Eddy Viscosity Research Articles

Semi-analytical modeling of sediment-laden open-channel flows with the effects of stratification, hindered settling, and eddy viscosities.

This study proposes semi-analytical models for simultaneous distribution of fluid velocity and suspended sediment concentration in an open-channel turbulent flow using three kinds of eddy viscosities. Apart from the classical parabolic eddy viscosity which is based on a log-law velocity profile, we consider two recently proposed eddy viscosities based on the concept of velocity and length scales. To deal with the flows with high sediment concentration, several turbulent features such as the hindered settling mechanism and the stratification effect are incorporated in the model. The governing system of highly nonlinear differential equations is solved using the homotopy analysis method (HAM), which produces solutions in the form of convergent series. Numerical and theoretical convergence analyses are provided for all three types of eddy viscosities. The effects of parameters on the derived models are discussed physically. Experimental data on both dilute and non-dilute flows are considered to verify the HAM-based solutions. The effects of the stratification correction factor (β) and the turbulent Schmidt number (α) reveal that they should be determined optimally for applicability of the proposed models in terms of accurate prediction with data. This optimal procedure required further investigation of these parameters, and, thus, an analysis of β and α is carried out, which linked them with the particle diameter through particle settling velocity, reference fluid velocity, and reference sediment concentration by proposing regression equations. Furthermore, using the optimal values of the parameters, the proposed models corresponding to the eddy viscosities based on the exponentially decreasing turbulent kinetic energy function and von Karman's similarity hypothesis are seen to be superior to the model corresponding to a parabolic eddy viscosity. Finally, a comment on the HAM is made where it is observed that the method can remove the numerical singularity of the governing equations at the water surface, which arises because of the consideration of vanishing eddy viscosity thereat.

Assessment of Turbulence Models for Unsteady Separated Flows Past an Oscillating NACA 0015 Airfoil in Deep Stall

This paper provides 2D Computational Fluid Dynamics (CFD) investigations, using OpenFOAM package, of the unsteady separated fully turbulent flows past a NACA 0015 airfoil undergoing sinusoidal pitching motion about its quarter-chord axis in deep stall regime at a reduced frequency of 0.1, a free stream Mach number of 0.278, and at a Reynolds number, based on the airfoil chord length, , of . First, eighteen 2D steady-state computations coupled with the SST model were carried out at various angles of attack to investigate the static stall. Then, the 2D Unsteady Reynolds-Averaged Navier-Stokes (URANS) simulations of the flow around the oscillating airfoil about its quarter-chord axis were carried out. Three eddy viscosity turbulence models, namely the Spalart-Allmaras, Launder-Sharma , and SST were considered for turbulence closure. The results are compared with the experimental data where the boundary layer has been tripped at the airfoil’s leading-edge. The findings suggest that the SST performs best among the other two models to predict the unsteady aerodynamic forces and the main flow features characteristic of the deep stall regime. The influence of moving the pitching axis downstream at mid chord was also investigated using URANS simulations coupled with the SST model. It was found that this induces higher peaks in the nose-down pitching moment and delays the stall onset. However, the qualitative behavior of the unsteady flow in post-stall remains unchanged. The details of the flow development associated with dynamic stall were discussed

An Alternative Reynolds Shear Stress Model and Wake Model for a Flat Plate Boundary Layer Flow

The well-known closure problem has led to many models for the Reynolds shear stress. But many of these models are based on the Boussinesq approximation involving the mean velocity gradient and an eddy viscosity, which, however, requires to be modeled. Here, for a steady, incompressible and constant-pressure boundary layer, the Reynolds shear stress is expressed as a product of the mean velocity and the local momentum-deficit. This closure model is then incorporated in the boundary layer equation for the outer layer to predict the streamwise velocity in the wake region.

Numerical Study of Supersonic Flow Past a Cylindrical Afterbody

The near wake of a cylindrical afterbody aligned with a uniform Mach 2.46 flow has been numerically investigated using Reynolds-Averaged Navier-Stokes equations (k-epsilon two equation model) and Large Eddy Simulation (dynamic sub-grid scale eddy viscosity model). Mean flow field properties obtained from numerical simulations, such as axial velocity, pressure on the base surface has been compared with the experimental results. It has been found that k-epsilon model fails to predict the flow properties in the recirculation region where better agreement has been observed between the data obtained from LES and measurements. Data obtained from LES has been further analyzed to investigate the turbulent flow field in the wake region. Parameters like turbulent kinetic energy and primary Reynolds stress have been calculated and compared with the results obtained from an experiment in order to achieve a better understanding of the role of turbulence in the flow field.

Dynamic Mixed Modeling in Large Eddy Simulation Using the Concept of a Subgrid Activity Sensor

Following the relative success of mixed models in the Large Eddy Simulation of complex turbulent flow configurations, an alternative formulation is suggested here which incorporates the concept of a local subgrid activity sensor. The general idea of mixed models is to combine the advantages of structural models (superior alignment properties), usually of the scale similarity type, and functional models (superior stability), usually of the eddy viscosity type, while avoiding their disadvantages. However, the key question is the mathematical realization of this combination, and the formulation in this work accounts for the local level of underresolution of the flow. The justification and evaluation of the newly proposed mixed model is based on a priori and a posteriori analysis of homogeneous isotropic turbulence and laminar–turbulent transition in the Taylor–Green vortex, respectively. The suggested model shows a robust and accurate behavior for the cases investigated. In particular, it outperforms the separate structural and functional base models as well as the simulation without an explicit subgrid-scale model.

Swash flows generated by a train of solitary waves on a planar slope

Six consecutive solitary waves with identical wave height and separation time are generated to study the flow structures during the uprush–downwash interactions in the swash zone. Using particle image velocimetry, the cross-shore velocity fields are captured. Two different wave conditions are examined with different wave-height-to-water-depth ratios, i.e.$H_o/h=0.11$and 0.22. The uprush–downwash interaction reaches quasi-steady state from the third solitary wave for both cases. For the former case, a weak non-stationary hydraulic jump appears during the downwash flow for all the six consecutive waves. The weak hydraulic jump evolves into a momentarily ‘stationary’ broken bore when the next wave arrives. For the latter case, the larger wave height generates stronger wave breaking. No non-stationary hydraulic jump is observed as the duration of downwash flow is relatively short. The flow reverses to the onshore direction before the downwash Froude number reaches the hydraulic jump condition. The temporal and spatial evolution of turbulence structure at the quasi-steady state is quantified using the spatial spectral analysis, the integral length scale and turbulence eddy viscosity. The results suggest that the large-scale energy generated during the uprush–downwash interaction modified the slope of the turbulence energy spatial spectrum in the inertial subrange from$-$5/3 to$-$1 in the larger length scale region, indicating the energy cascade depends not only on the dissipation rate, but also on the turbulent kinetic energy from the large-scale turbulence structure because of the large-scale energy injection in the inertial subrange.

A Study of Wind Stress Effects on the Vertical Eddy Viscosity Coefficient Using the Ekman Model with Data Assimilation

The vertical eddy viscosity coefficient (VEVC) is an important parameter used in ocean dynamics studies to describe the intensity of mixing in the vertical direction and the process of momentum transport. In this paper, an adjoint assimilation method was used to invert the VEVC, based on the Ekman model, with the measured wind field and current data. The main purpose was to study the effect of changes in the background wind field on VEVC, and thus investigate the role of wind stress in the inversion process. The results indicated that the inverse vertical eddy viscosity coefficient increased slightly at the surface layer, reaching its maximum at around 10–12 m, and then decreased monotonically with depth. The maximum VEVC value corresponds to different depths for different wind speed ranges. Additionally, wind steering could affect the VEVC inversion curve, causing it to deviate from the general trend. The kinetic energy ratio increased with depth, peaked at 18–20 m, and then rapidly decreased to nearly zero beyond 24 m. The impact of wind field strength and steering on VEVC was observed in the kinetic energy ratio curve, which confirms the speculative wind stress effect. This study revealed the characteristics and mechanisms of VEVC in coastal waters under different wind conditions, which could provide a reference for further research on physical oceanography.

Variational RANS modeling of hydraulic jumps

The hydraulic jump is a phenomenon which frequently appears in nearshore, estuarine and riverine regions; it plays a major role for example in the development and migration of sand waves in mobile beds, and in controlling the propagation of infra-gravity water waves from the nearshore into the river through bar-built, shallow estuaries. Although the prediction of these natural flows becomes crucial in many circumstances, it is difficult because hydraulic jumps are complex flows, and their modeling is far from simple. The steady-state hydraulic jump may be undular or broken, and their physical properties differ significantly. Undular jump profiles are dominated by the deviation of the fluid pressure from hydrostatic conditions, with turbulence playing a secondary role. In contrast, broken hydraulic jumps are determined by the differential advection of momentum due to breaking, and turbulence stresses. Between both types, there is a complex transition involving the interaction of non-hydrostaticity and turbulence, which is difficult to mimic with depth-averaged models. Simulations of these flows are typically conducted by resorting to vertically-averaged models either using the Saint-Venant or the Serre-Green-Naghdi equations, but none of them are fully satisfactory. In this work, an alternative approach for hydraulic jumps is proposed by developing a variational Reynolds-Averaged Navier-Stokes (RANS) model which uses Kantorovich-Krylov expansions of the flow variables, and a novel depth-averaged, eddy viscosity approach suitable for non-hydrostatic flows. The model is validated through results of simulations of both undular and broken jumps compared against detailed experiments. New experiments were conducted in a simplified bar-built shallow river inlet; the variational RANS model was found to reproduce well the flow profile over the bar, the hydraulic jump position and dynamic pressures over the bar surface.

Turbulence near a sandbar island in the lower Missouri River

AbstractRiver turbulence is spatially variable due to interactions between morphology of rivers and physical mechanics of flowing water. Understanding the variation of turbulence in rivers is important for characterizing transport processes of soluble and particulate materials in these systems. We present an exploratory effort to understand ecologically relevant flow patterns using measurements of mean flow and turbulence in a highly engineered river channel around an island in the lower Missouri River. Specifically, the profiles of mean river velocities were investigated to examine the logarithmic relation and associated parameters, including shear velocity and bed roughness. Turbulence intensity and Reynolds shear stress were compared with classic open‐channel profiles and previously reported river data in the hydraulics literature. With the capability of pulse‐to‐pulse coherent Doppler velocity profiling in high spatial resolution, we estimated the profiles of turbulence dissipation rate using resolved one‐dimensional velocity spectra. These measurement data allow us to examine the validity of turbulence production‐dissipation balance and the classic open‐channel profiles of turbulence statistics, including turbulence intensity, Reynolds shear stress, dissipation rate, and eddy viscosity. The field data show a strong variation of turbulence profiles in close vicinity of the river island. In shallow water depths close to the island, turbulence is substantially enhanced in comparison with classic open‐channel profiles. Such turbulence enhancement is likely attributed to non‐uniformity of the flow structures.

Large Eddy Simulation of Bubble Column Bubbly Flow Considering Subgrid‐Scale Turbulent Diffusion Effects and Bubble Oscillation

AbstractThrough Euler/Euler large eddy simulation (LES) modeling, it is demonstrated that turbulent dispersion of bubbles can effectively indicate the impact of turbulent eddies on the bubble dynamics, i.e., the bubble oscillation behavior. This finding builds on previous work using the Euler/Lagrange LES modeling approach and leads to a significant improvement in predicting bubble lateral dispersion. Spatially filtered terms were proposed for the subgrid‐scale (SGS) turbulent dispersion and added mass stress force models, with a modification made to the SGS eddy viscosity to reflect bubble turbulent dispersion and oscillations. The proposed model substantially improves the prediction of bubble volume fraction distribution, bubble and liquid phase velocity profiles, the turbulent kinetic energy spectrum, and mass transfer.

Machine learning from RANS and LES to inform coarse grid simulations

Nuclear system thermal hydraulic analysis has historically relied on computationally inexpensive 1D codes. However, such tools are unable to capture multiscale multidimensional effects in large nuclear reactor enclosures. On the other hand, simulations with higher fidelity can be too expensive for such purposes. One of the ways to reduce computational cost is to perform simulations on a coarse grid, which, unfortunately, introduces large discretization errors.In this paper, two high-to-low data-driven approaches are investigated: (1) a coarse grid turbulence model to predict eddy viscosity and (2) correction of errors in coarse grid velocity fields. The approaches aim to reduce grid- and turbulence model-induced errors in coarse grid Reynolds-averaged Navier–Stokes (RANS) simulations. Two sources of high-fidelity data, RANS and large eddy simulations (LES), are explored.To extract the eddy viscosity from the LES data, an inverse optimization problem is solved. However, the LES eddy viscosity is shown to be comparable to the RANS eddy viscosity in terms of error reduction. Therefore, the directly available RANS eddy viscosity was used to develop a coarse grid data-driven turbulence model. Additionally, error correction in velocity is used to reduce the remaining uncertainties and bring the results closer to reality. The performance of the frameworks is demonstrated for a scaled upper plenum of a gas-cooled reactor facility.

О СВЯЗИ ПЕРИСТОЙ ОБЛАЧНОСТИ И ПОЛЕЙ ОТНОСИТЕЛЬНОЙ ВЛАЖНОСТИ В ВЕРХНЕЙ ТРОПОСФЕРЕ С СИММЕТРИЧНОЙ НЕУСТОЙЧИВОСТЬЮ

The relationship of cirrus clouds and areas of increased relative humidity in the upper troposphere with symmetric instability developing on the subtropical (anticyclonic) side of jet streams are explored. Theoretical arguments are supplemented by case studies related to the Northern and Southern hemispheres. Using satellite images of cirrus clouds and reanalysis data, the values of the vertical eddy viscosity, which compensates for the inviscid growth of disturbances in case of symmetric instability, are estimated. At the levels of 250 and 200 hPa, a significant correlation between high relative humidity and potential vorticity characteristic of symmetric instability is confirmed.

A posteriori error estimates for the Large Eddy Simulation applied to incompressible fluids

We study the two dimensional time dependent Large Eddy Simulation method applied to the incompressible Navier–Stokes system with Smagorinsky’s eddy viscosity model and a filter width that depends on the local mesh size. The discrete model is based on the implicit Euler scheme and a conforming finite element method for the time and space discretizations, respectively. We establish a reliable and efficient a posteriori error estimation between the numerical LES solution and the exact solution of the original Navier–Stokes system, which involves three types of error indicators respectively related to the filter and to the discretizations in time and space. Numerical results show the effectiveness of adaptive simulations

Numerical simulations of cavitating water jet by an improved cavitation model of compressible mixture flow with an emphasis on phase change effects

Focusing on cavitation phenomena caused by high-speed submerged water jets, this paper presents an improved cavitation model for a compressible fluid mixture based on a concise estimation of fluid compressibility that considers phase change effects. The homogeneous two-phase flow assumption is adopted, and the gas phase is assumed to consist of vapor and non-condensable components. Equations of state for a pure liquid and an ideal gas are employed to evaluate the compressibility of the liquid and non-condensable components, and the compressibility of the vapor is treated semi-empirically as a constant. The model is embedded in an unsteady Reynolds-averaged Navier–Stokes solver, with the realizable k-ε model employed to evaluate the eddy viscosity. The turbulent cavitating flow caused by an impulsively started submerged water jet is treated. The pattern of periodic cavitation cloud shedding is acceptably captured, and the mass flow rate coefficient and its fluctuation frequency evaluated by simulations agree with the experimental results well. The validity of the proposed method is confirmed. The results reveal that cavitation occurs when pin/Pin reaches 0.65 and fluid flow begins to pulsate. In the well-developed stage, the leading cavitation cloud and a subsequent cloud are successively shed downstream, and this process is repeated. The subsequent cloud catches the leading cloud, and they coalesce in the range x/d≈ 2–3. The pressure fluctuations concentrate in the range of x/d≈2–5 corresponding to the periodic shedding of cavitation clouds. The mass flow rate coefficient pulsates from 0.59–0.66 under the effect of cavitation.

Effect of constant eddy viscosity assumption on optimization using a discrete adjoint method

This paper presents a thorough study of the effect of the Constant Eddy Viscosity (CEV) assumption on the optimization of a discrete adjoint-based design optimization system. First, the algorithms of the adjoint methods with and without the CEV assumption are presented, followed by a discussion of the two methods’ solution stability. Second, the sensitivity accuracy, adjoint solution stability, and Root Mean Square (RMS) residual convergence rates at both design and off-design operating points are compared between the CEV and full viscosity adjoint methods in detail. Finally, a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization. Two gradient-based optimizers, the Sequential Least-Square Quadratic Programming (SLSQP) method and Steepest Descent Method (SDM) are respectively used to draw a firm conclusion. The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability, especially at a near stall point. Compared with the steady cases, the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger. Nevertheless, the CEV adjoint solver is still capable of achieving optimization goals to some extent, particularly if the flow under consideration is benign.

Semi-analytical method to estimate boundary shear stress in smooth rectangular and trapezoidal open channels

The boundary shear stress is obtained from the continuity and momentum equations in smooth prismatic open channels. Shear stress is a function of gravity, secondary flow, and gradient velocity. In this research, semi-analytical equations were obtained to estimate the average boundary shear stress in smooth rectangular and trapezoidal open channels by 0.5, 1, 1.5 and 2 side slopes using conformal mapping techniques. After dividing the channel cross section into the bed and wall sub-sections, the present study used the method by Guo and Julien (2005) to estimate the average shear stress of the bed and the wall in open channels. For the studied case, by the first shear stress estimation, secondary current and constant eddy viscosity were neglected. After recommending two secondary current and eddy viscosity correction factors, a second approximation well matched the experimental measurements of the average shear stress for all aspect ratios.

Direct numerical simulations of oscillatory boundary layers over rough walls

Numerical simulations of turbulent oscillatory flow over a bed made of fixed, identical spherical particles have been performed. Oscillations are imposed through a shear-driven forcing by means of an harmonic velocity boundary condition on the bed. A parametric study on the effect of the particle size and Reynolds number, spanning from the laminar to the fully-turbulent regimes, has been performed. Results show that the temporal evolution of the flow over the rough bed is modified compared to the classical Stokes’ solution for smooth-wall laminar boundary layers. In the turbulent cases, the phase shift in the velocity at various distances from the bed is reduced compared to the laminar case. The phase shift reduction seems predominantly dependent on the Reynolds number rather than the bed morphology, which suggests that a unique curve may exist for large enough values of the Reynolds number. Turbulence is observed during the deceleration phases of the cycle, with the presence of a logarithmic layer in the velocity profile, and “canonical” distribution of Reynolds shear stress and turbulent kinetic energy production. During the acceleration, the logarithmic region is suppressed and the Reynolds shear stress changes sign. Nevertheless, the turbulent kinetic energy production becomes only slightly negative, because the shear is also very small during these phases. A good correlation between Reynolds and shear stresses is also evidenced from the eddy viscosity profiles, which show the presence of plateau in the outer layers approximately constant across the phases of the cycle. Turbulence in the outer layers is related to structures which develop during previous cycles and propagate from the bed to the bulk of the channel. When the flow is fully developed, anisotropy maps show similar distributions to unidirectional wall-bounded flows, while departures from canonical distributions during the transient phases are mild, because the flow responds very rapidly to the time-varying forcing.

A Linear Theory of Local Thermal Circulations

The dynamics of local thermal circulations (LTCs) are examined by constructing a linear theory based on Boussinesq equations in the planetary boundary layer (PBL). Linear theory arranges LTCs into a sixth-order partial differential equation of temperature, which can be solved by using the Fourier transform and inverse Fourier transform. The analytic solution suggests that the horizontal distribution of the temperature anomaly is basically determined by surface heating, while the vertical distribution of the temperature anomaly is a combination of exponential decay and an Ekman spiral. For shallow PBL cases where the Ekman elevation is much smaller than the vertical scale of motion, the higher-order partial differential terms that represent the Ekman spiral structure can be ignored so that the equations reduce to a second-order partial differential equation. Compared with the numerical results, this so-called low-order approximation does not induce dramatic errors in the temperature distribution. However, to avoid distortion in the forced atmospheric circulation, the eddy viscosity in the motion equations should be replaced with the Rayleigh form, which is the common practice in LTCs. For deep PBL cases where the Ekman elevation is comparable to the vertical scale of motion, both the exponential decay and Ekman spiral structure play roles in the forced atmospheric circulation. The most significant influence is that there exist three additional compensating forced circulation cells that surround the direct forced circulation cell.

NUMERICAL SIMULATION OF GAS FLOW PATTERNS AND CHARACTERISTICS OF A CENTRIFUGAL COMPRESSOR

The design of compressors based on one- and two-dimensional flow calculations has reached the limit of development by achieving the maximum efficiency of devices. Further development is possible by conducting expensive experiments or by analyzing the results of the modern mathematical models of three-dimensional flow calculations based on the Reynolds-averaged Navier-Stokes equations. To improve the characteristics of the centrifugal compressor and improve the flow part with the subsequent change in the shape and location of the compressor impeller blades, threedimensional flow calculations were performed. Calculations were made using the Ansys CFX software with a student license. The effect of mesh on the integral flow characteristics is analyzed. There is a qualitative coincidence of the compressor performances, as well as gas flow patterns, and separation zones with experimental data. The research uses a sectoral approach to modeling. Calculations based on different turbulence models were compared and it was found that the models based on the "k – ε" model give practically identical results. Compared with the experimental data of the compressor, the Eddy Viscosity Transport Equation turbulence model showed a significant error in determining the total-to-total polytropic efficiency, this value is underestimated by more than 7 %. Due to an increase in the mass flow rate in the compressor, the velocity in the blade diffuser increases significantly and the flow separates from the blades. Based on the comparison of Mach number contours, in addition to flow separation from the diffuser blades, flow separation from the splitter walls and impeller blades are also observed, even at low mass flow rates. The influence of the sector approach on the calculation results and the mutual location of the impeller sector and the blade diffuser sector require further research, which will be the subject of further research.

Eddy viscosity enhanced temporal direct deconvolution models for temporal large-eddy simulation of turbulence

A dynamic eddy viscosity (DEV) model and a constant eddy viscosity (CEV) model are proposed for stabilizing the temporal direct deconvolution model (TDDM) in temporal large-eddy simulation of turbulence. Compared to the original unresolved subfilter-scale model used in TDDM, the new eddy viscosity models reduce the number of empirical coefficients and make TDDM more convenient to be applied in practice. The DEV model does not have any empirical coefficients, and the CEV model has only one constant model coefficient that is independent of the filter width and insensitive to the grid resolution. To solve the stability issue of TDDM, an algorithm called the variable filter-width method (VFM) is proposed. In VFM, the filter width is initialized by a small value or 0 and then grows linearly in a small number of time steps until it reaches the target filter width. The three dimensional homogeneous isotropic turbulence is applied to investigate the performance of the proposed models. In the a posteriori testing at different grid resolutions, eddy viscosity enhanced temporal direct deconvolution models show a good accuracy in predicting various statistics and instantaneous spatial structures of turbulence, and they perform better than the original model, especially in the prediction of subfilter-scale (SFS) stress and SFS energy flux. Moreover, the energy spectrum and other flow statistics predicted by the CEV model with a fixed model coefficient 0.03 are in a good agreement with the filtered DNS.