In fluid engineering, the turbulence problem is the longstanding challenge of obtaining accurate predictions of engineering quantities at affordable computational cost. Viewed through computational complexity, a practical algorithm requires cost growth no worse than O(N), where N denotes problem size. For turbulent flows, the problem size may be approximated by the number of dynamically relevant scales and hence by the Reynolds number Re. We propose a multi-fidelity, physics-constrained, data-driven framework designed to meet this criterion under stated assumptions. We augment the Spalart–Allmaras model through field inversion and machine learning using a constrained formulation that preserves the law of the wall. The model is trained at a low Reynolds number, where high-fidelity data are affordable, and deployed at higher Reynolds numbers. For a mean-flow-aligned grid in a wall-bounded flow, fixed spanwise resolution, and steady-solver cost linear in grid-point count, the low-fidelity RANS prediction scales as O(log(Re)). The high-fidelity calculation and learning stage each contribute O(Re^0) relative to the target Reynolds number, giving an overall formal cost of O(log(Re)). In plane channel flow, a model trained at Re_τ=1000 corrects the wake-layer error of the baseline model and retains the improvement at Re_τ=5200. In the periodic hill, a model trained at Re_b=5600 is tested at Re_b=10595, 19000, and 37000. The constrained formulation preserves separation and recovery behavior as Reynolds number increases, yields the lowest root-mean-square error across all tests, and exhibits nearly Reynolds-number-independent error, indicating robust extrapolation.

Error diagnostics for turbulence models have traditionally focused on engineering quantities of interest, such as the skin-friction coefficient $C_{\!f}$ , most often by comparing the predicted $C_{\!f}$ against reference data. In wall-bounded turbulent boundary layers, however, $C_{\!f}$ results from several physical mechanisms - viscous effects, turbulence, pressure gradients and mean-flow development - whose relative importance depends on the flow conditions. Modelling errors in these mechanisms vary across turbulence closures, and identifying them offers valuable physical insight for model evaluation and improvement. We propose a diagnostics framework that systematically isolates and quantifies such errors using the angular momentum integral formulation. The method is applied to five transport-type Reynolds-averaged Navier-Stokes models in two test cases: a canonical zero-pressure-gradient flat-plate boundary layer, and flow over a three-dimensional hill. For the flat-plate case, comparison with direct numerical simulations data shows that all models reproduce $C_{\!f}$ reasonably well, but often through strong error cancellation, particularly between the turbulent torque and mean-flux contributions; individual terms can deviate by more than 20 % of $C_{\!f}$ . For the hill case, where wall-resolved large-eddy simulations are used as the reference, errors are significantly larger. The dominant erroneous contribution differs by model and may exceed several times the local $C_{\!f}$ , depending on streamwise position. In separated-flow regions, the error cancellation that was observed in the flat-plate case largely disappears for the hill case, and the leading source of error shifts between mechanisms. These results highlight the value of mechanism-resolved diagnostics, and provide guidance for targeted turbulence model improvements.

Flow separation is often characterized by mean-flow quantities such as flow reversal and reattachment and eigen values of the Reynolds stress tensor. In this study, we examine the turbulent characteristics of three canonical flows using direct numerical simulation: pressure-driven Couette–Poiseuille flow, backward-facing step, and periodic hills. The results first show that mean-flow alone is not sufficient to define separation. In particular, although the Couette–Poiseuille flow exhibits reversed mean flow, its turbulence statistics remain consistent with those of attached wall-bounded flows. By contrast, the backward-facing step and periodic hill flows exhibit common turbulence features associated with genuine separation, including the breakdown of near-wall scaling, energy transfer from the detached shear layer toward the wall, enhanced turbulent kinetic energy, an overshoot of Reynolds shear stress relative to turbulent kinetic energy, and a transition of Reynolds stress anisotropy from a near-wall two-component state toward a more isotropic state within the separation region. Motivated by these observations, we construct a profile-based identification criterion for separation that combines wall shear stress with the deviations of mean velocity, turbulent kinetic energy, Reynolds shear stress, and turbulence anisotropy from their corresponding equilibrium, attached reference profiles. A logistic-regression model based on these quantities accurately identifies separated and attached profiles in the canonical flows considered. The results show that separation cannot be identified reliably from wall shear stress or any other single quantity alone; instead, it is more robustly characterized by the combined departure of both the mean flow and the turbulence field.

In the field of scientific computing, complex matrices arise from Laplace, Burgers, Kuramoto-Sivashinsky, and Allen-Cahn equations that are not necessarily symmetric positive definite. Computational fluid dynamics, in particular, often deals with pressure Poisson equation. For iterative solvers, time complexity is one of the most critical properties, if not the most critical. Its notation is O(N-alpha) with N denoting the size of the discretized system and alpha the scaling exponent. This property indicates how an iterative method's performance scales with the size of the discretized system. Due to the large size of systems in today's scientific computing, methods with lower time complexity are almost always preferred over those with higher time complexity, regardless of the prefactor. This emphasis on time complexity reveals a significant gap in the literature: although the integration of data-enabled methodologies in scientific computing has led to the developments of convergence accelerators and the observation of a speedup of O(10) or so, the reported reductions in cost predominantly concern the prefactor rather than the time complexity. This paper aims to explore reduction in time complexity. The accelerator developed in this paper involves projecting the intermediate solution, which is otherwise only used to assess the residual in the baseline iterative method, onto a low-dimensional Hilbert subspace and directly solving the discretized system there. The solver alternates between the baseline iterative method and the accelerator. Our scaling analysis, which is usually not possible for data-based methods, shows a O(N-i) reduction in the time complexity for N(i)(d-)sized problems in d-dimensional space. Here, N-i is the number of grids in each dimension, and the system size is N=N-i(d). Consolidated by tests up to 10(9) degrees of freedom, the present method is shown to offer increasingly more acceleration as the problem size increases, up to 200 times speedup for systems of size 10(9). Moreover, we demonstrate that the accelerator remains effective for highly nonlinear equations and unstructured grids, yielding similar speedup as for Poisson equation.

Flow over rough surfaces has been studied and modeled for many decades, due to its important role in turbulent boundary layer evolution and attendant drag and heat transfer amplification. While explicit resolution of deterministic and random roughness morphologies is often feasible given a geometry specification (i.e., CAD and/or optical scanning), CFD modeling of these roughness resolved configurations can be cost prohibitive in a design environment for DNS, LES and even sublayer resolved RANS. For this reason, surface parameterization based modeling is widely used to reduce computational cost. However, this approach suffers from many deficiencies, including ambiguity in determining the appropriate representative roughness length scale, and limitations associated with correctly predicting friction and heat transfer simultaneously. An alternative to surface parametrization is volumetric parameterization. Distributed Element Roughness Modeling (DERM) is an example of such a method. In this work, a DERM model based on the Double-Averaged Navier-Stokes (DANS) equations is developed. This formulation represents a complete treatment in that the three unclosed momentum transport processes that arise are each modeled; the roughness induced drag, the dispersive stress and the spatially averaged Reynolds stress. The models presented here are formulated based on physical and dimensional arguments, and are calibrated and validated using roughness resolved DNS, and neural network based machine learning. Three classes of surface topology are considered. These include cube arrays of varying packing density, sinusoidal roughness patterns of varying wavelengths, and random distributions associated with real additively manufactured surfaces. While DERM models are typically calibrated to specific deterministic roughness shape families, the results shown here demonstrate the wider range of applicability for the present, more generalized formulation.

This paper presents detailed analyses of the Reynolds stresses and their budgets in temporally evolving stratified wakes using direct numerical simulation. Ensemble averaging is employed to mitigate statistical errors in the data, and the results are presented as functions of both the transverse and vertical coordinates – at time instants across the near-wake, non-equilibrium, and quasi-two-dimensional regimes for wakes in weakly and strongly stratified environments. Key findings include the identification of dominant terms in the Reynolds stress transport equations and their spatial structures, the generation and destruction processes of the Reynolds stresses, and the energy transfer between the Reynolds stress and the mean flow. The study also clarifies the effects of the Reynolds number and the Froude number. Additionally, we assess the validity of the eddy-viscosity type models and some existing closures for the Reynolds stress model, highlighting the limitations of isotropy and return-to-isotropy hypotheses in stratified flows.

The baseline Launder-Spalding k- epsilon model cannot be integrated to the wall. This paper seeks to incorporate the entire law of the wall into the model while preserving the original k-epsilon framework structure. Our approach involves modifying the unclosed dissipation terms in the k and epsilon equations specifically within the wall layer according to direct numerical simulation (DNS) data. The resulting model effectively captures the mean flow characteristics in both the buffer layer and the logarithmic layer, resulting in robust predictions of skin friction for zero-pressure-gradient (ZPG) flat-plate boundary layers and plane channels. To further validate our formulation, we apply our model to boundary layers under varying pressure gradients, channels experiencing sudden deceleration, and flow over periodic hills, with highly favorable results. Although not the focus of this study, the methodology here applies equally to the k-omega formulation and yields improved predictions of the mean flow in the viscous sublayer and buffer layer.

Predicting drag caused by turbulent flow over rough surfaces remains one of the most challenging problems in engineering due to the intricate interactions between turbulent flow and surface roughness. This complexity is further heightened by the wide variability in rough-wall topographies and the roughness statistics required to accurately represent rough surfaces. In this study, we propose an approach to rough-wall modeling for drag prediction using deep convolutional autoencoders. We first compress complex data of 93 distinct rough surfaces into a low-dimensional space of just three variables using an autoencoder. These rough-wall topographies are obtained from various experiments and direct numerical simulation studies. We then utilize this reduced-order representation and relate these to the equivalent sandgrain roughness height (ks) with the help of a feedforward neural network. The predictive accuracy of the resulting rough-wall model is further assessed against unseen rough surfaces generated from reconstructions obtained from the latent space in unexplored areas. We observe that the formulated rough-wall model predicts ks values for these previously unseen surfaces to a reasonable accuracy. The present findings suggest the potential for ultra-low-dimensional data-driven representations of complex surface roughness and demonstrate their relevance in constructing generalizable predictive models for rough-wall bounded turbulence.

We report direct numerical simulations results of the rough-wall channel, focusing on roughness with high k(rms)/k(a) statistics but small to negative Sk statistics, and we study the implications of this new dataset on rough-wall modelling. Here, k(rms) is the root mean square, k(a) is the first-order moment of roughness height, and $Sk$ is the skewness. The effects of packing density, skewness and arrangement of roughness elements on mean streamwise velocity, equivalent roughness height (z(0)) and Reynolds and dispersive stresses have been studied. We demonstrate that two-point correlation lengths of roughness height statistics play an important role in characterizing rough surfaces with identical moments of roughness height but different arrangements of roughness elements. Analysis of the present as well as historical data suggests that the task of rough-wall modelling is to identify geometric parameters that distinguish the rough surfaces within the calibration dataset. We demonstrate a novel feature selection procedure to determine these parameters. Further, since there is no finite set of roughness statistics that distinguish between all rough surfaces, we argue that obtaining a universal rough-wall model for making equivalent sand-grain roughness (k(s)) predictions would be challenging, and that each rough-wall model would have its applicable range. This motivates the development of group-based rough-wall models. The applicability of multi-variate polynomial regression and feedforward neural networks for building such group-based rough-wall models using the selected features has been shown.

The conventional k-epsilon model accurately predicts the slope of the logarithmic law but falls short in estimating its intercept as well as the buffer layer. This limitation can be addressed either through a two-layer formulation or by introducing additional terms. However, both strategies necessitate extra adjustable constants and ad-hoc functions. In contrast, this paper introduces a novel one-layer k-epsilon model, which seamlessly integrates the law of the wall while preserving the essential structure of the k-epsilon framework. Our approach modifies the unclosed dissipation terms in the k and epsilon equations specifically within the wall layer. We invoke no other assumption than the general law of the wall and the assumptions that led to the k- epsilon model. Neither do we resort to ad hoc source terms. The revised model yields the following physical scalings in the viscous sublayer: k similar to y(2), epsilon similar to y(0). In addition, we demonstrate analytically the in-feasibility of sustaining the v(t) similar to y(3) scaling. Beyond the sublayer scalings, our model effectively captures the mean flow characteristics in both the buffer layer and the logarithmic layer, resulting in robust predictions of skin friction for zero-pressure-gradient flat-plate boundary layers and plane channels. To further validate our one-layer formulation, we apply our model to boundary layers under varying pressure gradients and channels experiencing sudden deceleration. Our model's results closely align with the reference direct numerical simulation and experimental datasets.

The constants and functions in Reynolds-averaged Navier Stokes (RANS) turbulence models are coupled. Consequently, modifications of a RANS model often negatively impact its basic calibrations, which is why machine-learned augmentations are often detrimental outside the training dataset. A solution to this is to identify the degrees of freedom that do not affect the basic calibrations and only modify these identified degrees of freedom when {\color{black}re-calibrating} the baseline model to accommodate a specific application. This approach is colloquially known as the ``rubber-band'' approach, which we formally call ``constrained model re-calibration'' in this article. To illustrate the efficacy of the approach, we identify the degrees of freedom in the Spalart-Allmaras (SA) model that do not affect the log law calibration. By subsequently interfacing data-based methods with these degrees of freedom, we train models to solve historically challenging flow scenarios, including the round-jet/plane-jet anomaly, airfoil stall, secondary flow separation, and recovery after separation. In addition to good performance inside the training dataset, the trained models yield similar performance as the baseline model outside the training dataset.

The BeVERLI Hill configuration has been studied using steady RANS at several tunnel orientations and Reynolds numbers. We report results using a sequence of (supplied) meshes, and five different RANS turbulence models: the standard Spalart-Allmaras one-equation model, a data-augmented Spalart-Allmaras model, the Menter k - omega SST two-equation model, the Chien k - epsilon. two-equation model, the seven-equation SSG-LRR Full Reynolds Stress Model (FRSM). Results for the 30 degrees orientation BeVERLI Hill bump are presented at two Reynolds numbers Re-H =250000 and 650000 per the "blind" model comparison call for this series of papers, using the data templates provided by Virginia Tech. There is in general good agreements among the two SA models, the SST k - omega model and the FRSM, with Chien k - epsilon model yielding somewhat different results.

Components with internal passages created using some laser-sintering based, additive manufacturing (AM) systems can exhibit anisotropic surface features with an appearance of three-dimensional roughness superimposed on two-dimensional, rib-like features. This paper presents an investigation of flow over roughness representing internal cooling passages printed at different angles to the AM printing plane. A roughness geometry was acquired using an X-ray tomography scan of a direct-metal-laser-sintering (DMLS) created coupon with internal cooling passages. The base surface scan was then used to create four surfaces with notional rib-like features positioned at different angles relative to the spanwise flow direction. The flow resistance of each surface was measured using the roughness internal flow tunnel. The mean flow velocity profiles for the cases with Re-Dh <= 30,000 were characterized using a four-camera, tomographic, and particle tracking system. The results demonstrate roughness orientation effects include (1) reduced bulk flow resistance as the alignment angle from the spanwise direction increases, (2) generated flow in the spanwise direction and increased tunnel flow swirl as the alignment angle increases, and (3) velocity profile changes as the flow migrates away from the rough side of the tunnel to the opposing smooth wall. The particle tracking system also demonstrates that the mean streamwise flow profiles change significantly between the 30 deg and 45 deg roughness orientations. Finally, the equivalent sandgrain roughness measurements for the four surfaces were found to follow the trends predicted using the correlations of Bons (2002, "St and c(f) Augmentation for Real Turbine Roughness With Elevated Freestream Turbulence," ASME J. Turbomach., 124(4), pp. 632-644.) and Sigal and Danberg (1990, "New Correlation of Roughness Density Effect on the Turbulent Boundary Layer," AIAA J., 28(3), pp. 554-556.).

The mean flow in a turbulent boundary layer (TBL) deviates from the canonical law of the wall (LoW) when influenced by a pressure gradient. Consequently, LoW-based near-wall treatments are inadequate for such flows. Chen et al. (J. Fluid Mech., vol. 970, 2023, A3) derived a Navier–Stokes-based velocity transformation that accurately describes the mean flow in TBLs with arbitrary pressure gradients. However, this transformation requires information on total shear stress, which is not always readily available, limiting its predictive power. In this work, we invert the transformation and develop a predictive near-wall model. Our model includes an additional transport equation that tracks the Lagrangian integration of the total shear stress. Particularly noteworthy is that the model introduces no new parameters and requires no calibration. We validate the developed model against experimental and computational data in the literature, and the results are favourable. Furthermore, we compare our model with equilibrium models. These equilibrium models inevitably fail when there are strong pressure gradients, but they prove to be sufficient for boundary layers subjected to weak, moderate and even moderately high pressure gradients. These results compel us to conclude that history effects in mean flow, which negatively impact the validity of equilibrium models, can largely be accounted for by the material time derivative term and the pressure gradient term, both of which require no additional modelling.

Direct numerical simulations are conducted for temporally evolving stratified wake flows at Reynolds numbers from $10\,000$ to $50\,000$ and Froude numbers from $2$ to 50. Unlike previous studies that obtained statistics from a single realization, we take ensemble averages among 80–100 realizations. Our analysis shows that data from one realization incur large convergence errors. These errors reduce quickly as the number of statistical samples increases, with the benefit of ensemble average diminishing beyond 40–60 realizations. The data with ensemble average allow us to test the previously established scalings and arrive at new scaling estimates. Specifically, the data do not support power-law scaling in the centreline velocity deficit $U_0$ beyond the near wake. Its decay rate increases continuously from 0.1 at the onset of the non-equilibrium regime until the end of our calculations without reaching any asymptote. Additionally, while no power-law scalings could be found in the wake width ( $L_H$ ) and wake height ( $L_V$ ) in the late wake, $L_H\sim (Nt)^{1/3}$ is a good working approximation of the wake's horizontal size, where $N$ is buoyancy frequency and $t$ is time. Besides the low-order statistics, we also report the transverse integrated terms and the vertically integrated terms in the turbulent kinetic energy budget equation as a function of the vertical and transverse coordinates. The data indicate that there are two peaks in the vertically integrated production and transport terms, and one peak when the two terms are integrated horizontally.

Metal additive manufacturing has enabled geometrically complex internal cooling channels for turbine and heat exchanger applications, but the process gives rise to large-scale roughness whose size is comparable to the channel height (which is 500 $\mathrm {\mu }$ m). These super-rough channels pose previously unseen challenges for experimental measurements, data interpretation and roughness modelling. First, it is not clear if measurements at a particular streamwise and spanwise location still provide accurate representation of the mean (time- and plane-averaged) flow. Second, we do not know if the logarithmic layer survives. Third, it is unknown how well previously developed rough-wall models work for these large-scale roughnesses. To answer the above practical questions, we conduct direct numerical simulations of flow in additively manufactured super-rough channels. Three rough surfaces are considered, all of which are obtained from computed tomography scans of additively manufactured surfaces. The roughness’ trough to peak sizes are 0.1 $h$ , 0.3 $h$ and 0.8 $h$ , respectively, where $h$ is the intended half-channel height. Each rough surface is placed opposite a smooth wall and the other two rough surfaces, leading to six rough-wall channel configurations. Two Reynolds numbers are considered, namely $Re_\tau =180$ and $Re_\tau =395$ . We show first that measurements at one streamwise and spanwise location are insufficient due to strong mean flow inhomogeneity across the entire channel, second that the logarithmic law of the wall survives despite the mean flow inhomogeneity and third that the established roughness sheltering model remains accurate.

The constants and functions in Reynolds-averaged Navier Stokes (RANS) turbulence models are coupled. Consequently, modifications of a RANS model often negatively impact its basic calibrations, which is why machine-learned augmentations are often detrimental outside the training dataset. A solution to this is to identify the degrees of freedom that do not affect the basic calibrations and only modify these identified degrees of freedom when re-calibrating the baseline model to accommodate a specific application. This approach is colloquially known as the "rubber-band" approach, which we formally call "constrained model re-calibration" in this article. To illustrate the efficacy of the approach, we identify the degrees of freedom in the Spalart-Allmaras (SA) model that do not affect the log law calibration. By subsequently interfacing data-based methods with these degrees of freedom, we train models to solve historically challenging flow scenarios, including the round-jet/plane-jet anomaly, airfoil stall, secondary flow separation, and recovery after separation. In addition to good performance inside the training dataset, the trained models yield similar performance as the baseline model outside the training dataset.

In the current research, different architectures of physics-informed neural networks (PINNs) are implemented to investigate the mixed electroosmotic pressure driven (EOF/PD) flow in microchannels with non-uniform zetapotential distribution on the walls. Through performing a detailed numerical simulation and PINN solutions based on Poisson-Boltzmann, Laplace, Navier-Stokes, concentration, and energy equations, we predict the nonuniformly distributed zeta-potential on the fluid dynamic, and heat transfer characteristics. It is observed that there is a good agreement between the Finite Volume Method (FVM) and segregated PINN approach. Comparing the PINN results with the numerical simulation, we show that amalgamating losses in all governing equations and applying a single PINN leads to higher training loss when compared to the multi-structured PINN, where a PINN is separately trained for each governing equation. Specifically, the normalized mean absolute percentage errors in the velocity prediction of the single and segregated PINNs are 20.17% and 9.2% respectively. In addition to the structure of the PINN, we also examine the effect of grid point distribution on PINNs. We demonstrate that using boundary layer collocation points can drastically improve the training efficiency and reduce the total loss for EOF/PD flow.

Techniques such as mechanical, electrochemical, and chemical etching are recognized as functional strategies for modifying surface roughness, which in turn manipulates surface energy. This research aims to delve into the influence of these factors on the heat transfer and pressure drop characteristics of two-phase R134a refrigerant flow during boiling inside horizontal annular concentric tubes with internal heat flux. The study scrutinizes the heat transfer coefficient and frictional pressure drop for R134a evaporating flow in an annular tube featuring an inner diameter of 28.6 mm and an outer diameter of 42 mm. The exploration spans a broad range of parameters, which include vapor quality from 0.15 to 0.8, mass flux from 6.72 to 20.16 kg/m2s, internal heat flux from 0.608 to 2.432 kW/m2, and varying surface characteristics. The results show that escalating surface roughness via mechanical techniques introduces greater resistance to flow shear stress in comparison to other methods. Moreover, an increase in surface roughness contributes to a reduction in contact angle and an expansion of the heating area, thereby enhancing both heat transfer coefficient and pressure drop. However, the implementation of low-energy coating holds an insignificant influence on R134a heat transfer characteristics, potentially due to its diminutive surface tension. This results in minor variations in the refrigerant contact angle in relation to the coatings. Performance evaluation criteria have been employed to assess the optimal thermal performance of the surface engineering method, with peak performance achieved at a mass flux of 20.16 Kg/m2s and an internal heat flux of 2.432 kW/m2. Ultimately, the study corroborates the experimental procedure by comparing the observed outcomes with established empirical correlations for R134a flow boiling.

The damage due to particulate matter ingestion by propulsion gas turbine engines can be significant, impacting the operability and performance of plant components. Here, we focus on the axial compressor whose blades become damaged when operated in dusty/sandy environments, resulting in significant performance degradation. In this work, CFD studies are performed to model the effects of airfoil damage on the first-stage rotor blading of a GE T700-401C compressor. We use thermoplastic additive manufacturing to construct representative physical models of three damage morphologies-ballistically bent/curved leading edges, cragged erosion of leading edges, and eroded leading/tailing edges at outer span locations. The resultant damaged plastic geometries, and a baseline undamaged configuration are then optically scanned and incorporated into sublayer resolved full stage, unsteady RANS analyses. Boundary conditions are imposed that conform to damaged compressor operation protocols, and this iterative process for accommodating corrected mass flow and off-design powering is presented. The results for the three damaged and one undamaged configuration are studied in terms of compressible wave field and secondary/tip flows, spanwise performance parameter distributions and efficiency. A method to estimate the effect of rotor damage on engine SFC is presented. The code, modeling, and meshing strategies pursued here are consistent with a validation study carried out for NASA Rotor 37 - these results are briefly included, and provide confidence in the predictions of the T700 geometry studied. The results provide quantitative comparisons of, and insight into, the physical mechanisms associated with damaged compressor performance degradation.