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Reassessment of the scale-determining equation for advanced turbulence models
Abstract
A comprehensive and critical review of closure approximations for two-equation turbulence models has been made. Particular attention has focused on the scale-determining equation in an attempt to find the optimum choice of dependent variable and closure approximations. Using a combination of singular perturbation methods and numerical computations, this paper demonstrates that: (1) conventional κ-ε and κ-ω+2$/ formulations generally are inaccurate for boundary layers in adverse pressure gradient; (2) using 'wall functions' tends to mask the shortcomings of such models; and (3) a more suitable choice of dependent variables exists that is much more accurate for adverse pressure gradient. Based on the analysis, a two-equation turbulence model is postulated that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows.
... The complexity of this perturbed eddy-viscosity approach strongly depends on the complexity of the considered turbulence model. A single supplementary transport equation arises in the linearized operator when considering the one-equation Spalart-Allmaras model, but additional equations would arise if other models were considered such as the k-ω model (Wilcox 1988). Note that algebraic linear eddy-viscosity models have been considered by Viola et al. (2014) to investigate the coherent fluctuations developing in a wind turbine wake. ...
... Note that the methodology is not restricted to one-equation turbulent models. Considering two-equation turbulence models such as the k-ω model (Wilcox 1988) is feasible but more tedious and out of the scope of the present paper. Now, any one-equation turbulence model is based on the transport of a turbulent quantity, denoted hereν, that is a scalar field related to the turbulent eddy viscosity ν t by an analytical relation of the form ...
Stall cells are transverse cellular patterns that often appear on the suction side of airfoils near stalling conditions. Wind-tunnel experiments on a NACA4412 airfoil at Reynolds number Re = 350 000 show that they appear for angles of attack larger than α = 11.5 (±0.5). Their onset is further investigated based on global stability analyses of turbulent mean flows computed with the Reynolds-averaged Navier-Stokes (RANS) equations. Using the classical Spalart-Allmaras turbulence model and following Plante et al. (J. Fluid Mech., vol. 908, 2021, A16), we first show that a three-dimensional stationary mode becomes unstable for a critical angle of attack α = 15.5 which is much larger than in the experiments. A data-consistent RANS model is then proposed to reinvestigate the onset of these stall cells. Through an adjoint-based data-assimilation approach, several corrections in the turbulence model equation are identified to minimize the differences between assimilated and reference mean-velocity fields, the latter reference field being extracted from direct numerical simulations. Linear stability analysis around the assimilated mean flow obtained with the best correction is performed first using a perturbed eddy-viscosity approach which requires the linearization of both RANS and turbulence model equations. The three-dimensional stationary mode becomes unstable for angle α = 11 which is in significantly better agreement with the experimental results. The interest of this perturbed eddy-viscosity approach is demonstrated by comparing with results of two frozen eddy-viscosity approaches that neglect the perturbation of the eddy viscosity. Both approaches predict the primary destabilization of a higher-wavenumber mode which is not experimentally observed. Uncertainties in the stability results are quantified through a sensitivity analysis of the stall cell mode's eigenvalue with respect to residual mean-flow velocity errors. The impact of the correction field on the results of stability analysis is finally assessed.
... The flow is computed using the incompressible Navier-Stokes equations through a cell-centered finite volume method (FVM) with the CFD software STAR-CCM+ 2020.2 (Siemens PLM Software, Plano, TX, USA). The modeling of the turbulent flow is based on the Reynolds-Averaged Navier-Stokes (RANS) equations in combination with an SST (Menter's Shear Stress Transport) k − ω turbulence model [26,27]. As previously described, two different time discretization setups are used. ...
River surfing has evolved from natural rivers to artificial standing waves, like the Fuchslochwelle in Nuremberg, where optimizing wave quality and safety remains a challenge. Key issues include recirculation zones that pose risks, particularly at higher inflows. This study addresses safety and performance improvements by introducing geometric modifications to reduce recirculation zones. Using STAR-CCM+ simulations, 16 configurations of baffles and inlays were analyzed. A 3D-CAD model of the Fuchslochwelle was developed to test symmetrical and asymmetrical configurations, focusing on reducing vorticity. Results showed that baffles placed 2 m from the inlay reduced recirculation zones by over 50%. Asymmetrical setups, combining wall and inlay baffles, also proved effective. Following simulations, a baffle was installed at 3 m, enhancing safety and quality. Previously, inflows above 7.5 m3/s caused dangerous backflow, requiring surfers to swim or dive to escape turbulence. With the baffle, safe operation increased to 9 m3/s, a 20% improvement, making the system suitable for surfers of all skill levels. These finding provide a novel approach to enhancing flow dynamics, applicable to a wide range of artificial standing waves. The valuable insights gained enable operators to optimize the dynamics and accessibility through geometric modifications while ensuring safety for users.
... • Turbulence models RANS and LES turbulence models are supported. RANS models available in the current release include: the standard Wilcox − model (Wilcox, 1988) and the transitional − LKE model (Medina et al., 2018). For LES simulations the classic Smagorinsky model (Smagorinsky, 1963) is available. ...
... The SST k-ω turbulence model [22] was applied to calculate the Reynolds stress. This model is a RANS eddy viscosity model that combines the advantages of the standard k-ω [23] and k-ε [24] turbulence model formulations to provide more accurate turbulent flow computations. That is, the SST k-ω model can better handle the near-wall flow as well as the far-field free stream that fits the flow conditions in this study, and is widely used for simulating turbulent flows over airfoils including rotating blades [13][14][15]25]. ...
A numerical study was conducted on winglet designs with multiple sweep angles for improving the performance of horizontal axis wind turbine (HAWT) blades, and their effect on reducing the wing tip vortex was investigated by CFD analysis. The effects of sweep angles were examined through NREL Phase VI turbine blades considering a wind speed range of 7 to 25 m/s. Numerical simulations were performed using RANS equations and the SST k–ω turbulence model. The interaction of the blade rotation and wind flow was modeled using a moving reference frame method. The numerical results were found to be in good agreement with the inferences drawn from the experiments for a baseline blade without a winglet, thereby validating the computational method. The investigations revealed that multi-swept winglets predicted a 14.6% torque increment, providing higher power output than single-swept winglets compared to the baseline blade at a wind speed of 15 m/s. Implementing multiple sweep angles in winglet design can improve the blade performance effectively without further increments in winglet length.
... He chose the turbulence kinetic energy k as one of the turbulence parameters and the turbulent eddy frequency of k, x, as the second parameter. 2 In 1988, Wilcox 100 proposed the standard k-x model, which includes the following two transport equations for k and x: ...
The flow patterns in the turbomachinery are highly turbulent, involving various types of unstable flow phenomena. In this paper, Reynolds-averaged Navier–Stokes simulation (RANS) as well as partially averaged Navier–Stokes (PANS), which is a hybrid turbulence model providing a balance solution between computational cost and simulation accuracy, are reexamined with a review toward developing a methodology for the simulation of complex turbulent flows. First, fundamental mathematical formulations and derivation on the RANS and PANS models are introduced systematically. Especially, the rationales of some newly proposed PANS models for simulating unstable flow involving with large-reverse flow, high-shear stress, and high-pressure gradient flows are introduced. Furthermore, recent progress for the unsteady flow using the PANS models in the turbomachinery is mainly summarized with an emphasis on their principles and applicability. Some representative engineering applications of PANS model in hydrofoil, pumps, turbines, and propellers are discussed. It is demonstrated that the PANS models show superior performance in predicting the cavitating flow, flow separation, tip leakage flow, and wake flow. Last, summary comments on the review are drawn. This paper attempts to provide insights and guidelines for the research utilizing the PANS model in turbomachinery. To further promote the study of PANS models in turbomachinery, some further validations on the newly proposed PANS model employed in the compressible turbulent flow in compressors or other turbomachinery are addressed for future research.
... It is used to capture turbulence behaviour near the walls, where velocity gradients are largest [49][50][51]. The SST k-ω is a combination of the techniques for capturing vorticity and far-wall mixing processes offered by the model k-ϵ, as well as capturing the logarithmic viscous layer and turbulence dissipation frequency near pipe walls, which the standard k-ω model provides [52]. This model fits well with adverse pressure gradients, can predict fluid separation, and is easily adaptable to flow regime changes [53]. ...
Pipeline filling and emptying are critical hydraulic procedures involving transient two-phase air–water interactions, which can cause pressure surges and structural risks. Traditional Digital Twin models rely on one-dimensional (1D) approaches, which cannot capture air–water interactions. This study integrates Computational Fluid Dynamics (CFD) models into a Digital Twin framework for improved predictive analysis. A CFD-based Digital Twin is developed and validated using real-time pressure measurements, incorporating 2D and 3D CFD models, mesh sensitivity analysis, and calibration procedures. Key contributions include a CFD-driven Digital Twin for real-time monitoring and machine learning (ML) techniques to optimise pressure surges. ML models trained with experimental and CFD data reduce reliance on computationally expensive CFD simulations. Among the 31 algorithms tested, decision trees, efficient linear models, and ensemble classifiers achieved 100% accuracy for filling processes, while k-Nearest Neighbours (KNN) provided 97.2% accuracy for emptying processes. These models effectively predict hazardous pressure peaks and vacuum conditions, confirming their reliability in optimising pipeline operations while significantly reducing computational time.
... However, it also encounters challenges in achieving quantitative accuracy. Additionally, various low-Reynolds number models such as AKN (Abe, Konhoh, and Nagano [196]), LS, CH (Chien [197]), LB (Lam and Bremhorst [198]), YS (Yang and Shih [199]), SA(Spalart and Allmaras [200]) and MK(Wilcox [201]) models have been employed in RANS simulations of turbulent heat transfer in SCFs. These models incorporate different damping functions to account for nearwall effects, partially addressing the sharp variations of thermophysical properties within boundary layer. ...
Supercritical fluids (SCFs) hold potential in the fields of energy and advanced propulsion, highlighting the significance of comprehensively investigating SCF flow and heat transfer characteristics. The intricate and nonlinear thermophysical property variations of SCFs coupled with the primitive variables in the conservation equations pose several challenges in effectively modeling and simulating SCF flows and heat transfer. This paper conducts a thorough assessment of commonly used equations of state and look-up tables for describing the thermophysical properties of SCFs. The data-driven methods based on machine learning for SCFs are also discussed. The challenges associated with direct numerical simulation, Reynolds-averaged simulation, and large-eddy simulation of SCFs are examined. Emphasis is placed on the evaluation and discussion of the issue of turbulence modeling strategies that stem from low-pressure or ideal-gas conditions directly applied to SCF flow and heat transfer. The primary objective is to provide guidance for future research, thereby advancing and promoting the modeling and simulations of SCF flows and heat transfer.
... • Standard ( − ) model [78]. ...
About the Research
This project explores hydrofoil technology, highlighting its impact on maritime transport by enhancing speed and efficiency. From pioneering work to modern applications in racing yachts and fast ferries, hydrofoils have shown transformative potential. However, ventilation inception poses risks to performance and safety.
This project addresses the unpredictable nature of ventilation onset in surface-piercing hydrofoils during initial design stages using RANS-VOF CFD simulations. The limitations of current CFD methods in predicting ventilation conditions necessitate further investigation to improve hydrofoil design reliability. The research objectives focus on evaluating RANS-VOF’s capabilities and limitations in predicting ventilation, identifying critical conditions, and developing recommendations to enhance simulation reliability. The project structure addresses the research question through a theoretical framework, computational simulations, and data analysis, aiming to advance design methodologies and ensure safer, more efficient maritime transport solutions.
Identifying the Critical Conditions of Ventilation inception
The theoretical framework analyzes the critical factors influencing ventilation inception on hydrofoils, emphasizing the importance of understanding and predicting this phenomenon to enhance hydrofoil efficiency and safety.
Ventilation occurs due to air cavities forming around the hydrofoil’s lifting surfaces when local pressure drops below atmospheric levels. This can be triggered by high speeds, high angles of attack, and proximity to the free surface, which create low-pressure zones conducive to air ingress, influencing pressure distribution and flow characteristics. Various experiments have identified critical conditions for ventilation onset, such as specific ranges of speed, angle of attack, and Froude numbers. For instance, experiments by by Wetzel [77] and Barden and Binns [5] have shown that higher speeds and angles of attack increase the likelihood of ventilation.
The framework also provided guidelines for recognizing different flow regimes around hydrofoils:
• Fully Wetted Flow: Complete submersion with stable lift and drag forces.
• Partially Ventilated Flow: Mixed surfaces with variable cavity lengths and flow separation.
• Fully Ventilated Flow: Stable cavity enveloping the suction surface with visible spray and tip vortices.
The review of experimental studies synthesizes findings on parameters influencing ventilation. The selection of the NACA 0010-34 profile dataset for numerical simulations was based on a multi-criteria decision analysis (MCDA). A detailed list of requirements for evaluating the fidelity of numerical simulations in predicting ventilation dynamics was established. These requirements guided the research
phases, ensuring accurate replication of experimental conditions and reliable predictions. It was concluded that all requirements for the evaluation of the numerical simulation’s fidelity are met.
Designing and Conducting Numerical Experiments
This methodology outlines an approach to designing and conducting numerical experiments on hydrofoil ventilation inception. The Design of Experiments (DOE) method was chosen for its effictive and systematic approach, enabling a thorough understanding with minimal tests, crucial for achieving research objectives and recommended for future studies.
The numerical experiment set-up was carfully planned, tailoring the DOE method to this research. The goal was to use Computational Fluid Dynamics (CFD) to replicate the hydrofoil’s geometry and compare numerical results with experimental towing test data, aiming to identify discrepancies and improve CFD accuracy. Independent variables, factor levels, and response metrics were systematically organized for a structured experimental design. Case selection was based on thorough experimental data analysis, ensuring comprehensive variable space coverage. An iterative approach ensured testing of the most critical conditions.
Reproducing experiments with CFD involved detailed pre-processing, processing, and post-processing steps. Using ReFRESCO as the CFD solver ensured accurate hydrodynamic performance simulation. Comparative analysis of different numerical methodologies highlighted the impact of various set-up choices. Post-processing with ParaView and JupyterLab Python scripts facilitated visualization and analysis of complex data, allowing extraction of key hydrodynamic performance metrics, providing a clear understanding of the foil’s hydrodynamic behavior.
In summary, this methodology provides a robust framework for conducting numerical experiments on hydrofoil ventilation inception. The systematic approach ensures research objectives are met, contributing to more accurate and reliable CFD simulations for maritime applications.
Comparing CFD Set-Ups
The results analyze different methods for predicting ventilation inception on hydrofoils, examining free surface and wave elevation, hydrodynamic forces, flow regime characteristics, and comparing CFD setups. Significant differences were found between Method 1 and Method 2, especially in mesh resolution and the ability to simulate vortices. Method 1, with its finer mesh and higher-order discretization, provided more accurate results. Numerical artifacts, like water patches, indicated the need for contact line correction.
Hydrodynamic forces analysis using the Transient Scanning Technique (TST) showed Method 1 produced more consistent results over longer computation times, while Method 2 had irregularities. Decomposing lift and drag forces into coefficients (CL and CD) allowed for a generalized comparison. High-speed flows were less sensitive to simulation settings, while low-speed flows were more affected by mesh
resolution, turbulence modeling, and numerical schemes. The characteristic check of flow regimes confirmedthatbothmethodsmetthecriteria for fully wetted flow regimes. However, for fully ventilated flow regimes, Method 1 succeeded in simulating ventilation, while Method 2 did not, highlighting the limitations of RANS-VOF in predicting complex flow phenomena associated with ventilation inception.
The Outcomes
While providing valuable insights the industry-standard RANS-VOF method falls short in accurately predicting the complex flow phenomena of ventilation inception. Undermining the reliability of the predictions necessary for effective modeling thechallenge lies not in the turbulencemodelingcapabilities of RANSbutinthe VOFmethod’s inability to accurately represent the water-air interface. RANS, which
averages the Navier-Stokes equations, captures the overall effect of turbulence without resolving all small-scale turbulent structures, making it suitable for general turbulence modeling. However, the VOF method lacks the precision needed to represent abrupt transitions at the fluid interface.
The comparative analysis highlights the strengths and limitations of each method. Method 1 excels in accuracy and precision, suitable for detailed hydrodynamic analyses but is computationally intensive. Method 2 is cost-effective and efficient for early design stages, providing quick estimates with reduced computational demands, though less accurate especially for low-speed flows.
A hybrid approach (Concept 3) is recommended, combining Method 2 for initial assessments and Method 1 for detailed analysis. This approach can enhance results by addressing boundary condition artifacts, wave damping, and adaptive grid refinements. The choice between Concepts 1, 2, and 3 depends on the specific needs of the hydrodynamic analysis:
• Concept 1: Method 1for high precision and reliability, despite higher computational demands and longer computation times.
• Concept 2: Method 2 for quick, efficient preliminary assessments, balancing accuracy and computational efficiency.
• Concept 3: Ahybrid approach, using Method 2 for initial assessments and Method 1 for detailed analysis, optimizing both accuracy and efficiency, providing a balanced solution for hydrodynamic performance prediction.
... The k-ω model [8] instead introduces the specific dissipation rate (ω) to govern the scale of turbulent eddies. This model provides improved accuracy in near-wall regions but can be sensitive to free-stream turbulence levels. ...
This paper extends the tuning parameters using Bayesian-optimization-RANS (turbo-RANS) methodology to enhance the predictive accuracy of Reynolds-averaged Navier-Stokes (RANS) turbulence models for flow over a converging-diverging channel, a benchmark case characterized by adverse pressure gradients and flow separation. Using Bayesian optimization, the generalized k-ω (GEKO) turbulence model was calibrated by tuning key coefficients (CSEP and CNW) with sparse reference data from direct numerical simulation (DNS) studies at Re = 12, 600. The optimized model was then tested on the streamwise velocity predictions against both DNS (Re = 12, 600) and large-eddy simulation (LES) data, along with the coefficient of friction (C f) predictions from the LES dataset at Re = 20, 580. Results with optimized coefficient values of CSEP = 0.489 and CNW = 1.778 demonstrate significant improvements in the prediction of wall quantities, addressing key limitations of default RANS coefficients. These findings provide further insight into the application of machine learning-assisted RANS calibration for adverse pressure gradient flows.
... where d ij is the identity tensor, and g t is turbulent/eddy viscosity, modeled using transport equations of turbulent kinetic energy (k), turbulent dissipation rate () and specific dissipation rate x (Wilcox, 1988). In the context of non-Newtonian fluids, these transport equations are further augmented to include non-Newtonian contributions, such as stress and diffusion effects caused by viscosity fluctuations. ...
This review explores recent advancements in modeling the flow behavior of Herschel–Bulkley (HB) fluids in pipes, discussing theoretical, semi-empirical, computational, and experimental methods. While the laminar flow of non-Newtonian HB fluids can be effectively modeled using first-principle physics, significant challenges remain in turbulent and transitional flow regimes. Existing turbulence models, though widely used, may not always fully align with experimental data, often requiring further validation or complex mathematical tuning, leading to higher computational costs. Further, the transition to turbulence in HB fluids is influenced by shear-thinning and yield stress, yet current models often fail to account for this delayed transition. Consequently, stability and Reynolds number-based transition models can exhibit inconsistencies, limiting their broader applicability. Progress is further hindered by limited experimental studies, constrained by resolution, attenuation, cost, and material combinations. Inaccuracies in rheological modeling—due to inappropriate shear rate ranges, curve-fitting techniques, or simplifying assumptions such as homogeneity and non-elasticity—further complicate flow predictions. Through this review, we delve deeper into the state-of-the-art modeling of HB fluids, highlighting progress and these challenges. Addressing these limitations requires advanced experimental and numerical studies, particularly for near-wall measurements, to better capture flow complexities and improve model predictions. This could also facilitate the development of data-driven approaches and operational envelopes that define their validity thresholds. Future research should also prioritize the independent effects of yield stress and shear-thinning properties while considering material attributes and settling phenomena in non-Newtonian suspensions. Ultimately, these advancements will enable more accurate flow predictions and practical solutions for industrial applications.
... The value of the specific rate of dissipation ω at the wall of the rough surface in the region of turbulence assumed by Wilcox [28] is determined by ...
The surface roughness during ablation significantly affects the hypersonic boundary-layer transition and heat transfer. In this study, an equation of transport for the roughness amplification factor [Formula: see text] is introduced based on the improved baseline hypersonic [Formula: see text] transition model by considering the effects of surface roughness. The roughness amplification factor [Formula: see text] is convected into the flow field to represent the disturbance induced by the surface roughness. It defines a region of influence of the surface roughness to locally modify the transition model and track changes in the momentum thickness to locally modify the criteria for the onset of transition. Moreover, the boundary condition of the wall is amended for the specific rate of turbulence dissipation to model the effects of roughness in the fully turbulent zone. The improved model can accurately predict the location of transition in roughness under different hypersonic flows.
... The typical k-ω model, created by Wilcox [29], is extremely sensitive to inlet free stream turbulence. Using a blending function F1, Menter and Esch [30] converted the standard k-ε model into the k-ω form. ...
... In the present work we employ the finite-volume spatial discretisation of the compressible RANS equations with Roe's upwind scheme [18], MUSCL extrapolation [14], and the van Albada limiter [1]. For the simulation results presented below, Wilcox' k-ω turbulence model [22] in combination with a Cauchy-Schwarz limiter is used. ...
... Those blades feature complex geometries: Both turbine blades have large fillets, the compressor stator has half-gaps (small gaps in the rear part at the hub and tip, see Figure 6) and the compressor rotor has a tip clearance (small gap between blade and casing). For each of these four original blades indexed by b, we sample its geometry vector θ b randomly M b times by varying radially among other things the leading edge blade angle, trailing edge blade angle and stagger angle, so as to construct a dataset of configurations were generated using the Navier-Stokes solver TRACE 3 (Becker et al., 2010) with the Wilcox k − ω turbulence model (Wilcox, 1988). For turbine blades, the γ − Re θ transition model (Langtry & Menter, 2009) was enabled. ...
... 6 by 2 ) and log layer (x ! us ffiffiffiffiffiffiffi b à jy p ). 46 No inflow turbulence fluctuations are provided. All the quantities are extrapolated at the outflow boundary using first-order extrapolation conditions. ...
The low-frequency unsteadiness associated with shock wave turbulent boundary layer interactions (STBLI) poses significant numerical challenges for Reynolds–averaged Navier–Stokes models as well as scale-resolving methods such as direct numerical simulations and large-eddy simulations and requires hybrid methods like detached eddy simulations (DES) from an engineering perspective. The recently developed adaptive DES (ADES) method [Yin and Durbin, “An adaptive DES model that allows wall-resolved eddy simulation,” Int. J. Heat Fluid Flow 62, 499 (2016)], which was originally introduced for low-speed flows and extended to transonic flows, is a promising approach. However, its efficacy for high-speed flows needs to be assessed. In the present study, a compressible version of the ADES method has been implemented and set to test for supersonic flows with complex STBLI features. Impinging STBLI at M∞=2.30 is investigated numerically using compressible ADES method. No inflow turbulence fluctuations are provided in the upstream (time-averaged) boundary layer profile so as to verify the ADES method's capability to predict the low-frequency “breathing” motion of the bubble and the shear layer flapping/shedding in the mid-frequency range, thus demonstrating the significance of downstream mechanism in the unsteadiness associated with STBLI. Time series analysis of surface pressure distribution and modal analysis of numerical schlieren using proper orthogonal decomposition are carried out to understand the dominant mechanisms and associated frequencies. In the interaction zone, starting from the intermittent region to the region downstream of reattachment, a transition of the dominant frequency from low-to-mid-to-high ranges is observed. The separation shock motion and the shear layer shedding are observed to have dominant Strouhal numbers [ Stδ0, defined based on the boundary layer thickness (δ0) upstream of interaction] in the ranges of 0.01 and 0.1, respectively, consistent with the literature. From correlation and coherence analyses, it is also observed that the separation bubble is exercising breathing motions at lower frequencies [ Stδ0∼O(0.01)] despite lacking the inflow turbulence, thus supporting the downstream influence arguments. The present study demonstrates that the ADES model is capable of predicting essential features of STBLI, including low- and mid-frequency events and a part of high-frequency phenomena.
... In a GLS model, we solve two prognostic equations, one for k and one for another variable that can be linked to ε. The choice of this second equation is the main difference between the different GLS models (k-ε: Hanjalić and Launder (1972); Rodi (1987), k-kl: Mellor and Yamada (1982), k-ω: Wilcox (1988), k-τ: Zeierman and Wolfshtein (1986); Thangam et al. (1992)). In this paper, we focus on the k-ε model which solves directly the equation for ε. ...
The representation of turbulent fluxes during oceanic convective events is important to capture the evolution of the oceanic mixed layer. To improve the accuracy of turbulent fluxes, we examine the possibility of adding a non‐local component in their expression in addition to the usual downgradient part. To do so, we extend the k k–ε algebraic second‐moment closure by relaxing the assumption on the equilibrium of the temperature variance θ′2‾ . With this additional evolution equation for the temperature variance, we obtain a k k–ε –θ′2‾ model (the “kεt ” model) which includes a non‐local term for the temperature flux. We validate this new model against Large Eddy Simulations (LES) in three test cases: free convection (FC), wind‐driven mixing, and diurnal cycle (DC). For wind‐driven mixing, kεt is equivalent to k k–ε . However, in the presence of a buoyancy flux (FC and DC), we find that the vertical profile of temperature of the LES is better captured by kεt than k k–ε . Particularly, the non‐local term increases the fraction of the mixed layer that is stably stratified. For FC, this fraction is near 50% for both kεt and the LES, whereas the k k–ε value is 20%. We show that this improvement is due to a better representation of the temperature variance in the inner part of the mixed layer. This better representation is mainly caused by the diffusion of temperature variance, which is described by kεt and not by k k–ε .
... boundary layer [18]. The simulations were conducted using the commercial solver ANSYS CFX v2022.2. ...
Innovative technology will be needed in future aero engine designs to solve environmental challenges and fulfill upcoming noise and emissions regulations. Current development trends aim to increase engine efficiency while reducing weight and size. This demands greater overall pressure ratios from the axial core compressors and a more compact, lightweight design. On one hand, weight reduction can be achieved through novel materials with higher yield strength. On the other hand, size reduction inherently implies weight reduction. The high-pressure compressor front stages are often equipped with variable stator vanes for the operability of the compressor. Decreasing the diameter of the core compressor necessitates design modifications to the variable stator vanes, specifically the stator hub region with the penny needs to be redesigned. This research examines the aerodynamic characteristics of a 1.5-stage transonic axial compressor featuring an engine like variable stator vane with a penny gap and penny cavity. In addition to the experiments, a detailed computational fluid dynamics simulation with the non simplified test rig geometry has been carried out. The investigated compressor stage serves as a reference for follow up investigations. Therefore the variable stator vane behaviour for the design speed line and the behaviour under different variable stator vane stagger angles are investigated in detail. The performance evaluation encompasses an in-depth analysis of various critical loss mechanisms inherent to each respective arrangement. Thereby, flow conditions within the penny and hub region of the shrouded stator are investigated. The conducted experimental studies enable a detailed insight into the pressure loss mechanisms of the variable stator vane. In conjunction with the computational fluid dynamics, an in-depth descripition of the flow regime at the hub-sided penny is possible.
... -Standart Modeli Wicox'unmodeli, türbülans kinetik enerjisi için bir denklemi ve spesifik türbülans dağılım oranı veya türbülans frekansı için ikinci bir denklemi çözer [WILCOX, 1980] [WILCOX, 1988]. Model,modelinden daha olumsuz basınç gradyanı koşullarında önemli ölçüde daha iyi performans gösterir, ancak yazarın deneyimi, daha güçlü olumsuz basınç gradyanlarına daha yüksek bir duyarlılığın arzu edileceğidir [Menter, 1992A]. ...
... The transport equations for the turbulent kinetic energy k and the turbulent dissipation ratio ω are defined by Eq. (6) and (7), respectively [1], [10] : ...
... Here, only the sensitivity to 3D LES Leray-α modeling is presented for one filter width (α = 5). The simulations were run using the hydrostatic and non-hydrostatic mode of CROCO in three dimensions, with two types of simulations: i) one uses the classical subgrid-scale LES Smagorinsky model (Smagorinsky, 1963) for horizontal mixing and the two-equations Reynolds Averaged Navier-Stokes (RANS) k-ω model (Wilcox, 1988) for the vertical mixing and, ii) another one uses the LES Leray-α model for both horizontal and vertical mixing. The results were compared as reported in Fig. 2, in which the barotropic current was computed as , with , components of the barotropic velocity; as, in this case, the sign of the current depends on the sign of the meridional velocity, we defined c as negative whenever u is negative, so to show the curve as reported in the Fig. 2. Leray-leads, as expected, to a reenergization of the flow with more intense barotropic velocity, which could suggest a better representation of turbulence with respect to Smagorinsky, and with comparable computational times. ...
... However, the medium-and high-fidelity simulations are often computationally too expensive for practical engineering design and optimization problems, which may need to run CFD hundreds of times. In the foreseeable future, engineering design will probably still rely on RANS simulations with turbulence models, such as the one-equation Spalart-Allmaras (SA) [1] model and two-equations − [2] and − [3] model. However, the RANS turbulence models have many assumptions, often resulting in inaccurate predictions of complex flows, particularly for those involving flow separation and strong adverse pressure gradients. ...
Reynolds-averaged Navier-Stokes (RANS) models are widely used in practical aerospace engineering designs because of their low computational costs. However, RANS models' assumptions and simplifications may not provide accurate predictions for complex flows with separation and strong adverse pressure gradients. This study uses field inversion machine learning (FIML) to improve RANS turbulence models' accuracy in predicting separated flows over airfoils. Most existing FIML studies focused on steady-state or time-averaged unsteady flows, and this paper is a step forward to improve prediction accuracy for time-resolved unsteady flows. We augment the Spalart-Allmaras (SA) turbulence model's production term with a spatial scalar field. We then compute the above scalar field using a solver-imbedded neural network (NN) model, which takes the local flow features as the inputs at each time step. We optimize the weights and biases of the built-in NN model to minimize the regulated temporal prediction error between the augmented flow solver and reference data. We consider the unsteady separated flow over a NACA0012 airfoil at a large angle of attack. We use only the time series of drag coefficient as the training data, and the trained model can accurately predict the spatial-temporal evolutions of other surface variables, such as lift, pitching moment, and pressure distribution around the airfoil, as well as field variables, such as velocity and pressure. The FIML-trained model has the potential to be generalized to accurately predict airfoil aerodynamics for different shapes and flow conditions.
... Although the k-ω model presents higher heat transfer, it exhibits minimal pressure drop variation compared to the k-ε model. This highlights the robustness of the k-ω model for areas of separation and reattachment created by irregular geometries (elliptical and circular tubes with baffles in our case) and its accuracy for near-wall flow [30].In contrast, the Realizable k-ε model shows the lowest heat transfer coefficient, with a difference ranging from 9.2% to 16.7% compared to the k-ω model, and an average increase of 13.8%. Nevertheless, when considering variations in mass flow rate and changes in baffle geometries, the Realizable k-ε model demonstrates the highest heat transfer coefficient as reported by [23,[31][32]. ...
Shell and tube heat exchangers are widely used for their thermal efficiency, and their performance is heavily influenced by the turbulence regime, which is primarily turbulent due to the complex geometry involved. This study evaluates the impact of three turbulence models—k-ε, k-ω, and realizable k-ε—on shell and tube heat exchangers with circular, 90° elliptical, and combined tube configurations, including line-line, random, and elliptical 90°-circular combinations. Using COMSOL Multiphysics software, we simulate thermofluidic flows and compare the results for each turbulence model and tube geometry. The results show that the k-ω model is the most effective for simple geometries, providing reliable and consistent outcomes, while the realizable k-ε model exhibits the lowest pressure drop, making it particularly suitable for configurations with varying mass flow rates and more complex geometries. The choice of different tube bundle combinations can significantly impact the overall thermal and hydraulic performance of the heat exchanger. For combined configurations, the k-ε model provides the best heat transfer performance, particularly in the STHE-E90°_C combination, where the strategic placement of circular tubes in the center and near the shell has a significant effect on optimizing both heat transfer efficiency and pressure losses. This study offers valuable insights for optimizing the design and performance of shell and tube heat exchangers, ultimately contributing to more efficient thermal systems.
... The flow simulation in this study is around an airfoil at low-Reynolds number where the wake region and free shear region need to be taken into account to accurately predict the aerodynamic characteristics. Wilcox [25] developed a model − which allows the treatment of rough walls and surface mass injection. The parameters and stand for turbulent kinetic energy and specific dissipation rate respectively. ...
Low Reynolds number flows are characterized by boundary layer separation due to the effect of viscous forces. The separated flow may reattach through the exchange between the molecules resulting in Laminar Separation Bubble LSB. LSB, known for its detrimental effect on the performance is sensitive to the airfoil geometry. The current paper describes the use of trip boundary on the suction surface of SD7003 airfoil as flow control technique to enhance the airfoil performance at and using numerical analysis. Due to the complexity of finding the trip size and location, Response Surface Methodology RSM is used to obtain the optimum combinations between the height, width and the position of the turbulator. Thus, the design variables are the height [0.2mm,0.6mm], width [140mm, 200mm] and the position of the trip away from the leading edge [10%c, 25%c]. Using the Desirability approach with Nelder-Mead simplex algorithm, the trip location is found to be effective downstream the separation location of the untripped airfoil. At the optimum design parameters, the length of LSB has decreased by 2.32% and the airfoil performance increased by 6%.
... The Wilcox k-ω turbulence model is well-suited for flows with Re values between 81,000 and 240,000 [30,31]. ...
Wastewater accumulates debris as it moves through sewage systems and must undergo purification at treatment plants, where insoluble debris is screened at the inlet. Previous studies have focused on screening mechanisms using rotating or ascendible sub-screens with vertical bars, and the effects of horizontal bars on structural integrity and fluid flow have not yet been explored. The present study addresses this gap by proposing a novel screening mechanism with horizontal bars and providing insights into flow behaviour and structural performance. The proposed mechanism consists of a main screen, an ascendible sub-screen and a rake system, and its effects on the flow distribution inside the channel and the resulting deformations and stress affecting the mechanism are computationally analysed. The problem is modelled as a fluid–structure interaction and solved using the arbitrary Lagrangian–Eulerian approach. Velocity distribution, structural deformation and stress are analysed for the various inlet flow velocities and critical configurations of the screening mechanism. The sub-screen in the proposed mechanism exhibited reduced deformation (0.9 mm for vertical bars and 0.2 mm for horizontal bars versus 2.2 mm in previous vertical-only designs). The maximum von Mises stress values were well below the 250 MPa yield strength, with peak stresses of 3.8 MPa in the sub-screen and 0.23 MPa in the main screen. Key operating conditions causing flow separation and velocity fluctuations are identified, and design improvements are suggested. The study provides guidelines for manufacturing and operating wastewater-screening mechanisms whilst mitigating undesirable performance and minimising deformation and stress in the mechanism.
... Blowing effects are introduced by altering the slope of the mean velocity profile in the inertial region through a blowing parameter v + w = v w /u τ , i.e. wall-normal velocity at the wall (also called blowing velocity). Wilcox's correction 165 to the k-ω model to account for the blowing effects has the same effect as the roughness corrections. It consists of a reduction of the specific dissipation rate wall condition given by ...
Hypersonic flow conditions pose exceptional challenges for Reynolds-Averaged Navier-Stokes (RANS) turbulence modeling. Critical phenomena include compressibility effects, shock/turbulent boundary layer interactions, turbulence-chemistry interaction in thermo-chemical non-equilibrium, and ablation-induced surface roughness and blowing effects. This comprehensive review synthesizes recent developments in adapting turbulence models to hypersonic applications, examining approaches ranging from empirical modifications to physics-based reformulations and novel data-driven methodologies. We provide a systematic evaluation of current RANS-based turbulence modeling capabilities, comparing eddy viscosity and Reynolds stress transport formulations in their ability to predict engineering quantities of interest such as separation characteristics and wall heat transfer. Our analysis encompasses the latest experimental and direct numerical simulation datasets for validation, specifically addressing two- and three-dimensional equilibrium turbulent boundary layers and shock/turbulent boundary layer interactions across both smooth and rough surfaces. Key multi-physics considerations including catalysis and ablation phenomena along with the integration of conjugate heat transfer into a RANS solver for efficient design of a thermal protection system are also discussed. We conclude by identifying the critical gaps in the available validation databases and limitations of the existing turbulence models and suggest potential areas for future research to improve the fidelity of turbulence modeling in the hypersonic regime.
The coaxial compound helicopter has enhanced the maximum flight speed, while retaining the hover and low-speed maneuvering capabilities of conventional helicopters. As a novel configuration, its coaxial rigid rotors and tail-mounted propeller introduce unique aerodynamic and flight dynamic characteristics. This paper reviews both the aerodynamic and flight dynamic characteristics of coaxial compound helicopters. This review begins with an introduction to the unique features of coaxial compound helicopters, such as lift offset and control redundancy. In the aerodynamic section, various numerical simulation methods used in aerodynamic research are summarized, containing Computational Fluid Dynamics trimming methods. The aerodynamic characteristics of coaxial compound helicopters are reviewed in terms of aerodynamic interference, aerodynamic loads, and aerodynamic noise. Aerodynamic interference between the rotors is a major cause of unsteady aerodynamic loads, which in turn lead to aerodynamic noise. Subsequently, this paper introduces the models involved in flight dynamics modeling, detailing the rotor aerodynamic model, inflow model, blade motion model, and aerodynamic interference model. Based on this, the trim, stability, controllability, and flight control design of coaxial compound helicopters are reviewed. The differences in trim results between coaxial compound helicopters and conventional helicopters, as well as the control coupling effects of coaxial compound helicopters, are addressed. Finally, this paper summarizes the aerodynamic and flight dynamic characteristics of coaxial compound helicopters and provides some suggestions for future research.
Hypersonic flow conditions pose exceptional challenges for Reynolds-averaged Navier–Stokes (RANS) turbulence modeling. Critical phenomena include compressibility effects, shock/turbulent boundary layer interactions, turbulence–chemistry interaction in thermo-chemical non-equilibrium, and ablation-induced surface roughness and blowing effects. This comprehensive review synthesizes recent developments in adapting turbulence models to hypersonic applications, examining approaches ranging from empirical modifications to physics-based reformulations and novel data-driven methodologies. We provide a systematic evaluation of current RANS-based turbulence modeling capabilities, comparing eddy viscosity and Reynolds stress transport formulations in their ability to predict engineering quantities of interest such as separation characteristics and wall heat transfer. Our analysis encompasses the latest experimental and direct numerical simulation datasets for validation, specifically addressing two- and three-dimensional equilibrium turbulent boundary layers and shock/turbulent boundary layer interactions across both smooth and rough surfaces. Key multi-physics considerations including catalysis and ablation phenomena along with the integration of conjugate heat transfer into a RANS solver for efficient design of a thermal protection system are also discussed. We conclude by identifying the critical gaps in the available validation databases and limitations of the existing turbulence models and suggest potential areas for future research to improve the fidelity of turbulence modeling in the hypersonic regime.
High specific speed impeller with low aspect ratio is ideal for high-speed pump-jet propulsion. This study investigates its unsteady hydrodynamic characteristics using experimental and numerical methods. At the design flow rate (Qd), the pressure amplitude at the blade’s leading and trailing edges varies significantly along the streamline. Blade shape minimally affects flow velocity but significantly impacts leading-edge pressure fluctuations. For the mainstream, the highest fluctuation peak occurs at 0.9Qd. The pressure amplitude gradually decreases from blade’s shroud to hub. Along circumferential direction the minimum pressure amplitude is located in the middle of the two blades. In gap flow, the leading-edge pressure peaks at Qd, with fluctuations primarily at 5 times and 10 times the rotation frequency. Meanwhile, pressure fluctuations in the tip clearance’s height direction exhibit a consistent distribution, reaching their maximum at the leading edge. Vortex structure analysis using various Q criteria reveals that increasing Q enhances vortexes in the impeller while reducing them in the diffuser. Moreover, flow rates result in a simultaneous decrease in vortexes within both components, while the pressure distribution on the isotropic vortex surface remained stable.
Kolmogorov-Arnold Networks (KAN) is a newly proposed deep learning technique. This novel architecture utilizes the Kolmogorov-Arnold representation theorem, which distinguishes this model with the conventional multi-layer perceptron (MLP) approach. Instead of training the weights and biases with fixed activation functions like the standard neural networks, KAN enables us to learn the activation functions that replace the weights in conventional neural networks. As a result, KAN has no trainable weights in its architecture. The KAN architecture relies on the superposition of several activation functions to model any arbitrary function. Additionally, this architecture can also provide a better interpretability aspect of our model. To examine the capability of this new architecture, this paper aims to implement this architecture for turbulence modelling. Therefore, in this paper, we developed a data-driven turbulence model using KAN. This model takes input features from the mean flow properties and predicts the Reynolds stresses as the main quantity of interest. Our findings indicate that KAN can rival the capabilities of a standard neural network model while employing simpler architecture, despite its more expensive training cost. In this work, we also tested the capability of KAN to perform symbolic regression.
Supersonic wind tunnels are important to build to experimentally investigate compressible flow phenomena. In the current paper, the design process of a supersonic wind tunnel, which is being constructed at Embry-Riddle Aeronautical University, has been discussed. Flow calculations and consequent selection of upstream wind tunnel components have been detailed. This is followed by a detailed design process for each of the wind tunnel components. This paper hopes to serve as a design guideline for building future supersonic wind tunnels.
We live in a turbulent world observed through coarse grained lenses. Coarse graining (CG), however, is not only a limit but also a need imposed by the enormous amount of data produced by modern simulations. Target audiences for our survey are graduate students, basic research scientists, and professionals involved in the design and analysis of complex turbulent flows. The ideal readers of this book are researchers with a basic knowledge of fluid mechanics, turbulence, computing, and statistical methods, who are disposed to enlarging their understanding of the fundamentals of CG and are interested in examining different methods applied to managing a chaotic world observed through coarse-grained lenses.
Numerical results are discussed for the flow in a horizontal plane channel filled with a novel sphere-rod spacer and exchanging heat from both the top and the bottom sides. Direct Numerical Simulations (DNS) are compared with RANS ones based on different turbulence models in the Reynolds number range 100∼2000. Preliminary comparisons for non-buoyant flow show that models using wall functions perform poorly, grossly overpredicting Nusselt numbers, while ω -based models resolving the viscous-conductive sublayer all yield satisfactory results. In the presence of buoyancy, simulations using either DNS or the k - ω model yield a thermal asymmetry between top and bottom wall, confirmed by experiments and related to the stable or unstable thermal stratification occurring in the lower and upper layers of the channel. The asymmetry, large at low Re, becomes negligible for Re≥1000. The Spalart-Allmaras model yields satisfactory results in the absence of buoyancy but grossly overpredicts Nu in buoyant flows.
The capabilities of modeling complex rotor aeromechanics, principally via Computational Fluid Dynamics-Computational Structural Dynamics (CFD-CSD) coupling, have significantly improved with advances in computational hardware and software. Of particular interest is the accurate turbulence modeling for rotating blades, which remains one of the primary uncertainties in rotorcraft aeromechanics predictions. Recent collaborative investigations, where different solvers and turbulence models were applied, yielded dissimilar results for some flight conditions, but comparable results in others. An investigation of the behavior of two popular two-equation models for rotorcraft aeromechanics has been undertaken to understand their similarity and differences in various applications. The two models are based on the Kok and Menter variants of the k- equations. The Kok model is evaluated with and without the Shear Stress Transport (SST) terms. The role of Delayed Detached Eddy Simulation (DDES) is also examined. These studies have been performed for rotating and non-rotating configurations, applying a common solver, mesh, and time step. Aerodynamic influences have been decoupled from aeroelastic applications to isolate causes of differences. The results indicate that differences arise when separated flows are present. The Kok model is unstable without the addition of a limiter on the production of the turbulent viscosity when a dynamic motion with large separation is modeled. Some of the aerodynamic differences in separated flows can be compensated for during fully trimmed aeroelastic predictions. Trim conditions arising from the use of different structural solvers or near dynamic stall conditions may introduce new variations.
In this paper, approaches to implement implicit pseudotime marching for harmonic balance are studied. We first give a motivation for using pseudotime stepping as a solution technique. It turns out that, when discretization errors are neglected, the harmonic balance solution converges to a stable periodic flow, provided that the initial solution is sufficiently close to that solution. This motivates the use of a robust implicit pseudotime marching approach. A central question is which simplifications of the residual Jacobian are appropriate in terms of the overall efficiency and robustness of the solver. As has been shown in the literature, the spectral time-derivative operator should be taken into account in the implicit system. On the other hand, the linearization of the flow residual can be simplified to a certain extent, especially if the system is solved in the frequency domain. In this paper, we show that, up to terms which scale with the amplitude of the disturbances, the linear system matrix is the sum of a scalar diagonal and a block diagonal matrix with identical blocks for each harmonic. We show that when the unsteadiness is small, the nonlinear coupling terms can be neglected in the implicit solver and the resulting special matrix structure can be exploited to massively speed up the solver. In contrast, when a strong disturbance is simulated, this simplification can lead to significant losses in robustness. Our findings are illustrated by means of the forced response analysis of a transonic compressor in off-design conditions.
https://www.sciencedirect.com/science/article/pii/S0360544224039707 Computational Fluid Dynamics is a crucial tool for designing Positive Displacement machines. In this regard, we have developed a Computational Fluid Dynamics in-house solver to predict phase change phenomena inside the chambers of twin-scroll compressors and expanders handling two-phase fluid flow. The main numerical assessments of this work include velocity and pressure distribution, average inlet and outlet mass flow rates, vapor condensation and liquid evaporation phenomena, as well as isentropic, adiabatic, and volumetric efficiencies. These are essential for determining whether the machines experience excessive vibrations or work under overloaded conditions. The numerical results obtained with our solver align well with those obtained with a Deterministic Model found in existing literature. Additionally, the pressure and velocity distribution inside the scroll machines are similar to those reported in numerical studies of single-phase flow scroll compressors and expanders available in the literature.
The paper presents a new model of turbulence in which the local turbulent viscosity is determined from the solution of transport equations for the turbulence kinetic energy and the energy dissipation rate. The major component of this work has been the provision of a suitable form of the model for regions where the turbulence Reynolds number is low. The model has been applied to the prediction of wall boundary-layer flows in which streamwise accelerations are so severe that the boundary layer reverts partially towards laminar. In all cases, the predicted hydrodynamic and heat-transfer development of the boundary layers is in close agreement with the measured behaviour.
The steady turbulent boundary layer structure above a flat plate is theoretically studied along a set of turbulence model equations. These equations are first integrated through the viscous sublayer by means of time-marching numerical integration techniques and the constant in the law of the wall is predicted as a function of wall roughness. The Van Driest compressible law of the wall is then deduced by classical mathematical methods. Finally, using the Van Driest law as a wall boundary condition, the model equations are integrated through the entire boundary layer, again by time-marching methods, for a freestream Mach number of 2. 96.
A set of model equations is given to describe the gross features of a
statistically steady or 'slowly varying' inhomogeneous field of
turbulence and the mean velocity distribution. The equations are based
on the idea that turbulence can be characterized by 'densities' which
obey nonlinear diffusion equations. The diffusion equations contain
terms to describe the convection by the mean flow, the amplification due
to interaction with a mean velocity gradient, the dissipation due to the
interaction of the turbulence with itself, and the diffusion also due to
the self interaction. The equations are similar to a set proposed by
Kolmogorov (1942). It is assumed that both an 'energy density' and a
'vorticity density' satisfy diffusion equations, and that the self
diffusion is described by an eddy viscosity which is a function of the
energy and vorticity densities; the eddy viscosity is also assumed to
describe the diffusion of mean momentum by the turbulent fluctuations.
It is shown that with simple and plausible assumptions about the nature
of the interaction terms, the equations form a closed set. The
appropriate boundary conditions at a solid wall and a turbulent
interface, with and without entrainment, are discussed. It is shown that
the dimensionless constants which appear in the equations can all be
estimated by general arguments. The equations are then found to predict
the von Karman constant in the law of the wall with reasonable accuracy.
An analytical solution is given for Couette flow, and the result of a
numerical study of plane Poiseuille flow is described. The equations are
also applied to free turbulent flows. It is shown that the model
equations completely determine the structure of the similarity
solutions, with the rate of spread, for instance, determined by the
solution of a nonlinear eigenvalue problem. Numerical solutions have
been obtained for the two-dimensional wake and jet. The agreement with
experiment is good. The solutions have a sharp interface between
turbulent and non-turbulent regions and the mean velocity in the
turbulent part varies linearly with distance from the interface. The
equations are applied qualitatively to the accelerating boundary layer
in flow towards a line sink, and the decelerating boundary layer with
zero skin friction. In the latter case, the equations predict that the
mean velocity should vary near the wall like the 5/3 power of the
distance. It is shown that viscosity can be incorporated formally into
the model equations and that a structure can be given to the interface
between turbulent and non-turbulent parts of the flow.
The k-epsilon model and a one-equation model have been used to predict adverse pressure gradient boundary layers. While the one-equation model gives generally good results, the k-epsilon model reveals systematic discrepancies, e.g., excessively high skin friction coefficients, for these relatively simple flows. These shortcomings are examined and it is shown by an analytical analysis for the log-law region that the generation term of the epsilon-equation has to be increased to conform with experimental evidence under adverse pressure gradient conditions. A corresponding modification to the epsilon-equation emphasizing the generation rate due to deceleration was employed in the present investigation and resulted in improved predictions for both moderately and strongly decelerated flows.
CONTENTS: THE STATISTICAL DESCRIPTION OF TURBULENT FLOW; THE EQUATIONS OF MOTION FOR TURBULENT FLOW; HOMOGENEOUS TURBULENT FLOWS; INHOMOGENEOUS SHEAR FLOW; TURBULENT FLOW IN PIPES AND CHANNELS; FREE TURBULENT SHEAR FLOWS; BOUNDARY LAYERS AND WALL JETS; TURBULENT CONVECTION OF HEAT AND PASSIVE CONTAMINANTS; TURBULENT FLOW WITH CURVATURE OF THE MEAN VELOCITY STREAMLINES.
Various aspects of a turbulence-model-transition-prediction method have been analyzed. The primary aims of the research have been to develop an understanding of the relation of model-predicted transition to the important physical processes and to improve, generalize, and further validate the model. Progress has been made in developing understanding of (1) low-Reynolds-number modifications to the turbulence generation terms and (2) the method used to simulate surface roughness effects on transition. As a result of this improved understanding, boundary conditions for the model have been revised and the model now predicts a roughness-dominated transition regime. To further validate the model, effects of pressure gradient on flat-plate boundary-layer transition have been analyzed. Generalization of boundary conditions for flows with mass injection has also been accomplished. The model has been generalized to include effects of coordinate-system rotation; the improved model accurately predicts laminarization in rotating channel flow.
The paper reviews the currently available models for calculating
turbulent stresses and heat or mass fluxes in incompressible flow and
reports on progress made in replacing the Prandtl mixing-length
hypothesis by more generally applicable models. These include models
employing transport equations for the intensity and the length scale of
the turbulent motion, notably the k-epsilon model, as well as second
order closure schemes based on transport equations for the turbulent
stresses and heat or mass fluxes themselves. The individual models are
described briefly, their merits and demerits are discussed, and typical
examples of calculations relevant to aerospace problems are presented.
An objective three-part comparison of the Jones-Launder, Ng-Spalding, Saffman-Wilcox, and Wilcox-Traci two-equation turbulence models has been conducted. Numerical computations were conducted in which the models were applied to four equilibrium boundary-layer flows including adverse, zero, and favorable pressure gradients. The models were tested using the same numerics, boundary conditions, and starting profiles. Overall, the Ng-Spalding and Wilcox-Traci models yielded results in closest agreement with experimental data. Computations of zero pressure gradient flow over a convex wall composed the final part of the comparison. With rationally devised streamline curvature modifications, the Jones-Launder, Ng-Spalding, and Wilcox-Traci models yielded excellent agreement with experimental data for the flow considered.
The closure approximations used in the two-equation turbulence models
are reviewed in order to determine an optimum choice of dependent
variables. By using a combination of perturbation methods and numerical
computations, it is shown that the conventional k-epsilon and
k-omega-squared formulations are generally inaccurate for boundary
layers in an adverse pressure gradient and that a more suitable choice
of dependent variables exists. As a result, a new two-equation model is
developed which shows promise of being more accurate for boundary layers
in an adverse pressure gradient than any other similar model.
The new 'multiscale' model for turbulent flows developed by Wilcox has
been subjected to a continuing series of rigorous applications,
including shock-induced boundary-layer separation, to test its accuracy
in simulating complex flow phenomena. While previous advanced
applications presented by Wilcox have been inconclusive regarding
superiority of the multiscale model over two-equation models, this paper
demonstrates a marked improvement in predictive accuracy for flows which
include boundary-layer separation. As speculated in the original
development of the model, results obtained demonstrate that the model is
superior because it accounts for disalignment of the
Reynolds-stress-tensor and the mean-strain-rate-tensor principal axes.
Effects of surface roughness, mass injection, streamline curvature and
the mixing layer are also analyzed.
Asymptotic expansion techniques are used, in the limit of large Reynolds number, to study the structure of fully turbulent shear layers. The relevant Reynolds number characterizes the ratio of the local turbulent stress to the local laminar stress, so that a relatively thick outer defect layer, in which, to lowest order, there is a balance between turbulent stress and convection of momentum, may be distinguished from a relatively thin wall layer, in which, to lowest order, there is a balance between turbulent and laminar stresses. The two cases examined are channel (or pipe) flow and two-dimensional boundary-layer flow with an applied pressure gradient, upstream of any separation. Attention, for these two cases, is confined to the flow of incompressible constant property fluids. Closure is effected through the introduction of an eddy-viscosity model formulated with sufficient generality for most existing models to be special cases. Results are carried to higher orders of approximation to indicate what properties for the friction velocity, integral thicknesses, and velocity profiles, and what conditions for similarity are implied by current eddy-viscosity closures.
The turbulent energy equation is converted into a differential equation for the turbulent shear stress by defining three empirical functions relating the turbulent intensity, diffusion and dissipation to the shear stress profile. This equation, the mean momentum equation and the mean continuity equation form a hyperbolic system. Numerical integrations by the method of characteristics with preliminary choices of the three empirical functions compare favourably with the results of conventional calculation methods over a wide range of pressure gradients. Nearly all the empirical information required has been derived solely from the boundary layer in zero pressure gradient.
A Crank-Nicolson type finite-difference scheme with a nonuniform grid spacing has been interpreted in terms of a coordinate stretching approach to show that it is second-order accurate. The variable grid scheme is applied to a flat plate laminar to turbulent boundary layer flow with a rapidly changing grid interval across the layer. The accuracy of the solution is determined for a different number of intervals and compared to results obtained with the Keller box scheme. The influence of changing the grid spacing on the accuracy of the solutions is determined for one coordinate stretching or grid spacing relation. The use of Richardson extrapolation is also investigated.
Two second-order-closure turbulence models were devised that are suitable for predicting properties of complex turbulent flow fields in both incompressible and compressible fluids. One model is of the "two-equation" variety in which closure is accomplished by introducing an eddy viscosity which depends on both a turbulent mixing energy and a dissipation rate per unit energy, that is, a specific dissipation rate. The other model is a "Reynolds stress equation" (RSE) formulation in which all components of the Reynolds stress tensor and turbulent heat-flux vector are computed directly and are scaled by the specific dissipation rate. Computations based on these models are compared with measurements for the following flow fields: (a) low speed, high Reynolds number channel flows with plane strain or uniform shear; (b) equilibrium turbulent boundary layers with and without pressure gradients or effects of compressibility; and (c) flow over a convex surface with and without a pressure gradient.
A set of constitutive equations suitable for a priori computation of turbulent shear flows has been developed. Since no properties of a given turbulent flow need be known in advance in order to obtain a solution, the equations comprise a complete model of turbulence. Perturbation analysis shows that the model predicts a composite five-layer structure for an incompressible turbulent boundary layer, viz, a defect layer, a law-of-the-wall layer, a viscous sublayer, a near-surface roughness layer, and a viscous superlayer at the boundary-layer edge. Analysis of the defect layer demonstrates the key improvement of the model over its predecessor, the Saffman-Wilcox two-equation model of turbulence. Examination of model-predicted sublayer structure yields model-parameter boundary conditions appropriate for surfaces with roughness and mass injection. Results of numerical computations of compressible and incompressible equilibrium boundary layers show that, for such flows, the model is as accurate as mixing-length theory. Applications to transitional boundary layers and to nonequilibrium relaxation of a boundary layer passing from a rough to a smooth surface indicate that the model's applicability extends far beyond that of mixing-length theory's.
While developing a three-dimensional boundary layer program using a standard parabolic matching scheme, the author has found computing time with the Wilcox-Rubesin (1979) two-equation turbulence model to be very lengthy. The long computing time occurs because converged solutions are possible only when very small streamwise steps are taken. The proposed remedy reduces computing time by increasing the maximum permissible step size.
Turbulent shear flows transport properties, computing atmospheric and vortex motions by invariant modeling of Reynolds stress term in boundary layer momentum equation
Although much progress has already been made In solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A new method for solving these equations has been devised that 1) is second-order accurate in space and time, 2) is unconditionally stable, 3) preserves conservation form, 4) requires no block or scalar tridiagonal inversions, 5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs), 6) is more efficient than present methods, and 7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 3 x 10(exp 5) to 3 x 10(exp 7) and at CFL numbers as high as 10(exp 3) are compared with theory and experiment.
Program EDDYBL User's GuideVariable Grid Scheme Applied to Turbulent Boundary Layers
- D C F G Wilcox
Wilcox, D. C., "Program EDDYBL User's Guide," DCW Indus-tries, La Canada, CA, Rept. DCW-R-NC-04, 1988. 24 Blottner, F. G., "Variable Grid Scheme Applied to Turbulent Boundary Layers," Computational Methods for Applied Mechanics and Engineering, Vol. 4, No. 2, 1974, pp. 179-194.