Reynolds Stress Components Research Articles

A new turbulent viscosity modification method based on multiphase compressibility for cavitating flows

The alteration of physical properties in cavitating flows due to phase transition presents challenges for accurately expressing turbulent viscosity in Reynolds-Averaged Navier–Stokes models. Addressing this issue is crucial for capturing cavitating flow characteristics effectively. This study introduces a modification to turbulent viscosity by considering the compressibility of the vapor–liquid mixture, applied within the k-omega Shear-Stress Transport (SST k−ω) model framework. Simulations of cavitating flow around the Clark-Y hydrofoil, National Advisory Committee for Aeronautics 66 hydrofoil, and wedge are conducted to validate the proposed method. Results indicate that the modified model can reproduce the cavity inception, development, cutoff, and shedding processes observed in the experiment. Notably, the modification model accurately reproduces distinct cavitating flow features such as U-shaped cavities, secondary shedding, and high-pressure phenomena resulting from collapse. Moreover, the predicted time-averaged velocity, time-averaged Reynolds stress component, and dominant frequencies of pressure and phase volume fraction surpass those of the original SST k−ω model, demonstrating improved performance. These findings highlight the enhanced accuracy and reliability of the proposed SST-multiphase compressibility modification k−ω model for simulating cavitating flows, thus contributing to improved understanding and prediction capabilities in relevant engineering applications.

Numerical study on the influence of riblet yaw-angle on drag reduction rate with different immersion heights

This paper uses the Large-eddy simulation method to investigate the influence of riblet yaw angle on drag reduction rate with different immersion heights, revealing that the vortex structure is gradually transformed from riblet-tip vortex pairs to boundary vortex forms with the increase of yaw angle, and analyzing the influence mechanism of immersion height on the critical yaw angle. The result shows that the drag-reduction rate of riblets decreases as the yaw angle increases, and the critical angle of the riblet varies with the different immersion heights. When the immersion height is 0.7, 0.5, and 0.3, the critical yaw angle is close to 30, 40, and 50°, respectively. As the immersion height increases, the cross-flow velocity at the riblet tip is more accessible to reach the critical velocity at which a boundary vortex can be formed, resulting in a smaller critical yaw angle for larger immersion heights. Furthermore, with the yaw angle increases, the normal diffusivity of normal vorticity and spanwise vorticity increase and decrease, respectively, which indicates that the lateral flow fluctuations decrease beyond a critical yaw angle, while the Reynolds stress component due to energy dissipation induced by the streamwise velocity decreases. Meanwhile, with the formation of a boundary vortex, the ability of the riblet to reduce viscous shear stresses increases due to the reduction of near-wall velocity gradient, and the emergence of a high-pressure region on the windward side of the riblet and a low-pressure region on the leeward side, consequently augmenting the differential pressure drag. This work provides valuable insights for designing riblets with heightened effectiveness at larger yaw angles, thereby extending their applicability under diverse flow conditions.

Consistent outer scaling and analysis of adverse pressure gradient turbulent boundary layers

Under adverse pressure gradient (APG) conditions, the outer regions of turbulent boundary layers (TBLs) are characterized by an increased velocity defect $U_{e}-U$ , an outwards shift of the peak value of the Reynolds shear stress $-\langle uv\rangle$ and an appearance of the outer peak value of the Reynolds normal stress $\langle uu\rangle$ . Here $U_{e}$ is the TBL edge velocity. Scaling APG TBLs is challenging due to the non-equilibrium effects caused by changes in the APG. To address this, the response distance of TBLs to non-equilibrium conditions is utilized to extend the Zagarola–Smits scaling $U_{zs} = U_{e}({\delta ^{*} }/{\delta })$ and ensure that the original properties of the Zagarola–Smits scaling are maintained as $Re \to \infty$ . Here $\delta ^{*}$ is the displacement thickness and $\delta$ is the boundary layer thickness. Based on the established correlation between $U_{e}-U$ and $-\langle uv\rangle$ , the scaling is extended to $-\langle uv\rangle$ . Furthermore, considering the coupling relationship between Reynolds stress components, the scaling is extended to encompass each Reynolds stress component. The proposed consistent scaling is verified using five non-equilibrium databases and five near-equilibrium databases, successfully collapsing the data of the TBL outer region. The pressure gradient parameter $\beta =({\delta ^{*} }/{\rho u_{\tau }^{2} }) ({\mathrm {d} P_{e} }/{\mathrm {d}\kern0.7pt x})$ of these databases spans two orders of magnitude. Here $P_{e}$ is the boundary layer edge pressure, $u_{\tau }$ is the friction velocity and $\rho$ is the density. Finally, the influence of the APG on the inner and outer regions of TBLs is analysed using the mean momentum balance equation. The analysis suggests that the shift of the $-\langle uv\rangle$ peak to the outer region under APG conditions is due to an insufficient inertia term near the inner region to balance the APG. It is observed that the APG promotes interaction between the inner and outer regions of TBLs, but the inner and outer regions still retain distinctive properties.

Study of compressible turbulent plane Couette flows via direct numerical simulation

Compressible turbulent plane Couette flows are studied via direct numerical simulation for wall Reynolds numbers up to $Re_w=10\ 000$ and wall Mach numbers up to $M_w=5$ . Various turbulence statistics are compared with their incompressible counterparts at comparable semilocal Reynolds numbers $Re^*_{\tau,c}$ . The skin friction coefficient $C_f$ , which decreases with $Re_w$ , only weakly depends on $M_w$ . On the other hand, the thermodynamic properties (mean temperature, density and others) strongly vary with $M_w$ . Under proper scaling transformations, the mean velocity profiles for the compressible and incompressible cases collapse well and show a logarithmic region with the Kárman constant $\kappa =0.41$ . Compared with wall units, the semilocal units yield a better collapse for the profiles of the Reynolds stresses. While the wall-normal and spanwise Reynolds stress components slightly decrease in the near-wall region, the inner peak of the streamwise component notably increases with increasing $M_w$ – indicating that flow becomes more anisotropic when compressible. In addition, the near-wall turbulence production decreases as $M_w$ increases – due to rapid wall-normal changes of viscosity caused by viscous heating. The streamwise and spanwise energy spectra show that the length scale of near-wall coherent structures does not vary with $M_w$ in semilocal units. Consistent with those in incompressible flows, the superstructures (the large-scale streamwise rollers) with a typical spanwise scale of $\lambda _z/h\approx 1.5{\rm \pi}$ become stronger with increasing $Re_w$ . For the highest $Re_w$ studied, they contribute about $40\,\%$ of the Reynolds shear stress at the channel centre. Interestingly, flow visualization and correlation analysis show that the streamwise coherence of these structures degrades with increasing $M_w$ . In addition, at comparable $Re^*_{\tau,c}$ , the amplitude modulation of these structures on the near-wall small scales is quite similar between incompressible and compressible cases – but much stronger than that in plane Poiseuille flows.

General attached eddies: Scaling laws and cascade self-similarity

We generalize Townsend's classical attached-eddy hypothesis (AEH) to eddies with arbitrary fractal dimension. New scaling laws for the Reynolds stress components are obtained. Preliminary evidence from high-resolution simulations of adverse-pressure-gradient turbulent boundary layers and wing boundary layers are provided. Furthermore, we investigate the cascade self-similarity of general attached eddies, and obtain relationships between the power-law exponents of the probability density, population density, area coverage, volume fraction, and fractal dimension of general attached eddies, leading to insights into the connection between turbulent coherent structures and statistics.

Large eddy simulation of shock wave/turbulent boundary layer interaction under incipient and fully separated conditions

Large eddy simulations of shock wave/turbulent boundary layer interaction on a compression ramp at the Mach number M a ∞ = 5 and Reynolds number R e ∞ = 14 000 are performed to investigate the impact of the incipient and fully separated conditions on the development of the flow field. The quasi-dynamic subgrid-scale kinetic energy equation model, which combines the benefits of the gradient model with the eddy-viscosity model, has been applied. Compared with the previous experimental and numerical results, the simulation was validated. The flow structures, turbulence properties, vortex structures, and low-frequency unsteadiness are all investigated. The flow field of the incipient separation is attached and rarely impacted by shock. An evident separation bubble and localized high wall temperatures in fully separated flow are caused by the separation shock's significant reverse pressure gradient. The Reynolds stress components exhibit significant amplification in both cases, and the peak outward shifts from the near-wall region to the center of the free shear layer. Turbulent kinetic energy terms were analyzed, and the two scenarios show a significant difference. The power spectral density of the wall pressure fluctuations shows that the low-frequency motion of the incipient separation is not apparent relative to the fully separated flow.

Characteristics of negative and undular surges in an impounded channel

ABSTRACT Physical experiments were conducted in an impounded channel to study the propagation of negative surges advancing in still water and evolution of undular surges against an instantaneously varied flow. The results demonstrated that the initial wave height of a negative surge could be accurately predicted using the simple wave solution of the Saint–Venant equations. Moreover, the wave height evolution during surge propagation could be satisfactorily described using the solution by Ippen and Kulin. The main characteristics of an undular surge were strongly correlated to the surge Froude number and turbulent boundary layer development at the toe of the surge. However, the occurrence of wave breaking at the first wave crest was independent of the inflow conditions and was close to the breaking criterion for solitary waves. Furthermore, the present results indicated the largest turbulent Reynolds stress components beneath the wave crests of an undular surge.

Anisotropic property of turbulent flow control through multiple stenosed microtubes

Rapid development in micro-scale systems and devices for drug delivery and biotechnical analysis has been scurrying. Microtube or microchannel roughness is another challenging factor in microfabrication technology. Moreover, the irregular roughness occurs due to the multiple depositions of cholesterol and other substances in the endothelium of the arterial wall. To investigate those issues, the effect of the irregular roughness of the wall on the anisotropy of the turbulent blood flow through an atherosclerotic artery has been analyzed. The flow through a stenosed microtube has been numerically simulated to understand the blood flow phenomenon through arterial stenosis and different turbulent states. The Reynolds stress components are estimated and plotted an invariant map and Eigen map to examine the turbulent states. The comparison results have been studied to show the nature of turbulent flow in low Reynolds numbers due to irregular roughness. The turbulent kinetic energy and turbulent viscous dissipation rate are also examined. Due to wall shear stress, the flow deserves the turbulence flow characteristic downstream of the abnormally narrowed regions of the microtube. The nature of the wall shear rate and the high viscosity of blood impact the disturbance in the flow velocity in microtubes.

Структура течения и теплоперенос в газокапельном течении за внезапным сужением канала

The numerical study of the local flow structure and heat transfer in a gas-droplet turbulent flow behind a forward-facing step in the two-phase gas-droplet flow is carried out. Two-dimensional steady-state Reynolds-averaged Navier-Stokes (RANS) equations are used for the numerical solution. They were written taking into account the presence of a dispersed phase. The Eulerian two-fluid approach is used to describe the flow dynamics and heat and mass transfer in the gaseous and dispersed phases. The turbulence of the carrier phase was described using an elliptical model for the transport of Reynolds stress components, taking into account the presence of droplets. The effect of evaporating droplets flowing after the forward-facing step on the local flow structure, turbulence, distribution of the dispersed phase, and heat transfer inetensification is analyzed.

Effect of interaction strength on recovery downstream of incident shock interactions

Direct numerical simulations of a supersonic turbulent boundary layer on a flat plate interacting with an impinging shock wave are carried out with two different incident shock angles at Mach 2.25. The effect of the interaction strength on the recovery process in the downstream region is systematically studied, including the turbulence evolution, the statistical and structural properties of wall pressure fluctuations, and the generation of mean skin friction and wall heat flux. The variations of the Reynolds stress components, the anisotropy tensor, and the turbulent kinetic energy budget in the two flow cases highlight a slow reversal tendency and an increasingly pronounced importance of the outer-layer large-scale structures in the relaxation region of the strong interaction. We find that the effect of increasing the interaction strength on the fluctuating wall pressure is reflected by a decrease in the characteristic frequencies, an increase in the spatial extent, and a decrease in the convection velocity. We decompose the mean skin friction and wall heat flux into different physically informed contributions and reveal that the mean wall heat flux generation is the same regardless of the interaction strength; in contrast, the generation mechanism of mean skin friction is found to be fundamentally changed. A novel scale-decomposition method is used to quantify the effect of the increased interaction strength on the leading components, and it is demonstrated that the energetic outer-layer large-scale structures are the dominant contributor in the recovery process as the interaction strength is increased.

Evolution of Turbulence and Its Modification by Axial Casing Grooves in a Multi-Stage Axial Compressor

Abstract This experimental study examines the evolution of turbulence across an axial compressor and its modification by semicircular axial casing grooves (ACGs) at the pre-stall and near the best efficiency (BEP) flowrates. The turbulence is highly anisotropic and spatially inhomogeneous, with each normal Reynolds stress component evolving differently. Most of the observed trends can be explained by examining the dominant production rate terms. At the pre-stall flowrate, the turbulence increases significantly upon entering the rotor with peak RMS values of axial velocity fluctuations reaching as high as 71% of the mean axial velocity. The region with elevated turbulent kinetic energy (TKE) covers 30% of the outer span near the rotor leading edge, expanding to 50% near the trailing edge. While the TKE in the outer span decays rapidly in the stator, the local turbulence production persists in the stator blade boundary layer. By stabilizing and homogenizing the flow, the ACGs reduce the turbulence production, hence the TKE, in the rotor and the stator. The only exception is an increase in turbulence in the region dominated by groove–passage flow interactions. Near BEP, the TKE is much lower everywhere, except for the region influenced by the outflow from grooves. Downstream of the rotor and the stator, the turbulence level with or without ACGs are similar. The large variations in the magnitude and even the sign of the measured eddy viscosity highlight the extreme non-equilibrium conditions over the entire machine, questioning the fundamental assumptions of local equilibrium in eddy viscosity-based Reynolds stress models.

Direct numerical simulation of supersonic bump flow with shock impingement

Direct numerical simulations are carried out to identify the effects of shock impingement on the behavior of bump flow at freestream Mach number of 2.25. Two cosine-shaped bump cases, with and without an impinging oblique shock at an angle of 33.2°, are compared. The shock impingement exhibits a remarkable influence on the pattern of the shock system and on the size of the separation region. A spectral analysis finds that low-frequency unsteadiness is significantly enhanced by the impingement interaction, and the proper orthogonal decomposition highlights the low-frequency breathing motion of the separation bubble, which is accurately reconstructed using only the first ten low-order modes. Downstream of the bump, both the Reynolds stress components and the turbulence kinetic energy exhibit a general amplification, with the peaks reoccurring at outer wall-normal locations. A turbulent kinetic energy budget analysis shows the greatly increased production in the outer layer which is balanced by turbulent transport and dissipation. An anisotropy-invariant map analysis identifies enhanced isotropic turbulence in the vicinity of the bump, which is qualitatively modified into a two-component axisymmetric state around the reattachment point. In addition, the mean skin friction decomposition suggests that the shock impingement has little influence on the predominant contribution of turbulence kinetic energy production, apart from the spatial growth dominance at the bump summit in the absence of the impinging shock. Interestingly, a scale-decomposed analysis quantitatively demonstrates that the contributions of small-scale structures are attenuated, but those of large-scale ones are relatively increased, with a contribution of more than 80% with shock impingement.

Numerical Prediction of Turbulent Drag Reduction with Different Solid Fractions and Distribution Shapes over Superhydrophobic Surfaces

The exploration of superhydrophobic drag reduction has been and continues to be of significant interest to various industries. In the present work, direct numerical simulation (DNS) is utilized to investigate the effect of the parameters on the drag-reducing performance of superhydrophobic surfaces (SHS). Simulations with a friction Reynolds number of 180 were carried out at solid fraction values of ϕs=116,111, and 14, and three distribution shapes: aligned, staggered, and random. The top wall is the smooth one, and the bottom wall is a superhydrophobic surface (SHS). Drag reduction and Reynolds stress profiles are compared for all cases. The turbulent kinetic energy budget, including production, dissipation, and diffusion, is presented with respect to the solid fraction and type of distribution to investigate the drag reduction mechanism. The sizes of the longitudinal vortices and formation of hairpin vortices are investigated through the observation of coherent structures. The simulation of a post model is a useful method to study the drag reduction for different solid fraction values and distribution geometries. Our study demonstrates that the drag reduction could acquire 42% with the solid fraction value ϕs=116 and an aligned distribution shape for post superhydrophobic surface geometry. Our study also showed the relationship of the Reynolds stress component (R11, R22, and R33) to the drag reduction with the differences in the solid fraction values and distribution geometry. In which, the R11 component has the most change between an aligned distribution and a random one. The peak value of R11 tends to shift away from the SHS wall. In addition, the analysis of the TKE budget over the superhydrophobic surface was performed, which can be adopted as a useful resource in turbulence modeling based on RANS methodology.

The scale-adaptive simulation of non-reacting and reacting jet in crossflow: Non-uniform fuel injection to decrease wall temperature

• SAS is employed to non-reacting and reacting jet-in-crossflow configurations. • SAS improves the prediction of Reynolds stress compared with RANS approaches. • SAS coupled with FRED model predicts velocity field well for reacting jet-in-crossflow. • Arrangement of non-uniform fuel injection is proposed to decrease wall temperature. Jet in crossflow (JIC) is widely used in many industrial applications, but the lack of accuracy on JIC flow prediction with Reynolds-averaged Navier Stokes (RANS) approaches has been recognized through some numerical studies. Thus new numerical method with both low computational cost and reasonable accuracy should be applied to JIC simulation, and the combustion systems also require reliable reacting JIC prediction specially. In the present study, the Scale-Adaptive Simulation (SAS) is employed to both non-reacting and reacting JIC configurations. Significant improvement has been observed on the prediction of Reynolds stress component with SAS as compared with RANS approaches, therefore the scalar field is also predicted accurately by SAS modeling. For the reacting JIC flow, the SAS model coupled with the finite-rate/eddy-dissipation (FRED) model is proved to predict the velocity field fairly well, and the flame-vertex interaction can be well captured by this method. The attachment of the heat release regions tends to increase the wall temperature near the nozzle. Subsequently an arrangement of non-uniform fuel injection is proposed and numerically studied, showing that this arrangement increases the flame lift-off and thus decreases the wall temperature effectively.

Hypersonic shock wave and turbulent boundary layer interaction in a sharp cone/flare model

A direct numerical simulation of hypersonic Shock wave and Turbulent Boundary Layer Interaction(STBLI) at Mach 6.0 on a sharp 7° half-angle circular cone/flare configuration at zero angle of attack is performed. The flare angle is 34° and the momentum thickness Reynolds number based on the incoming turbulent boundary layer on the sharp circular cone is Reθ = 2506. It is found that the mean flow is separated and the separation bubble occurring near the corner exhibits unsteadiness. The Reynolds analogy factor changes dramatically across the interaction, and varies between 1.06 and 1.27 in the downstream region, while the QP85 scaling factor has a nearly constant value of 0.5 across the interaction. The evolution of the reattached boundary layer is characterized in terms of the mean profiles, the Reynolds stress components, the anisotropy tensor and the turbulence kinetic energy. It is argued that the recovery is incomplete and the near-wall asymptotic behavior does not occur for the hypersonic interaction. In addition, mean skin friction decomposition in an axisymmetric turbulent boundary layer is carried out for the first time. Downstream of the interaction, the contributions of transverse curvature and body divergence are negligible, whereas the positive contribution associated with the turbulence kinetic energy production and the negative spatial-growth contribution are dominant. Based on scale decomposition, the positive contribution is further divided into terms with different spanwise length scales. The negative contribution is analyzed by comparing the convective term, the streamwise-heterogeneity term and the pressure gradient term.

Physics-informed neural networks for solving Reynolds-averaged Navier–Stokes equations

Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations. We employ PINNs for solving the Reynolds-averaged Navier–Stokes equations for incompressible turbulent flows without any specific model or assumption for turbulence and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier–Stokes equations for laminar flows by solving the Falkner–Skan boundary layer. We then apply PINNs for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows over a NACA4412 airfoil and the periodic hill. Our results show the excellent applicability of PINNs for laminar flows with strong pressure gradients, where predictions with less than 1% error can be obtained. For turbulent flows, we also obtain very good accuracy on simulation results even for the Reynolds-stress components.

Aerodynamic and Acoustic Simulations of Thick Flatback Airfoils Employing High Order DES Methods

AbstractThe results of high fidelity aerodynamic and acoustic computations of thick flatback airfoils are reported in the present paper. The studies are conducted on a flatback airfoil having a relative thickness of 30% with the blunt trailing edge thickness of 10% relative to chord. Delayed Detached‐Eddy Simulation (DDES) approaches in combination with high order (5th) flux discretization WENO (Weighted Essentially Non‐Oscillatory) and Riemann solver are employed. Two variants of the DES length scale calculation methods are compared. The results are validated against experimental data with good accuracy. The studies provide guideline on the mesh and turbulence modeling selection for flatback airfoil simulations. The results indicate that the wake breakdown is strongly influenced by the spanwise resolution of the mesh, which directly contributes to the prediction accuracy especially for drag force and noise emission. The Reynolds normal stress and the Reynolds stress component have the largest contributions on the mixing process, while the contribution of the component is minimal. Proper orthogonal decomposition is further performed to gain deeper insights into the wake characteristics.

Turbulence statistics from three different nacelle lidars

Abstract. Atmospheric turbulence can be characterized by the Reynolds stress tensor, which consists of the second-order moments of the wind field components. Most of the commercial nacelle lidars cannot estimate all components of the Reynolds stress tensor due to their limited number of beams; most can estimate the along-wind velocity variance relatively well. Other components are however also important to understand the behavior of, e.g., the vertical wind profile and meandering of wakes. The SpinnerLidar, a research lidar with multiple beams and a very high sampling frequency, was deployed together with two commercial lidars in a forward-looking mode on the nacelle of a Vestas V52 turbine to scan the inflow. Here, we compare the lidar-derived turbulence estimates with those from a sonic anemometer using both numerical simulations and measurements from a nearby mast. We show that from these lidars, the SpinnerLidar is the only one able to retrieve all Reynolds stress components. For the two- and four-beam lidars, we study different methods to compute the along-wind velocity variance. By using the SpinnerLidar's Doppler spectra of the radial velocity, we can partly compensate for the lidar's probe volume averaging effect and thus reduce the systematic error of turbulence estimates. We find that the variances of the radial velocities estimated from the maximum of the Doppler spectrum are less affected by the lidar probe volume compared to those estimated from the median or the centroid of the Doppler spectrum.

Hot-Wire Investigation of Turbulence Topology behind Blades at Different Shape Qualities

The scope of this paper is to perform a detailed experimental investigation of the shape error effect on the turbulence evolution behind NACA 64-618 airfoil. This airfoil is 3D-printed with predefined typical shape inaccuracies. A high-precision optical 3D scanner was used to assess the shape and surface quality of the manufactured models. The turbulent flow was studied using hot-wire anemometry. The developed force balance device was provided to measure the aerodynamic characteristics of the airfoil. Experimental studies were carried out for three angles of attack, +10∘, 0∘, −10∘, and different chord-based Reynolds numbers from 5.3×104 to 2.1×105. The obtained results show that the blunt trailing edge and rough surface decline the aerodynamic performance of the blades. In addition, the experimental results revealed a strong sensitivity of the Taylor microscale Reynolds number to the type of shape inaccuracy, especially at Re≈1.7×105. We also discuss the evolution of the Reynolds stress components, the degree of flow anisotropy, and the power spectrum distributions depending on the airfoil inaccuracies.

Study on bubble-induced turbulence in pipes and containers with Reynolds-stress models

Bubbly flow still represents a challenge for large-scale numerical simulation. Among many others, the understanding and modelling of bubble-induced turbulence (BIT) are far from being satisfactory even though continuous efforts have been made. In particular, the buoyancy of the bubbles generally introduces turbulence anisotropy in the flow, which cannot be captured by the standard eddy viscosity models with specific source terms representing BIT. Recently, on the basis of bubble-resolving direct numerical simulation data, a new Reynolds-stress model considering BIT was developed by Ma et al. (J Fluid Mech, 883: A9 (2020)) within the Euler—Euler framework. The objective of the present work is to assess this model and compare its performance with other standard Reynolds-stress models using a systematic test strategy. We select the experimental data in the BIT-dominated range and find that the new model leads to major improvements in the prediction of full Reynolds-stress components.