructured grids. Calculations were compared with experiments and Large Eddy Simulations (LES). Predicted overall flow parameters and mean flow velocities exhibited a very satisfactory agreement with experiments and LES, while the agreement of predicted turbulent stresses was satisfactory. Calculations showed that SAS is an efficient and relatively fast turbulence modeling approach, especially in relevant practical problems, in which the very high accuracy that can be achieved by LES at the expense of large computational times is not required.</p> </div> <p> </p>

Extending the bounds of ‘steady’ RANS closures: Toward an instability-sensitive Reynolds stress model

The incapability of the conventional Unsteady RANS (Reynolds–Averaged Navier Stokes) models to adequately capture turbulence unsteadiness presents the prime motivation of the present work, which focuses on formulating an instability-sensitive, eddy-resolving turbulence model on the Second-Moment Closure level. The model scheme adopted, functioning as a ‘sub-scale’ model in the Unsteady RANS framework, represents a differential near-wall Reynolds stress model formulated in conjunction with the scale-supplying equation governing the homogeneous part of the inverse turbulent time scale ωh (ωh=ɛh/k). The latter equation was straightforwardly obtained from the model equation describing the dynamics of the homogeneous part of the total viscous dissipation rate ɛ, defined as ɛh=ɛ−0.5ν∂2k/(∂xj∂xj) (Jakirlic and Hanjalic, 2002), by applying the derivation rules to the expression for ωh. The model capability to account for vortex length and time scales variability was enabled through an additional term in the corresponding length-scale determining equation, providing a selective enhancement of its production, pertinent particularly to the highly unsteady separated shear layer region, modeled in terms of the von Karman length scale (comprising the second derivative of the velocity field) in line with the SAS (Scale-Adaptive Simulation) proposal (Menter and Egorov, 2010). The present model formulation, termed as SRANS model (Sensitized RANS), does not comprise any parameter depending explicitly on grid spacing. The predictive capabilities of the newly proposed length-scale determining model equation, solved in conjunction with Jakirlic and Hanjalic’s (2002) Reynolds stress model equation, are presently demonstrated by computing the flow configurations of increasing complexity featured by boundary layer separation from sharp-edged and continuous curved surfaces: backward-facing step flow, flow over a wall-mounted fence, flow over smoothly contoured periodically arranged hills and flow in a 3-D diffuser. The model performances are also assessed in capturing the natural decay of the homogeneous isotropic turbulence and the near-wall Reynolds stress anisotropy in a plane channel. In most cases considered the fluctuating velocity field was obtained starting from steady RANS results.

Convective heat transfer predictions in an axisymmetric jet impinging onto a flat plate using an improved [formula omitted] model

The paper discusses the convective heat transfer prediction in turbulent axisymmetric jets impinging onto a flat plate using the k – ω model of Wilcox (2006) [14]. Improvements to the heat transfer predictions are obtained in flow regions characterized by large level of strain using an impingement invariant proposed by Manceau (2003) [19]. The Manceau term is carefully introduced in order to leave the k – ω model predictions unmodified for reference flows. This is obtained using a blending function of Menter (1994) [12]. As an alternative, a modification based on the von Karman length scale is also discussed. Both modifications do not lead to stability problems in flow regions characterized by large levels of strain, preserving the robustness of the k – ω model. Improvements have been obtained for convective heat transfer prediction in stagnation flow regions of axisymmetric jets impinging onto a flat plate with nozzle-plate distances H / D = 2 , 6 , 10 and Reynolds numbers R e = 23000 , 70000 .

Application of the SAS turbulence model to predict the unsteady flow field behaviour in a forward centrifugal fan

In the current study, the unsteady flow in a centrifugal fan is carried out using Computational Fluid Dynamics calculation based on the Scale Adaptive Simulation (SAS) approach to model the turbulence phenomenon. The SAS concept is based on the introduction of the von Karman length scale into the turbulence scale equation. The information provided by the von Karman length scale allows SAS models to dynamically adjust to resolved structures in an Unsteady Reynolds-Averaged Navier–Stokes (URANS) simulation, which results in a Large Eddy Simulation-like behaviour in unsteady regions of the flow field. At the same time, the model provides standard RANS capabilities in stable flow regions. The introduction of the von Karman length scale is based on the reformulation of Rottas's equation for the integral length scale. To validate the numerical results, the overall performances of the fan and the wall pressure fluctuations computed upon the volute casing surface are compared with the unsteady measured data.

Blending Functions in Hybrid Large-Eddy/Reynolds-Averaged Navier-Stokes Simulations

Several blending functions for use in hybrid large-eddy simulation (LES)/Reynolds-averaged Navier-Stokes (RANS) simulations of shock-separated flows are tested. The blending functions shift the turbulence closure from a k-ζ turbulence model near solid surfaces to a k − ∆ subgrid closure away from the wall. Three distinct forms for the blending function are developed: one that depends on the ratio of the von Karman length scale and the Taylor microscale, another that depends on the ratio of the RANS eddy viscosity to the subgrid eddy viscosity, and a third which replaces the von Karman length scale in the first form with the distance to the nearest wall. Comparisons are made for two cases: Mach 2.79 flow over a 20-deg compression corner and Mach 2.88 flow over a 25-deg compression/expansion corner. boundary conditions for all calculations employ the rescaling/reintroducing procedure developed by Xiao et al. (Xiao, X., Edwards, J. R., Hassan, H. A., and Baurle, R. A., Inflow Boundary Conditions for Hybrid Large Eddy/Reynolds Averaged Navier-Stokes Simulations, AIAA Journal ,V ol. 41, No. 8, 2003, pp. 1481-1489) for hybrid LES/RANS simulations of wall-bounded flows. In general, the blending function based on the von Karman length scale gives the best results when compared with measured data. The skin friction predictions show the highest sensitivity to the various blending functions.

Mellor-Yamada simplified second-order closure models: Analysis and application of the generalized von Karman local similarity hypothesis

Mellor-Yamada's “superequilibrium” Level 2 and Level 1 models are analyzed using the Monin-Obukhov theory framework. Yamada's (1975) analysis is supplemented by a discussion of the realizability requirements posed on model constants and by the inclusion of the master-length scale problem. The generalized von Karman local similarity hypothesis (Laikhtman, 1979) is examined as an alternative closure hypothesis for second-order models. A systematic method of model examination is used. First, a family of models, consisting of Level 1 and Level 2 Reynolds-stress equation sets and different length-scale hypotheses (Prandtl's, generalized von Karman's), is built. Next, asymptotic characteristics of individual models are investigated and compared with similarity predictions. Monin-Obukhov universal functions for turbulent energy, space scale and temperature variance, derived from the models, are compared with experimental surface-layer data. Generally, models employing the stability-dependent generalized von Karman hypothesis perform better than those that use the conventional Prandtl mixing-length concept. The choice amongst the von Karman type models is still ambiguous. However, the Level 1 model with a stability-dependent generalized von Karman length scale seems to be the best of those considered.