Efficacy of several large-scale flow parameters as transition onset markers are evaluated using direct numerical simulation (DNS) of boundary layer bypass transition. Preliminary results identify parameters and to be a potentially reliable transition onset marker, and their critical values show less than 15% variation in the range of Re and turbulence intensity (TI). These parameters can be implemented into general-purpose physics-based Reynolds-averaged Navier–Stokes (RANS) models for engineering applications.
Engineering applications often involve bypass transition, which occurs due to the presence of strong disturbances (high freestream turbulence, large wall roughness elements, flow separation, pressure gradient effects, etc.) 1. The bypass transition process entails strongly nonlinear phenomena leading to boundary layer breakdown, hence it cannot be well described by linear theory and remains a significant modeling challenge , particularly for Reynolds-averaged Navier-Stokes (RANS) based computational fluid dynamics. Currently available general-purpose transition-sensitive RANS turbulence models can be loosely classified as either correlation based  or physics based . Correlation-based models typically solve for an intermittency transport equation, which is the fraction of time the flow is turbulent during the transition phase and is used as a turbulent eddy viscosity multiplier. Several studies have reported that the intermittency distribution shows a universal behavior upon normalization . The transition onset location is either specified explicitly based on empirical correlations or solved for using additional transport equations. These models rely directly on empirical correlations to specify model parameters.
Physics-based models  though still highly empirical in nature aim for a more generalized approach wherein the evolution of turbulent fluctuations is predicted in the pretransitional and transitional regions. Development of such models requires: (1) a proper understanding of turbulence production processes such as entrainment of freestream turbulence, development of fluctuations in the pretransitional regions including turbulence damping (shear sheltering), boundary layer breakdown (turbulent spot formation), turbulent energy production dynamics, and overshoot of turbulent fluctuations in the post-transition region and (2) evaluation and/or identification of flow parameters that can be used as a marker for turbulence onset/growth in low-fidelity RANS simulations. Previously documented large eddy simulation (LES) and direct numerical simulation (DNS) studies have helped in highlighting some of the underlying transitional flow physics [6–8] to address the above requirements. Studies agree that freestream disturbances induce low-frequency streamwise vortices or streaks in the pretransitional region (referred to as Klebanoff modes), which lift from the wall causing ejection events. Transition occurs due to the formation of turbulent spots, which are associated with multiple head hairpin-type vortices with U- or Λ-shaped structures underneath them. However, the energy transfer pathway from freestream disturbances to pretransitional (nonturbulent) fluctuations to turbulent fluctuations remains somewhat unclear. Mayle and Schulz  identified the pressure-diffusion terms as the driver of the growth of Klebanoff modes from freestream disturbance and pressure-strain terms as the driver of the energy redistribution from the Klebanoff modes to the other components. The latter is also supported by Lardeau et al. , wherein it was indicated that unlike the fully developed turbulent region, the pressure-strain terms are negligible in the pretransition regime. Walters  hypothesized that the absence of the pressure-strain inhibits nonlinear turbulence breakdown and is closely related to shear-sheltering as proposed by Jacobs and Durbin .
Jacobs and Durbin  reported that the near-wall streamwise streaks (Klebanoff modes) extend throughout the transition region, i.e., λ ∼ x, therefore, the Praisner and Clark  transition onset parameter is potentially consistent with the parameter used in other studies.
The objective of this study is to investigate the transition onset markers discussed above and analyzed to evaluate their potential efficacy as transition onset criteria in RANS simulations of transitional and turbulent flow. Temporally developing DNS are performed for channel flow at Reτ = 180 (ReH = 3300) (Re180 henceforth) and 590 (ReH = 12,656) (Re590 henceforth), and a flat-plate boundary layer flow at zero pressure gradient flat plate (FP), with different initial TI = 1% to 5%.
Simulation Setup and Conditions
The simulations are performed using a pseudo-spectral code which solves incompressible Navier-Stokes equations using Fast Fourier Transforms along periodic streamwise (x) and spanwise (z) directions, and Chebyshev's polynomials along the non-homogeneous wall normal (y) direction . For channel flow cases, Reynolds number Reτ is defined based on mean friction velocity in the fully developed turbulence region, where is the mean wall shear stress and is the density. ReH is defined based on initial centerline velocity and half channel height H. As summarized in Table 1, channel flow Re180 simulations (i.e. Reτ = 180) are performed for TI =1%, 2.5%, and 5% (cases 1-3); channel flow Re590 simulations are performed for TI =1%, 2%, 3%, and 5% (cases 4-8); and the FP simulations are performed for TI =2.1%, 2.8%, and 3.5% (cases 9-11). In the channel flow simulations, a body force term is applied in the -direction to balance the momentum loss via wall friction expected in the fully developed turbulent region, i.e., .
The channel flow simulations are started with a fully developed mean turbulent velocity profile, expected for the flow, superimposed with turbulence fluctuations. The FP simulations are started from a Blasius mean streamwise velocity (U0) profile at Reθ0=106 or Rex0=2.55 × 104, superimposed with turbulence fluctuations. To obtain the initial turbulence fluctuations, a separate simulation was performed in which isotropic turbulence was generated using Rogallo's  approach, mapped to the simulation domain, and damped near the wall using an exponential damping function. The initial fluctuations were allowed to evolve, such that the prescribed random phases adjusted to satisfy the Navier-Stokes equations. The velocity skewness, kurtosis, and velocity derivative skewness were monitored in regions away from the wall, and the simulation was continued until they converged to 0, 3, and -0.29, respectively, expected for isotropic turbulence. Figure 1 shows the converged initial turbulence and TI variation along the wall normal direction for Re590, TI = 2%.
Bhushan et al.  provide detailed validation of the channel flow simulations in the fully developed turbulent region using M-DNS, and Muthu and Bhushan  provide detailed validation of the FP transition flow predictions using JD-DNS and experimental data . Therefore, only key validation results are presented herein. The discussion below first focuses on the analysis of transient flow in the channel to evaluate similarities with FP bypass transition, followed by an analysis of large scale turbulence structures in the channel and FP simulations to identify appropriate potential transition onset markers for use in RANS modeling.
Bypass Transition Behavior of Transient Channel Flow
The comparison between transition in channel flow and flat plate bypass transition is in part inspired by the study of He and Seddighi , which demonstrated that the evolution of turbulent structures from low-Re (or low turbulence level) conditions to high-Re (high turbulence level) conditions in channel flow exhibits bypass transition behavior. During the simulations, the flow was accelerated such that the bulk velocity increased by 250%. Further, the near-wall region included an initial disturbance, and “quasi-laminar” near-wall streaky structures during pre-transition (similar to Klebanoff modes) were caused by the stretching of initial turbulence by the increased flow rate. The simulations performed herein differ from Ref.  both for the temporal variation of mean flow and for near-wall turbulence growth. In the current study, the mean flow variations are expected to be small, as the flow is started with a fully developed turbulent mean velocity profile, though the mean flow does adjust to the difference in applied body force and wall-shear stress during turbulence growth. All simulations, except case #4, show < 6% variation in the mean center line velocity (Uc). Case #4 showed 17% variation in Uc as the turbulence growth region was about two times longer than for the other cases. In addition, the simulations are set up such that the near-wall flow is devoid of resolved turbulence. Thus, the growth of turbulence is expected to be similar to those starting from laminar conditions.
The variation of wall shear stress (τw) for the channel flow simulations (Fig. 2(a)) shows four distinct regions: (a) an initial sharp drop, (b) a slow-steady decrease, (c) a steady increase, and (d) quasi-steady values. Zone (a) is due to adjustment of the turbulent flow to the initial conditions, as the turbulence fluctuations in the near-wall region are not adequate to account for the expected turbulent stresses, resulting in a stress depletion. Zone (d) is the fully developed turbulent flow region, which shows a quasi-steady wall stress with small variation about . The flow in this region is different from FP, since the boundary layer ceases to grow while it grows continuously during the FP simulations. Figure 2(b) shows that zones (b) and (c) are qualitatively similar to the laminar and transition flow regions, respectively, obtained in the FP simulations.
For channel flow simulations, an initial temporal location (t0) was chosen in zone (b) (as demonstrated in Fig. 2(a) for Re590, TI = 1%), and and were obtained from the corresponding using the Blasius solution.4 For flat plate simulations, and were based on the initial Blasius velocity profile. The domain velocity VD was chosen to match the Cf variation in the pretransition/laminar region with the Blasius profile, following . was estimated to be 0.5, 0.75, and 0.5 for Re180, Re590, and FP simulations, respectively. The estimate for Re590 is consistent with . Muthu and Bhushan  reported that is mostly constant (0.5) in the laminar region, but is expected to vary in the transition and turbulent regions as the turbulence level increases. They proposed the estimation of VD using a momentum integral method, which provided better agreement with the spatial DNS than the constant velocity formulation. The use of a constant velocity is expected to affect the intermittency profile predictions, but not the growth of flow structures during pre-transition or transition initiation, which is the focus of this study.
For the channel flow simulations, variation of with as shown in Fig. 2(c) compares well with the Blasius profile, as expected. As TI decreases, the length of the pretransition region increases, Cf increases more rapidly during transition, and the overshoot becomes more prominent. These trends are consistent with those of bypass transition. As shown in Fig. 2(d), Reθ shows a linear increase with Rex during transition and converges to a constant value in the fully developed region. The Re590, TI = 2% predictions compare reasonably well with the T3A data . The Re590, TI = 3% case shows a faster Reθ growth than the TI = 2% case and lies in between flat plate data for TI = 3% and 6% (T3B). A faster growth rate is expected for higher TI with earlier transition. The FP predictions also show an effect of TI consistent with bypass transition, and Cf and Reθ predictions for TI = 2.8% are compared well with T3A data.
The streamwise mean velocity and turbulent fluctuation profiles predicted for Re590, TI =2% and FP, TI = 2.8% are compared with JD-DNS in Fig. 3. Both the channel and the FP velocity predictions show a large laminar-like sublayer up to y+ ∼60 in the pretransition region. As the flow transitions to turbulence, the sublayer extent decreases and the log-law extent increases. The predictions agree very well with each other and JD-DNS in the fully developed turbulent region. Channel and FP results show some differences in the pretransition/transition region. The FP results do not show a well-defined log-layer in the transition region, whereas the channel flow does due to the initial condition. Channel flow streamwise turbulent velocity ( predictions also show variation trends similar to the FP results. In wall coordinates, a peak is observed at y+∼ 15-20 for all regimes, but in terms of wall distance it moves closer to the wall as the flow transitions. The highest value is obtained during transition and is higher than the peak values in the fully developed turbulent region.
The near-wall vortical structures predicted for Re590, TI = 2% are shown in Fig. 4 using isosurfaces of the second-invariant of the velocity gradient (Q). Counter-rotating longitudinal structures are visible in the pretransition region, similar to Klebanoff modes. Note that these structures in case #5 extend the entire streamwise length of the domain and similar behavior is observed for the longer streamwise domain 2Lx (case #8), which is consistent with the previous observations . Cases #5 and #8 both show similar Cf distributions (Fig. 2), mean and turbulent velocity predictions (figure not shown), and similar growth of the large-scale structure during transition. This suggests that the use of streamwise periodic boundary conditions does not significantly alter the dynamics of the transition process. Figures 4(c) and 4(d) show that as the transition starts, the longitudinal streaks in the pretransition region induce lifting on each other to form hair-pin structures, consistent with . The frequency of the hair-pin generation increases, and the structures are most intense (and numerous) in the overshoot region.
As shown in Fig. 5(a), the peak value of in the pretransition region shows linear growth, which is expected for bypass transition . The fluctuations reach a maximum midway through transition, then decrease to a quasi-steady value in the turbulent region. Fluctuation components and show a slow increase during early transition and a steeper increase during latter part of transition, and show limited overshoot compared to . The TKE and stress budget in the pretransition region (figure not shown) shows that is transported away from the wall at the sub- and log-layer interface due to pressure-diffusion, which in return is induced by the initial turbulence, consistent with . The fluid momentum is carried away from the wall-layer by fluctuations, which leads to the production of shear stress and subsequently production of near-wall fluctuations. The pressure-strain terms are negligible in this region, and redistribution of energy from streamwise to spanwise and wall-normal components is absent. This is consistent with the LES study of Voke and Yang  for the flat-plate boundary layer, and He and Seddighi  channel flow results. In the transition and turbulent regions (plots in the transition region are shown in Figs. 5(b)–5(d)), generation (via ejection events) is preceded by the lifting induced by the longitudinal vortices. Similar to the pretransition region, the fluctuations lead to the production of shear stress, and subsequently, fluctuations in the sublayer. The fluctuations are then transported away from the sublayer by the ejection events, and the energy is distributed to the and components via pressure strain. This results in the growth of the log-layer region. Both the transition and turbulent regions show similar energy generation and transfer processes, except that in the former the pressure-strain is more prominent and the energy dissipation is primarily from the component.
Overall, the growth of Cf and its variation with TI, growth of near-wall vortical structures, growth of during transition, and energy redistribution mechanisms confirm that the growth of turbulence in the channel flow simulations exhibits bypass transition behavior similar to that observed in flat plate boundary layers.
Identification of Transition Onset Markers
A viable large-scale transition onset parameter for the use in RANS transition-sensitive turbulence models should satisfy the following key criteria:
The dimensionless transition onset marker should be a function of local, statistical quantities that are available within the framework of Reynolds-averaged simulations, e.g., mean velocity gradient, turbulent kinetic energy, and fluid viscosity.
It should show a well-defined peak in the near wall region. A well-defined peak is essential, so that the onset parameter can be easily identified during simulations. The location of the peak is expected to coincide with the developing lower log layer or buffer region, where significant energy transfer from streamwise fluctuations to other components is expected via the action of the pressure strain terms.
The peak value should monotonically increase when the flow transitions from laminar to turbulent flow. A monotonic trend is important for modeling purposes so that there is no ambiguity regarding the continuity of the transition process from onset to fully turbulent flow. Likewise, the marker should obtain a value well above the critical value over most of the fully turbulent boundary layer, with values below the critical value only appearing in the viscous dominated region very close to the wall.
The critical value of the transition marker should be independent of Re and TI, so that the onset parameter is generally applicable over a relatively wide range of flow conditions.
Key indicators of transition onset are (a) a local minimum in the wall shear stress or friction velocity ; (b) rapid growth of streamwise turbulence fluctuations ; and (c) transfer of energy to other components and . As discussed earlier, studies have identified , , , and as potential markers that obtain universal peak values at locations corresponding to these indicators. In this study, each of these except , along with their streamwise normal planar counterparts , , and , where and , and local turbulent Reynolds number are investigated. The Praisner and Clark  formulation is not investigated due to the ambiguity in estimating the relevant local turbulence length scale.
The critical value of the onset parameters is summarized in Table 1, and their variation in pretransition, transition, and turbulent regimes is discussed below using key results in Fig. 6. The critical values of the parameters are identified using Cf and peak variation profiles. For FP, the local minimum of Cf and start of linear growth were coincident. However, for channel flows, growth was found to be a better indicator of the transition onset than Cf as shown in Fig. 5(a).
The value of showed a well-defined peak at y+∼40 for pretransition, transition, and turbulent regions. The critical value was for both channel flow and FP simulations for the range of TI investigated, which compares well with values reported in the literature. The marker satisfies criteria 1, 2, and 4 above, but fails to satisfy criterion 3, as its value in the turbulent region for low TI cases is close to the critical value.
The value of also shows a well-defined peak at y+∼40 in all the flow regimes, similar to . Its critical value decreases with the increase in TI for both the channel flow and FP simulations. For TI >2%, the critical value is ≈4.3, somewhat higher than that reported by Sharma et al. . In addition, its value falls below the critical value in the turbulent region. Therefore, fails to satisfy criterion 3. It also fails to satisfy criterion 1 since is not easily obtained in general RANS simulations at locations away from the wall boundary.
Both and show well-defined peaks at y+∼100 close to the transition onset, and the peak moves to lower y+∼60 in the turbulent region. The peak magnitudes are small and start to grow almost exponentially at transition onset, increase monotonically throughout transition, and show quasi-steady values in the turbulent region. The values in the turbulent region are about three times as large as those at onset. These parameters satisfy criteria 2 and 3, but do not satisfy criterion 4 as the critical values increase with increasing TI and show around 30% variation for the cases considered herein.
The parameter shows a well-defined peak around y+ ∼60–80 at transition onset and shows an almost constant value in the log-layer. The value increases sharply in the pre-transition region and shows quasi-steady values within ±10% of the critical value in the turbulent region. The critical value is ≈127 ± 14% for Re590 and FP but shows around 20% lower values for Re180. Thus, this parameter does not satisfy criterion 4 very well and may have limitations for low-Re flows.
The peak location for the parameter is similar to that for. The peak value shows a sharp increase in the pre-transition/transition regions and shows quasi-steady values in the turbulent region that are 3-4 times larger than the critical value. The critical value is of ≈ 20 ± 18% for all the cases. Re180 shows somewhat smaller values than the other cases. Excluding Re180 case, the critical value is ≈ 22 ± 14%. Overall, this parameter appears to satisfy all the criteria reasonably well, but may have limitations for low-Re flows.
The parameter shows a well-defined peak at around y+∼80 in the pretransition region, and the peak shifts to larger y+ as the flow transitions. In the turbulent region, the peak is observed at y+∼300 for the flat-plate cases, but occurs at the centerline (or peak y+) for the channel flow cases. The critical value is considering the average over all cases. The values are somewhat smaller for the Re180 case. Excluding Re180, the critical value is . The parameter shows a steady increase from the pretransition to the turbulent region, where the values in the turbulent region are almost three times the critical value. However, the parameter only partially satisfies criterion 3, as it is apparent that values above critical are only present in the fully turbulent boundary layer above y+∼100, as shown in Fig. 6.
Overall, all the transition onset parameters investigated satisfy some of the necessary criteria for the use in RANS-based transition models. Only satisfies all of the parameters, however even for the relatively limited set of cases tested here, there is up to 18% variation in its critical value, which may suggest limited universality over a wide range of conditions. The parameter showed the least variation in critical value for different test cases and is consistent with transition markers that have been similarly used by several previous investigators, although its failure to satisfy criterion 3. This suggests that it may be problematic for the use in a single-point RANS model. Interestingly, these two parameters suggest fundamentally different mechanisms responsible for suppression of pressure-strain energy redistribution in the pre-transitional boundary layer, with the former indicating a viscous damping and the latter an inviscid effect perhaps related to wall blocking. It is recommended that future studies examine each of these parameters more closely and seek to tie their performance as a transition marker to sound phenomenological reasoning.
Efficacy of several large-scale flow parameters as transition onset markers was evaluated for boundary layer bypass transition using temporally evolving DNS. The reliability of the parameters as effective markers was judged based on the appearance of well-defined peaks, monotonicity of the value during transition, and independence on TI and Re variations. Preliminary results identify parameters and to be potentially reliable transition onset markers. Their critical values were estimated to be ≈ 22±14% and Future work will focus on investigating the underlying physical mechanisms represented by these parameters, as well as increasing the DNS database for intermediate and lower TI, and for flows with pressure gradients. The eventual goal of the comprehensive effort is to incorporate one or more of the investigated transition markers into general-purpose physics-based RANS models for the use in engineering computational fluid dynamics predictions of transitional flows.
Air Force Office of Scientific Research (Air Force SFFP Program FY 2015 and FY 2016).
NASA EPSCoR (Project No. 80NSSC17M0039).
The parameter value at the transition onset is referred to as the critical value.
, where δ is the boundary layer thickness, and x is the distance from the plate leading edge.