Amplification Factor Transport Research Articles

Improvement and application of local-variable-based transition models to complex three-dimensional configurations

The hypersonic local-correlation-based transition modeling (LCTM) and amplification factor transport (AFT) transition models are extended to crossflow-induced transitions and applied to three-dimensional hypersonic vehicles. A new crossflow Reynolds number correlation is proposed and implanted to an existing hypersonic LCTM to extend it to take account of the crossflow effect. In the hypersonic AFT model, a new transport equation is added to predict the transported crossflow amplification factor. The two improved transition models are first validated using the HIFiRE-5 configuration. The results demonstrate that both transition models predict the transition on HIFiRE-5 with reasonable accuracy. Next, the two models are applied to the X-51A forebody and the X-33 vehicle. The results show that both models are capable of predicting transition behaviors and aeroheating environments on hypersonic complex geometries. The two models also provide reasonable reflections of the effects of various factors that influence transition, including the freestream Reynolds number, turbulence level, and angle of attack.

Natural laminar flow airfoil design via adjoint-based transition onset delay

This work is focused on the aerodynamic shape optimization (ASO) of airfoils with the aim of attaining natural laminar flow (NLF) designs. The two-equation Amplification Factor Transport (AFT) model is utilized for an accurate prediction of the transitional boundary layer flows. The NLF designs are achieved by utilizing two different objective functions. In the first approach, the conventional drag minimization problem is solved with a target lift coefficient. In addition, a second approach is proposed in this work that directly aims at delaying the transition onset on both the pressure and suction sides, thus expanding the natural laminar flow region around the airfoil. It is shown that by utilizing a sigmoid fitting of the turbulence index profile, the transition onset locations can be accurately predicted in a differentiable and smooth fashion, which is essential to the adjoint-based sensitivity analysis of the RANS solver. The efficacy of the proposed technique for NLF design has been tested for NACA 0012 and RAE 2822 airfoils at moderate lift coefficients. Results have shown that a significantly lower drag count can be achieved by delaying the transition onset locations compared to the case where only a drag minimization is sought. Additionally, the transition onset locations on both sides of the airfoil are delayed using our proposed technique which results in a significant expansion of the natural laminar flow region.

Amplification factor transport transition model for high-speed flows

A fully local amplification factor transport model is developed for high-speed transitional flows. On the basis of the approximate eN envelope method of linear stability theory, two transport equations are established to describe the evolution of amplification factors for first- and second-mode instabilities in high-speed boundary layers. An intermittency factor transport equation is then constructed based on the transported amplification factors and is coupled with Menter’s k−ω shear-stress transport eddy-viscosity turbulence model. The new model is validated through several test cases under different flow conditions, including flat plates, straight and flared cones, and a double ramp configuration. Comparisons with linear stability theory and experimental data demonstrate the ability of the model to predict the transition behavior induced by first- or second-mode instabilities. The model provides a reasonable reflection of the effects of different parameters that influence transitions, including the Mach number, temperature, nose bluntness, and pressure gradient.

Improved Amplification Factor Transport Transition Model for Transonic Boundary Layers

Linear-stability-theory-based method plays an important role in boundary-layer transition prediction of aeronautical flows. Based on the simplification of linear stability theory (Drela and Giles, AIAA J, 1987), the amplification factor transport (AFT) equation for Tollmien–Schlichting waves using local variables was first proposed in 2013 (Coder and Maughmer, AIAA J, 2014). However, in transonic high-Reynolds-number flows, the AFT model seriously overestimates the value of . To fix this problem, we have conducted many linear stability analyses to rebuild the formulations of AFT model. Compressibility effect, which has a significant impact on the prediction of value, has been considered in the improved version. Coupled with the model for crossflow instabilities (Xu et al., CJA, 2020), the improved AFT model is established for transonic boundary layers. Several classic test cases are successfully employed to validate present AFT model, including a flat plate, an airfoil, an infinite swept wing, a sickled-shape wing, a 6:1 inclined prolate spheroid, the NASA Common Research Model, and ARA Transonic Swept Wing. Our prediction results show that both Tollmien–Schlichting instabilities and stationary crossflow instabilities can be captured and predicted well by the improved model in subsonic and transonic boundary layers.

Numerical investigation of a pitching airfoil undergoing dynamic stall using Delayed Detached Eddy Simulations

This paper examines the detailed flow physics and predictive capabilities of Delayed Detached Eddy Simulations (DDES) related to the dynamic stall onset process during a pitching motion at a transitional Reynolds number of 200,000 and Mach number of 0.10. A comparison of the detailed flow physics of the onset of dynamic stall process between DDES and Large Eddy Simulations (LES) is first presented in this paper. State-of-the-art transition models will be tested in terms of their capabilities in capturing complex flow physics. The overall flow physics is captured in detail with a mesh composed of roughly 16 million grid points. A leading-edge separation region and formation of a dynamic stall vortex is examined in detail. The onset of lift stall was predicted slightly earlier using DDES with all turbulence transition models studied, and the peak lift coefficient was also slightly underpredicted compared to LES. However, DDES captures the important phenomena that are presented in LES including the formation and collapse of a Laminar Separation Bubble (LSB) and the subsequent formation of the Dynamic Stall Vortex (DSV). Coder’s SA Amplification Factor Transport (AFT) transition model is found to yield the most accurate results when compared with LES results and shows promising results for future use in more high-fidelity case studies. Langtry and Menter’s SST correlation-based transition model, its SA counterpart developed by Medida and Baeder, and a fully turbulent SA turbulent model are also investigated.

Calibration of the Critical Amplification Factor for Reynolds-Averaged Navier–Stokes-Based Transition Modeling

The present study uses an adjoint-based gradient optimization framework to perform calibration of the critical amplification factor for transition models based on the linear stability theory. More specifically, the two-equation amplification factor transport model is used, and the critical amplification factor, which directly controls the onset of the transition via the source term of the intermittency equation, is calibrated for a set of canonical flat plate test cases in both bypass and natural transitional regimes. It is shown that, by utilizing a sigmoid fitting of the turbulence index profile, the transition onset location can be accurately predicted in a differentiable and smooth fashion, which is essential to the adjoint-based sensitivity analysis of the Reynolds-averaged Navier–Stokes solver. Subsequently, the results of these calibration studies are used for obtaining a new relation via a high-order polynomial regression model relating the critical amplification factor to the freestream turbulence intensity. Finally, the prediction capability of the calibrated relation is tested for natural transitional flows past NLF(1)-0416 and S809 airfoils. The numerical results show significant improvements in predicting the transition onset location as well as lift and drag predictions.

Computational Fluid Dynamics Transition Models Validation for Rotors in Unsteady Flow Conditions

Computational fluid dynamics transition models are evaluated for rotors in unsteady axial and in forward flight conditions. The study is carried out using CREATETM-AV Helios. Three transition models are considered: amplification factor transport (AFT), Langtry–Menter (LM), and the LM with a crossflow transition model. The LM model is modified to allow for Galilean invariance. The first rotor configuration is the German Aerospace Center (DLR) rotor tested in the Rotor Test Facility in Göttingen (RTG). It is a four-blade, 2.13-foot-radius rotor in axial flow with pitching blades, operating at Reynolds numbers of and at three quarter radius, for the two test cases studied. The second configuration is the pressure-sensitive-paint rotor. It is a model-scale, 5.58-foot-radius, three-blade rotor operating at a high-advance-ratio, high-thrust forward flight condition and a hover Reynolds number of at three quarter radius. For both rotors, good agreement is found between the computed and the measured transition locations, obtained using the differential infrared thermography technique. The AFT and LM models produced remarkably consistent predictions. The crossflow model predicted an early transition onset compared with the test data.

Linear Stability-Based Smooth Reynolds-Averaged Navier–Stokes Transition Model for Aerodynamic Flows

The inclusion of transition to turbulence effects in computational fluid dynamics simulations makes it possible to design laminar flow airframes. It also increases the level of physical representations for simulations of standard airframes because laminar flow regions may be present in parts of the aircraft in multiple flight conditions. Modified Reynolds-averaged Navier–Stokes (RANS) models that consider transition effects became popular in the last decade and indicated favorable agreement with experimental data. However, these models present more difficult convergence behavior when compared with fully turbulent RANS approaches. To address this issue, an approximate Newton–Krylov solver is leveraged to solve the transitional flow over aeronautical configurations using a flow-stability-based, smooth RANS transition model. The amplification factor transport (AFT) model is modified to create a smooth variant, referred to as AFT-S. Strategies have been developed to obtain representative physical solutions that exhibit good agreement with experimental data. The convergence behavior for the transition RANS model is also addressed, and its impact on the numerical results is assessed. These results constitute the first in-depth investigation on the numerical behavior of an AFT-type transition model.

Effect of Crossflow Transition on the Pressure-Sensitive-Paint Rotor in Hover

The amplification factor transport transition model has been extended to include crossflow transition and applied to a rotor in hover. Validation of the crossflow model implementation has been performed using standard test cases. The pressure-sensitive-paint rotor is simulated in hover for various collective pitch angles as a test case for the transition model. The OVERFLOW 2.2n solver is used with laminar–turbulent transition modeling and hybrid Reynolds–averaged Navier–Stokes/large-eddy simulation turbulence modeling enabled. Grid generation, numerical methods, and transition modeling are described. The effects of crossflow transition are qualitatively and quantitatively assessed through comparisons to experiments, to a transition model without crossflow, and to a fully turbulent model. Additionally, several techniques for evaluating unsteady boundary-layer transition are described and are applied to solutions for the pressure-sensitive-paint rotor in hover.

Assessing Numerical Aspects of Transitional Flow Simulations Using the RANS Equations

ABSTRACT This paper addresses the influence of the scheme used in the convective terms of turbulence and transition transport equations on the numerical accuracy of transitional flow simulations. Three transition models are combined with the Shear Stress Transport turbulence model: the model, the γ model and the Amplification Factor Transport model. Two airfoils and a 6:1 prolate spheroid are used as test cases. Using first-order upwind in the convective terms of the k and ω transport equations has a weak effect on the numerical accuracy of the simulations when there is no transition model. However, when combined with the transition models, there is a significant improvement if second-order schemes are used. The iterative convergence obtained for the first-order approach was similar to that observed with the second-order schemes. Therefore, common ‘rules of thumb’ from RANS simulations at high Reynolds numbers may fail when transition models are included.

Adjoint-based aerodynamic shape optimization including transition to turbulence effects

The inclusion of transition to turbulence effects in aerodynamic shape optimization makes it possible to use it as a tool for the design of airframes with laminar flow. Modified Reynolds-Averaged Navier–Stokes (RANS) models that consider transition to turbulence have gained traction in the computational fluid dynamics (CFD) community. These models enable the computation of transitional flows without the need for external modules. In this work, we use a smooth version of the amplification factor transport (AFT) model, called AFT-S, to perform gradient-based aerodynamic shape optimization (ASO) of airfoils in subsonic and transonic flow conditions. We investigate the benefits of including transition effects into the optimization process and assess the impact of losing laminar flow when early transition to turbulence occurs due to surface contamination. Our results indicate that our design optimization approach yields lower drag airfoils when transition effects are considered. For the transonic case, the optimizer trades between shock wave strength and laminar flow extension to minimize drag.

Linear amplification factor transport equation for stationary crossflow instabilities in supersonic boundary layers

Based on the database from linear stability theory (LST) analysis, a local amplification factor transport equation for stationary crossflow (CF) waves in low-speed boundary layers was developed in 2019. In this paper, the authors try to extend this transport equation to compressible boundary layers based on local flow variables. The similarity equations for compressible boundary layers are introduced to build the function relations between non-local variables and local flow parameters. Then, compressibility corrections are taken into account to modify the source term of the transport equation. Through verifications of different sweep angles, Reynolds numbers, angles of attack, Mach numbers, and different cross-section geometric shapes, the rationality and correctness of the new transport equation established in this paper are illustrated.

Improved local amplification factor transport equation for stationary crossflow instability in subsonic and transonic flows

Transition prediction is a hot research topic of fluid mechanics. For subsonic and transonic aerodynamic flows, eN method based on Linear Stability Theory (LST) is usually adopted reliably to predict transition. In 2013, Coder and Maughmer established a transport equation for Tollmien-Schlichting (T-S) instability so that the eN method can be applied to general Reynolds-Average-Navier-Stokes (RANS) solvers conveniently. However, this equation focuses on T-S instability, and is invalid for crossflow instability induced transition which plays a crucial role in flow instability of three-dimensional boundary layers. Subsequently, a transport equation for crossflow instability was developed in 2016, which is restricted to wing-like geometries. Then, in 2019, this model was extended to arbitrarily shaped geometries based on local variables. However, there are too many tedious functions and parameters in this version, and it can only be used for incompressible flows. Hence, in this paper, after a large amount of LST analyses and parameter optimization, an improved version for subsonic and transonic boundary layers is built. The present improved model is more robust and more concise, and it can be applied widely in aeronautical flows, which has great engineering application value and significance. An extensive validation study for this improved transition model will be performed.

A CFD-Compatible Amplification Factor Transport Equation for Oblique Tollmien-Schlichting Waves in Supersonic Boundary Layers

Boundary layer transition is a hot research topic in fluid mechanics and aerospace engineering. In low-speed flows, two-dimensional Tollmien-Schlichting (T-S) waves always dominate the flow instability, which has been modeled by Coder and Maughmer from 2013. However, in supersonic flows, three-dimensional oblique Tollmien-Schlichting waves become dominant in flow instability. Inspired by Coder and Maughmer’s NTS amplification factor transport equation for two-dimensional Tollmien-Schlichting waves in low-speed flows and Kroo and Sturdza’s linear stability theory (LST) analysis results for oblique Tollmien-Schlichting waves in supersonic flows, a new amplification factor transport equation for oblique Tollmien-Schlichting waves has been developed based on LST. The compressible Falkner-Skan similarity equations are introduced to build the relationships between nonlocal variables and local variables so that all the variables used in the present model can be calculated using local variables. Applications of this new transport equation to the flows over supersonic flat plate, 3% thick biconvex airfoil, and one modified supersonic laminar airfoil show promising results compared with the standard LST analysis results.

On the Numerical Behavior of RANS-Based Transition Models

Abstract A comparison of several Reynolds-averaged Navier–Stokes (RANS) based transition models is presented. Four of the most widespread models are selected: the γ−Reθ, γ, amplification factor transport (AFT), and kT−kL−ω models, representative of different modeling approaches. The calculations are performed on several geometries: a flat plate, the Eppler 387 and NACA 0012 two-dimensional (2D) airfoils at two angles of attack, and the SD7003 wing. Distinct features such as the influence of the inlet boundary conditions, discretization error, and modeling error are discussed. It is found that all models present a strong sensitivity to the turbulence quantities inlet boundary conditions, and with the exception of the AFT model, are severely influenced by the decay of turbulence predicted by the underlying turbulence model. This makes the estimation of modeling errors troublesome because these quantities are rarely reported in experiments. Despite not having specific terms in their formulation to deal with separation-induced transition, both the AFT and kT−kL−ω models manage to predict it for the Eppler 387 foil, although presenting higher numerical uncertainty than the remaining models. However, both models show difficulties in the simulation of flows at Reynolds numbers under 105. The γ−Reθ and γ models are the most robust alternatives in terms of iterative and discretization error. The use of RANS compatible transition models allows for laminar flow and features such as laminar separation bubbles to be reproduced and can lead to greatly improved numerical solutions when compared to simulations performed with standard turbulence models.

Boundary Layer Transition Models for Naval Applications: Capabilities and Limitations

We describe the implementation of several recently developed boundary layer transition models into the overset computational fluid dynamics code, REX, developed at the University of Iowa, together with an evaluation of its capabilities and limitations for naval hydrodynamics applications. Models based on correlations and on amplification factor transport were implemented in one- and two-equation Reynolds-averaged Navier-Stokes turbulence models, including modifications to operate in crossflow. Extensive validation of the transition models implemented in REX is performed for several 2- and 3-dimensional geometries of naval relevance. Standard tests with extensive available experimental data include flat plates in zero pressure gradient, an airfoil, and sickle wing. More complex test cases include the propeller, P4119, with some experimental data available, and the generic submersible, Joubert BB2, with no relevant experimental data available, to validate the transition models. Simulations for these last two cases show that extensive regions of laminar flow can be present on the bodies at laboratory scale and field scale for small vessels, and the potential effects on resistance and propulsion can be significant. 1. Introduction Progress for prediction of attached, fully turbulent flows for practical aerodynamic and hydrodynamic applications has reached a relatively mature plateau. However, according to a recent comprehensive review of pacing items (Slotnick et al. 2014), the single largest hurdle for incorporating computational fluid dynamics (CFD) into the design process in the near future is the ability to accurately predict turbulent flows with boundary layer transition and separation. Transition can impact skin friction, heat transfer, noise, propulsion efficiency, and maneuverability. This is especially true at model scale and for small craft such as unmanned or autonomous surface and underwater vehicles.

Fully Local Amplification Factor Transport Equation for Stationary Crossflow Instabilities

The traditional linear stability analysis for transition prediction includes the following steps: solving velocity profiles using boundary-layer equations or a computational fluid dynamics solver, guessing and searching operations for the eigenvalues, computing the linear perturbation equations, and integrating the amplification factor . It is difficult to implement into modern computational fluid dynamics solvers with parallel computations because it requires many nonlocal variables and coordinate system transformations. In this Paper, an amplification factor transport equation, which can be coupled with a modern computational fluid dynamics parallel solver, is established based on the analysis results through linear stability theory for stationary crossflow instabilities. According to the analysis results of Falkner–Skan–Cooke similarity velocity profiles using linear stability theory analysis, the source term of the present transport equation is formulated locally to describe the growth of stationary crossflow waves in the laminar boundary layer. This new equation is added to the equation proposed by Coder and Maughmer for Tollmien–Schlichting instabilities. Finally, coupling of these two amplification factor transport equations and Menter’s shear stress transport turbulence model completes an effective transition turbulence model. Comparisons between the present predictions, wind tunnel experiments and the standard linear stability theory analysis on an NLF(2)-0415 infinite swept wing, Petzold’s sickle-shaped wing and 6:1 inclined prolate spheroid validate the reasonable establishment of the present amplification factor transport equation for stationary crossflow instabilities.

Transitional Delayed Detached-Eddy Simulation of Multielement High-Lift Airfoils

A strategy for predicting high-lift aerodynamic flows is presented that employs laminar–turbulent transition modeling, based on amplification factor transport, coupled with a hybrid Reynolds-averaged Navier–Stokes (RANS) and large-eddy simulation (LES) methodology. This modeling has been implemented in an overset, structured, finite difference computational fluid dynamics solver, and the predictive capabilities are demonstrated for the widely studied three-element MD high-lift airfoil. Lift forces, surface pressure distributions, and velocity profiles are compared for fully turbulent and transitional hybrid RANS/LES simulations. The inclusion of transition prediction has a net favorable effect when compared to an experimental reference; however, some discrepancies between measurement and prediction are not fully reconciled.

Transition study of 3D aerodynamic configures using improved transport equations modeling

As boundary layer transition plays an important role in aerodynamic drag prediction, the proposal and study of transition prediction methods simulating the complex flow phenomena are prerequisite for aerodynamic design. In this paper, with the application of the linear stability theory based on amplification factor transport transition equations on the two-equation shear stress transport (SST) eddy-viscosity model, a new method, the SST-NTS-NCF model, is yielded. The new amplification factor transport equation for the crossflow instability induced transition is proposed to add to the NTS equation proposed by Coder, which simulates Tollmien–Schlichting wave transition. The turbulent kinetic energy equation is modified by introducing a new source term that simulates the transition process without the intermittency factor equation. Finally, coupled with these two amplification factor transport equations and SST turbulence model, a four-equation transition turbulence model is built. Comparisons between predictions using the new model and wind-tunnel experiments of NACA64(2)A015, NLF(2)-0415 and ONERA-D infinite swept wing and ONERA-M6 swept wing validate the predictive quality of the new SST-NTS-NCF model.