A physics-based modification to the $kT−kL−ω$ transition-sensitive eddy-viscosity model is presented. The modification corrects an anomaly related to the physical mechanism of production of laminar kinetic energy for regions far from the wall in fully turbulent flows, by limiting the production of natural modes in the large-scale eddy-viscosity term by a rescale of the wall-limited turbulent length scale. Round jet and backward facing step test cases are used to reveal the relevant issue and to demonstrate that the new modification successfully addresses the problem.

## Introduction

The prediction of transitional flow using Reynolds-average Navier–Stokes (RANS) models is critical for many complex fluid flow applications. Most popular transition-sensitive models have been developed by coupling an empirical transition correlation to a fully turbulent RANS model [1,2] or including additional transport equations to the RANS-based turbulence models [310].

Efforts remain underway to improve the ability of RANS models to properly account for other complex physical mechanisms, for example, rotation and curvature effects [1113]. In all the cases, however, new models must be thoroughly validated and their range of applicability must be properly determined. For example, Ghahremanian and Moshfegh [14] recently demonstrated that transition-sensitive models, while improving predictions in attached boundary layers, may perform less well than the traditional fully turbulent models (e.g., see Ref. [15]) in separated shear flows.

In this paper, a modification to the popular single-point, physics-based $kT−kL−ω$ transition-sensitive eddy-viscosity model developed by Walters and Cokljat [7] is proposed and tested in order to correct nonphysical behavior that has been identified in the model. The modified form of the model presented here is recommended as a more appropriate form of the original model in Ref. [7].

## A Physics-Based Correction to the kT−kL−ω Model

The single-point, physics-based $kT−kL−ω$ transition-sensitive eddy-viscosity model (also referred to as the $k−kL−ω$ model) is based on a concept originally presented by Walters and Leylek [6] and further developed by Walters and Cokljat [7]. It incorporates an additional transport equation for laminar kinetic energy ($kL$) into a modified form of a two-equation eddy-viscosity turbulence model. Several modifications to the original version of the $kT−kL−ω$ model have been developed to include complex physical mechanisms, improve the results of the model in complex geometries, or develop new transitional models [1619].

### Model Equations.

The complete presentation of the model equations can be found in Ref. [7]. However, the reference contains typographical errors that have been previously identified and corrected [20], and these will be outlined in this section. One additional change, not yet reported in the open literature, will also be made in this section. This last modification corrects the behavior of the production of laminar kinetic energy away from the wall.

The general form of the model equations in their incompressible form is given by
$DkTDt=PkT+RBP+RNAT−ωkT−DT+∂∂xj[(ν+αTσk)∂kT∂xj]$
(1)

$DkLDt=PkL−RBP−RNAT−DL+∂∂xj[ν∂kL∂xj]$
(2)

$DωDt=Cω1ωkTPkT+(CωRfW−1)ωkT(RBP+RNAT)−Cω2ω2fW2+Cω3fωαTfW2kTd3+∂∂xj[(ν+αTσω)∂ω∂xj]$
(3)

Note that the first typographical correction made to Ref. [7] is the third term on the right-hand side of Eq. (3). This term appears as $−Cω2ω2$ in Ref. [7].

Also, Eqs. (11) and (16) in Ref. [7] should be, respectively, corrected to
$fW=(λeffλT)23$
(4)

$fINT=min(kTCINTkTOT,1)$
(5)

The previous three corrections are all typographical errors in the text of Ref. [7] and should be made in order to reproduce the simulation results shown in that paper.

This brief presents a further modification to the $kT−kL−ω$ model in order to avoid nonphysical production of laminar kinetic energy in regions far from walls. The production of $kL$ is defined as the interaction of Reynolds stresses that are associated with the pretransitional velocity fluctuations and mean shear, and it is conceptually governed by the large-scale near-wall turbulent fluctuations interacting with the mean velocity gradients in the boundary layer.

The production of $kL$, $PkL$, is defined in Ref. [7] by Eqs. (17)–(22) as
$PkL=νT,lS2$
(6)

$νT,l=min{fτ,1C11(Ωλeff2ν)kT,lλeff+βTSC12(d2Ων)d2Ω,0.5(kL+kT,l)S}$
(7)

The first term inside the brackets is comprised of the sum of two parts: the first addresses the development of Klebanoff modes and the second addresses self-excited (i.e., natural) modes [7].

Note that as the model term is currently expressed, the second part of that sum is proportional to the wall distance raised to a power of four. This formulation works well in describing the correct physical dependence for boundary layer flows (wall-bounded flows), but the entire term can be dominated by the distance from the wall for nonboundary layer flows. In fact, the formulation can be completely incorrect for fully turbulent free shear flows, as evidenced by the results presented below. To correct this, the term should be made proportional to a length scale that is equal to the wall distance in near-wall flows and scales proportional to the turbulent integral length scale in farfield flows.

To limit the production of natural modes in zones far from the wall in fully turbulent flows where this mechanism is not active, the second part of the large-scale eddy viscosity should be modified. The proposed modification to Eq. (7) is to simply adopt an effective wall distance analogous to the effective length scale used in Eq. (4)
$νT,l=min{fτ,1C11(Ωλeff2ν)kT,lλeff+βTSC12(deff2Ων)deff2Ω,0.5(kL+kT,l)S}$
(8)
Note that instead of the wall distance term $d$, $deff$ is used and it is defined as
$deff=λeffCλ$
(9)

where $λeff=min(Cλd,λT)$, $Cλ=2.495$, and $λT=(kT/ω)$ are identical to those given in Ref. [7]. All the other terms in the model are correctly defined in Ref. [7].

## Numerical Results

The model was implemented as a user-defined function (UDF) in the commercial finite volume based cfd solver ANSYS FLUENT® version 14.0 [21]. The pressure-based solver option was used with the semi-implicit method for pressure-linked equations (SIMPLE) method [22] for pressure–velocity coupling. All the discretization schemes were second-order accurate, and a mesh sensitivity study was performed to ensure that the results were grid independent (cf. Refs. [7,13]). This approach has been well demonstrated to be appropriate for incompressible single-phase flows. Specifically to the present study, it was verified that the nonphysical behavior documented for the original $kT−kL−ω$ was consistently observed regardless of the details of mesh, numerical scheme, or boundary conditions.

The original version of the $kT−kL−ω$ model presented in Ref. [7] (with the typographical errors corrected as discussed in the “Model Equations” section and the version with the large-scale eddy viscosity modified by Eq. (8) have been tested using a round jet flow and a backward facing step geometries.

### Round Jet Flow.

There are several experimental and numerical studies involving axisymmetric round jets [14,23,24], in which the performance of RANS models is mixed. Of particular interest are the results of Ghahremanian and Moshfeg [14]. Their results show that the $kT−kL−ω$ transitional model performs poorly for this particular case.

Figure 1(a) shows a schematic of the two-dimensional computational domain used for the round jet flow simulations. Only half of the physical domain was used in the calculations, taking the centerline of the jet as a symmetry axis (see Fig. 1(a)). At the jet inlet, the velocity $U∞$ was $56.2 m/s$, and turbulence intensity was approximately 0.6%. In order to ensure fully turbulent flow at the jet exit, a preliminary simulation was first run using a fully turbulent eddy-viscosity model. The results were used to obtain inlet boundary conditions for the simulation with the $kT−kL−ω$ model.

Figure 1(b) shows the contours of velocity computed with the $kT−kL−ω$ model. The maximum velocity is reached at the jet exit and decreases as the flow moves downstream. The predicted velocity field did not show any significant alteration by the change made to the large-scale eddy viscosity. Other test cases run by our group, including both wall-bounded and separated flows, were consistent with this result. This suggests that the changes outlined in this paper, while making the model more physically sound, do not fundamentally change the mean flow behavior predicted by the model.

On the other hand, the distribution of laminar kinetic energy far from the wall shows significant changes that are well demonstrated by the jet case presented here. Figure 2 shows the contours of laminar kinetic energy normalized by $U∞2$ for the fully turbulent jet flow, using the original [7] and the modified form of the model. It is apparent from the figures that the levels of laminar kinetic energy increase after the outlet of the jet for the original version of the $kT−kL−ω$ model. This is clearly an incorrect behavior, which is mitigated by the modification of the large-scale eddy viscosity as demonstrated in Fig. 2(b). The maximum value of $(kL/U∞2)$ for the original model is 0.0137, and it is reached downstream of the jet outlet, while the modified version reproduced a maximum of 0.00316, with the maximum located in the attached boundary layer.

### Backward Facing Step.

The backward facing step is a widely used benchmark test case for turbulence model validation. In this test case, the flow separates at the step with a reattachment farther downstream. In order to let the flow develop, the domain upstream of the step was built to measure $100D$, where $D=1.27 cm$ is the height of the step. With an inlet velocity of $44.2 m/s$ and $Tu∞=3.0%$, the flow is fully turbulent at the step location. The details of the experimental configuration can be found in Ref. [25].

Figure 3(a) confirms the nonphysical production of laminar kinetic energy in the original version of the model [7], in this case right after the step when the flow separates. Figure 3(b) also demonstrates that the correction proposed in Eqs. (10) and (11) effectively resolves the issue.

The normalized laminar kinetic energy is plotted in Fig. 4 along a plane downstream of the step corner and parallel to the bottom wall. The corrected version of the model shows the expected decay of laminar kinetic energy after the separation and away from the wall, while the original version on the model shows a nonphysical jump downstream of the step on the evaluation plane.

## Conclusions

A simple modification to the $kT−kL−ω$ model developed by Walters and Cokljat [7] was proposed and demonstrated for a fully turbulent jet flow and a backward facing step flow. The new version of the model does not appear to affect the transition prediction behavior of the original model for wall-bounded flows and shows no significant change for prediction of the mean velocity field in separated flow regions. However, the corrected model does correct the excessive, nonphysical production of laminar kinetic energy in regions far from the wall which was demonstrated by the original model [7]. This paper also identifies typographical errors in Ref. [7], which had been previously documented elsewhere [20].

## Acknowledgment

This work was partially funded by the U.S. National Aeronautics and Space Administration under Grant No. NNX10AN06A. The authors are grateful for the support.

## Nomenclature

• $d$ =

wall distance

•
• $DL$ =

anisotropic (near-wall) dissipation term for $kL$

•
• $DT$ =

anisotropic (near-wall) dissipation term for $kT$

•
• $fINT$ =

intermittency damping function

•
• $FW$ =

inviscid near-wall damping function

•
• $kL$ =

laminar kinetic energy

•
• $kT$ =

turbulent kinetic energy

•
• $kT,l$ =

effective large-scale turbulent kinetic energy

•
• $kTOT$ =

total fluctuation kinetic energy, $kT+kL$

•
• $PkT$ =

production of turbulent kinetic energy by mean strain rate

•
• $RBP$ =

bypass transition production term

•
• $RNAT$ =

natural transition production term

•
• $S$ =

magnitude of mean strain rate tensor

•
• $ν$ =

kinematic viscosity

•
• $νT,l$ =

large-scale turbulent viscosity contribution

•
• $αT$ =

effective diffusivity for turbulence-dependent variables

•
• $λeff$ =

effective (wall-limited) turbulence length scale

•
• $λT$ =

turbulent length scale

•
• $ω$ =

inverse turbulent time-scale

•
• $Ω$ =

magnitude of mean rotation rate tensor

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