Turbulent Equation Research Articles

Numerical simulation of condensation inside an inclined smooth tube

In this study the condensation phenomena inside an inclined smooth tube were investigated. The tube internal diameter was 8.38 mm, and its length was 1488 mm. The condensing fluid was R134a at a saturation temperature of 40 °C. The Volume of Fluid (VOF) multiphase-flow formulation was utilized to solve the governing equations of continuity, momentum, energy and turbulence. The flow field was assumed to be unsteady, turbulent and three-dimensional. The fluid properties were considered to be constant as temperature changes were negligible. The effects on heat transfer coefficients of various parameters were investigated. These parameters included tube inclination angle, vapour quality, refrigerant mass flux, and flow regimes. Simulations were conducted at a heat flux of approximately 5 kW/m2, mass fluxes of 100–600 kg/m2·s, while the inclination angles of flow were varied between vertical downward, to vertical upward. The results of the simulations were compared to an experimental database and a good agreement was found with the experimental data. The numerical simulations gave new perspectives and complementary information that was not determined experimentally. The results showed that an optimum downward inclination angle of between −30° and −15° exists, for the heat transfer coefficients.

Turbulence in the rotating-disk boundary layer investigated through direct numerical simulations

Direct numerical simulations (DNS) are reported for the turbulent rotating-disk boundary layer for the first time. Two turbulent simulations are presented with overlapping small and large Reynolds numbers, where the largest corresponds to a momentum-loss Reynolds number of almost 2000. Simulation data are compared with experimental data from the same flow case reported by Imayama et al. (2014), and also a comparison is made with a numerical simulation of a two-dimensional turbulent boundary layer (2DTBL) over a flat plate reported by Schlatter and Örlü (2010). The agreement of the turbulent statistics between experiments and simulations is in general very good, as well as the findings of a missing wake region and a lower shape factor compared to the 2DTBL. The simulations also show rms-levels in the inner region similar to the 2DTBL. The simulations validate Imayama et al.’s results showing that the rotating-disk turbulent boundary layer in the near-wall region contains shorter streamwise (azimuthal) wavelengths than the 2DTBL, probably due to the outward inclination of the low-speed streaks. Moreover, all velocity components are available from the simulations, and hence the local flow angle, Reynolds stresses and all terms in the turbulent kinetic energy equation are also discussed. However there are in general no large differences compared to the 2DTBL, hence the three-dimensional effects seem to have only a small influence on the turbulence.

Simulation of chemical reaction process in gas-particle CFB downers by anisotropic turbulent mass transfer model

A recently developed model, called the Reynolds mass flux (RMF) model is adopted for the simulation of multiphase flow involving chemical reaction in gas-particle CFB downers. The newly deduced Reynolds mass flux equation is utilized for closing the turbulent mass transfer equation, thereby realizing the simulation of anisotropic turbulent mass transfer. The simulations are validated with experimental data in literatures. The results indicate that the turbulent mass diffusion in downers would show significant anisotropic character.

Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics

First-principles simulations of tokamak turbulence have proven to be of great value in recent decades. We develop a pseudo-spectral velocity formulation of the turbulence equations that smoothly interpolates between the highly efficient but lower resolution three-dimensional (3-D) gyrofluid representation and the conventional but more expensive 5-D gyrokinetic representation. Our formulation is a projection of the nonlinear gyrokinetic equation onto a Laguerre–Hermite velocity-space basis. We discuss issues related to collisions, closures and entropy. While any collision operator can be used in the formulation, we highlight a model operator that has a particularly sparse Laguerre–Hermite representation, while satisfying conservation laws and the H theorem. Free streaming, magnetic drifts and nonlinear phase mixing each give rise to closure problems, which we discuss in relation to the instabilities of interest and to free energy conservation. We show that the model is capable of reproducing gyrokinetic results for linear instabilities and zonal flow dynamics. Thus the final model is appropriate for the study of instabilities, turbulence and transport in a wide range of geometries, including tokamaks and stellarators.

Numerical simulations of hydrodynamics and concentration distribution in a stirred tank during startup stage for producing phenylboronic acid ester

AbstractIn this work, a kinetic model of phenylboronic acid ester production process is firstly derived using the power law analysis based on experimental data and then combined with computational fluid dynamics models to simulate the concentration distributions of phenylmagnesium as reactant and phenylboronic acid ester as product during the stirred tank startup stage. Computational grid refinements as well as transient analysis are applied to confirm hydrodynamic and concentration results. Different factors including turbulent equations, wall functions, and rotational speed are investigated with different computational fluid dynamics methods to predict both the phenylmagnesium and phenylboronic acid ester concentration distributions. Simulation results show good agreements with experimental results, and thus, the diffusion and distribution of phenylmagnesium and phenylboronic acid ester inside the reactor can be successfully predicted. To achieve good mixing performance, high rotational speed should be provided with considering the balance between the mixing time and the energy consumption.

An immersed boundary/wall modeling method for RANS simulation of compressible turbulent flows

SummaryIn this paper, an immersed boundary (IB) method is developed to simulate compressible turbulent flows governed by the Reynolds‐averaged Navier‐Stokes equations. The flow variables at the IB nodes (interior nodes in the immediate vicinity of the solid wall) are evaluated via linear interpolation in the normal direction to close the discrete form of the governing equations. An adaptive wall function and a 2‐layer wall model are introduced to reduce the near‐wall mesh density required by the high resolution of the turbulent boundary layers. The wall shear stress modified by the wall modeling technique and the no‐penetration condition are enforced to evaluate the velocity at an IB node. The pressure and temperature at an IB node are obtained via the local simplified momentum equation and the Crocco‐Busemann relation, respectively. The SST k − ω and S‐A turbulence models are adopted in the framework of the present IB approach. For the Shear‐Stress Transport (SST) k − ω model, analytical solutions in near‐wall region are utilized to enforce the boundary conditions of the turbulence equations and evaluate the turbulence variables at an IB node. For the S‐A model, the turbulence variable at an IB node is calculated by using the near‐wall profile of the eddy viscosity. In order to validate the present IB approach, numerical experiments for compressible turbulent flows over stationary and moving bodies have been performed. The predictions show good agreements with the referenced experimental data and numerical results.

Significance of buoyancy in turbulence closure for computational fluid dynamics modelling of ultraviolet disinfection in maturation ponds.

Buoyancy-driven turbulent dispersion in a maturation pond is studied using a combination of field measurements and computational fluid dynamics. Modelling flow in maturation ponds requires turbulent closure models because of the large physical size and the need to model on diurnal timescales. Simulation results are shown to be more sensitive to the inclusion of a buoyancy production term appearing in the turbulent transport equations than to the model choice. Comparisons with experimental thermal profiles show that without this term, thermal mixing is over-predicted. When including the term, stratification occurs but thermal mixing is under-predicted in the lower water column. In terms of pond performance, the effect of this term is to cause increased surface die-off of Escherichia coli during sunlight hours due to the generation of stratification. It is recommended that future modelling consider and implement this term.

A novel laminar kinetic energy model for the prediction of pretransitional velocity fluctuations and boundary layer transition

Boundary layer transition onset estimation and modelling are essential for the design of many engineering products across many industries. In this work, a novel model for predicting pretransitional boundary layer fluctuations is proposed. The laminar kinetic energy (LKE) concept is used to represent such fluctuations. The new LKE model is implemented in OpenFOAM within the Reynolds-Averaged Navier-Stokes (RANS) framework. Only two approaches for modelling the LKE can be found in the literature. Mayle and Schulz (1997) has the limitation of requiring an initial LKE profile. Walters and Cokljat’s (2008) approach has been found to significantly overpredict the growth of the LKE. In addition, their model is tightly coupled with the specific dissipation rate and turbulent kinetic energy equations. The new model proposed here can act as a stand-alone equation for the LKE, making it portable and potentially facilitating the development of new transition models tailored to various industrial applications. Comparison with experiments shows that the new LKE model correctly predicts the growth of pretransitional velocity fluctuations and skin friction for a flat plate at zero-pressure gradient. To illustrate its practical application for transitional flows, the LKE model is coupled with an existing k−ω model using a new approach that requires minimal modifications. The resulting model (k−ω LKE) demonstrates excellent predictive capabilities when applied to a number of validation test cases.

A rigorous model for the simulation of chemical reaction in gas-particle bubbling fluidized bed: I. Modeling and validation

A rigorous model is presented for simulating the gas-particle two phase flow and the involved chemical reaction in bubbling fluidized beds (BFB). The present model employs the recently developed c2¯−εc closures for the turbulent mass transfer equation, thus enables the prediction of species concentration distributions without relying on empirical turbulent Schmidt number. The Two-Fluid model (TMF) is used to represent the fluid mechanics and the coupled chemical reaction. In mathematical expressions, the kg−εg−kp−εp−Θ equations for the description of phases turbulence is adopted. To test the validity of the present model, simulations are conducted for reactive multiphase flow in a BFB. The model predictions show good agreement with the experimental data in literatures. Part II of these serial works will further extend the application of this model to simulating a gas combustion process with strong heat effect in BFB.

Parametric study on the heat transfer of a blunt body with counterflowing jets in hypersonic flows

A quantified parametric study for the heat transfer acting on a hypersonic blunt body with counterflowing jets is presented. Three-dimensional turbulent Navier-Stokes equations are solved to simulate freestream-jet interactive flowfields. The freestream and jet controlling parameters are treated as input sources of variation, and a point-collocation non-intrusive polynomial chaos (NIPC) method is utilized to quantify the variations in the output surface heat flux and total surface heat load acting on the blunt body by identifying the maximum and minimum of surrogate response values predicted by the NIPC. All of the sample cases are confirmed to form steady jet structures. Furthermore, through a global sensitivity analysis, Sobol indices evaluated by the NIPC, are used to rank the contributions of each input parameter to the variation in output quantities of interest. It is found that the designed upstream injection of baseline case effectively reduces the heat transfer to body surface compared with the no-jet case. The variations of input parameters induce remarkable variations of output heat flux and total heat load. The sensitivity analysis indicates that the jet-to-freestream total-pressure ratio is the top contributor to variations in heat flux, followed by the freestream Mach number. The jet total temperature is mainly important on the front part of forebody, while the contributions of jet Mach number and freestream temperature slightly increase downstream. The freestream density has the smallest effect. The sensitivity of total heat load to input parameters coincides with that of heat flux. This parametric study is expected to illustrate the significance of flow-controlling parameters to the heat transfer over blunt body and provide insight for aerothermal management by using counterflowing jets in hypersonic flows.

Database of Near-Wall Turbulent Flow Properties of a Jet Impinging on a Solid Surface under Different Inclination Angles

In the present paper, direct numerical simulation (DNS) and particle image velocimetry (PIV) have been applied complementarily in order to generate a database of near-wall turbulence properties of a highly turbulent jet impinging on a solid surface under different inclination angles. Thereby, the main focus is placed on an impingement angle of 45 ∘ , since it represents a good generic benchmark test case for a wide range of technical fluid flow applications. This specific configuration features very complex flow properties including the presence of a stagnation point, development of the shear boundary layer and strong streamline curvature. In particular, this database includes near-wall turbulence statistics along with mean and rms velocities, budget terms in the turbulent kinetic energy equation, anisotropy invariant maps, turbulent length/time scales and near-wall shear stresses. These properties are useful for the validation of near-wall modeling approaches in the context of Reynolds-averaged Navier–Stokes (RANS) and large-eddy simulations (LES). From this study, in which further impingement angles ( 0 ∘ , 90 ∘ ) have been considered in the experiments only, it turns out that (1) the production of turbulent kinetic energy appears negative at the stagnation point for an impingement angle other than 0 ∘ and is balanced predominantly by pressure-related diffusion, (2) quasi-coherent thin streaks with large characteristic time scales appear at the stagnation region, while the organization of the flow is predominantly toroidal further downstream, and (3) near-wall shear stresses are low at the stagnation region and intense in regions where the direction of the flow changes suddenly.

On the topological degree of the mean field equation with two parameters

We consider the following class of equations with exponential nonlinearities on a compact surface M : −∆u = ρ1 ( h1 eu M h1 e u − 1 |M | ) − ρ2 ( h2 e−u M h2 e −u − 1 |M | ) , which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here h1, h2 are smooth positive functions and ρ1, ρ2 are two positive parameters. We start by proving a concentration phenomena for the above equation, which leads to a-priori bound for the solutions of this problem provided ρi / ∈ 8πN, i = 1, 2. Then we study the blow up behavior when ρ1 crosses 8π and ρ2 / ∈ 8πN. By performing a suitable decomposition of the above equation and using the shadow system that was introduced for the SU(3) Toda system, we can compute the Leray-Schauder topological degree for ρ1 ∈ (0, 8π) ∪ (8π, 16π) and ρ2 / ∈ 8πN. As a byproduct our argument, we give new existence results when the underlying manifold is a sphere and a new proof for some known existence result.

Control problems and invariant subspaces for sabra shell model of turbulence

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models and their existence theory along with numerical simulations have been studied earlier. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in [11]. We follow the same functional setup given in [11] and study control problems related to it. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the flow near a priori known state. We derive optimal controls in terms of the solution of adjoint equations for corresponding linearized problems. Another interesting problem studied in this work is to establish feedback controllers which would preserve prescribed physical constraints. Since fluid equations have certain fundamental invariants, we would like to preserve these quantities via a control in the feedback form. We utilize the theory of nonlinear semi groups and represent the feedback control as a multi-valued feedback term which lies in the normal cone of the convex constraint space, under certain assumptions. Moreover, one of the most interesting result of this work is that we can design a feedback control with only finitely many modes, which is able to preserve the flow in the neighborhood of the constraint set.

On the Kinetic Equation in Zakharov's Wave Turbulence Theory for Capillary Waves

The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectra in weakly nonlinear and dispersive media. Such a kinetic model was derived by physicists in the 1960s, though the well-posedness theory remains open due to the complexity of resonant interaction kernels. In this paper, we provide a global unique radial strong solution---the first such result---to the wave turbulence equation for capillary waves.

МОДЕЛИРОВАНИЕ ПРОЦЕССОВ РАСПРОСТРАНЕНИЯ ВЗВЕШЕННОГО ВЕЩЕСТВА В МОРСКИХ ПРИБРЕЖНЫХ АКВАТОРИЯХ: 1. ОПИСАНИЕ ВВ-МОДЕЛИ

A methodology is considered for developing a mathematical model to study the distribution of suspended matter concrentrations in coastal marine areas. This model assesses the consequences of dredging (namely the disturbance of the natural state of marine environment for bioresources), it reproduces the formation of suspended matter and its subsequent transport in the aquatic environment in sea bays and offshore water areas. This problem is solved in two basic stages: at the first stage – the system of equations of hydrodynamics (the equations of the theory of shallow water) is numerically integrated and the values of the mean components of the flow velocity are determined; and at the second stage – the system of turbulent diffusion equations is integrated, the processes of sedimentation of suspended matter, resuspension of bottom sediments, arrival of suspended matter from distributed point sources, their redistribution in the aquatic environment and the accumulation of residual quantities on the bottom are taken into account.

Retrospective cost adaptive Reynolds-averaged Navier–Stokes k–ω model for data-driven unsteady turbulent simulations

This paper presents a data-driven computational model for simulating unsteady turbulent flows, where sparse measurement data is available. The model uses the retrospective cost adaptation (RCA) algorithm to automatically adjust the closure coefficients of the Reynolds-averaged Navier–Stokes (RANS) k–ω turbulence equations to improve agreement between the simulated flow and the measurements. The RCA-RANS k–ω model is verified for steady flow using a pipe-flow test case and for unsteady flow using a surface-mounted-cube test case. Measurements used for adaptation of the verification cases are obtained from baseline simulations with known closure coefficients. These verification test cases demonstrate that the RCA-RANS k–ω model can successfully adapt the closure coefficients to improve agreement between the simulated flow field and a set of sparse flow-field measurements. Furthermore, the RCA-RANS k–ω model improves agreement between the simulated flow and the baseline flow at locations at which measurements do not exist. The RCA-RANS k–ω model is also validated with experimental data from 2 test cases: steady pipe flow, and unsteady flow past a square cylinder. In both test cases, the adaptation improves agreement with experimental data in comparison to the results from a non-adaptive RANS k–ω model that uses the standard values of the k–ω closure coefficients. For the steady pipe flow, adaptation is driven by mean stream-wise velocity measurements at 24 locations along the pipe radius. The RCA-RANS k–ω model reduces the average velocity error at these locations by over 35%. For the unsteady flow over a square cylinder, adaptation is driven by time-varying surface pressure measurements at 2 locations on the square cylinder. The RCA-RANS k–ω model reduces the average surface-pressure error at these locations by 88.8%.

Visualization of Process of Wheel Steel High Ingots Simulation

The mathematical model for computation of formation of wheel steel high ingots has been formulated based on the generalized system of equations consisting of the Navier-Stokes equation, the turbulent heat and mass transfer equation and the continuity equation. It is suggested to use a pattern when designing software for simulation of hydrodynamic and thermo-physical processes.A software complex with friendly input and output data flows is provided for technologists of metallurgical production.

Physical and Mathematical Modeling of Metal-Slag Exchanges in Gas-Stirred Ladles

AbstractLadle refining plays a key role in achieving the quality of the steel since in this reactor temperature and chemical composition is adjusted, elimination of non-metallic inclusions is performed, and also deoxidation and desulphurization are operations taking place in the refining process. Specifically, the metal-slag mass exchanges have not received much attention through scientific studies. In this work, a rigorous study on the mass exchange between metal and slag is presented through a scaled water physical model. In the model, thymol (playing the role of a solute such as sulfur) is added to the water (playing the role of steel) and silicon oil (playing the role of slag) picks up the thymol, while the ladle is agitated with the central injection of gas. The evolution of thymol concentration in time was measured. Also, a mathematical model was developed and cast into the commercial CFD code Fluent Ansys to represent the fluid flow phenomena and the mass transfer through the solution of the continuity equation, the turbulent momentum conservation equations and the species mass conservation equation. There is a good agreement between the measured and the computed results regarding the thymol concentration evolution in water and consequently the mathematical model was validated regarding the mass species metal-slag exchanges and it may be used to study metal-slag exchanges in the steel ladle such as deoxidation or desulphurization.

Turbulent boundary layers absent mean shear

We perform an experimental study to investigate the turbulent boundary layer above a stationary solid glass bed in the absence of mean shear. High Reynolds number $(Re_{\unicode[STIX]{x1D706}}\sim 300)$ horizontally homogeneous isotropic turbulence is generated via randomly actuated synthetic jet arrays (RASJA – Variano & Cowen J. Fluid Mech. vol. 604, 2008, pp. 1–32). Each of the arrays is controlled by a spatio-temporally varying algorithm, which in turn minimizes the formation of secondary mean flows. One array consists of an $8\times 8$ grid of jets, while the other is a $16\times 16$ array. Particle image velocimetry measurements are used to study the isotropic turbulent region and the boundary layer formed beneath as the turbulence encounters a stationary wall. The flow is characterized with statistical metrics including the mean flow and turbulent velocities, turbulent kinetic energy, integral scales and the turbulent kinetic energy transport equation, which includes the energy dissipation rate, production and turbulent transport. The empirical constant in the Tennekes (J. Fluid Mech. vol. 67, 1975, pp. 561–567) model of Eulerian frequency spectra is calculated based on the dissipation results and temporal frequency spectra from acoustic Doppler velocimetry measurements. We compare our results to prior literature that addresses mean shear free turbulent boundary layer characterizations via grid-stirred tank experiments, moving-bed experiments, rapid-distortion theory and direct numerical simulations in a forced turbulent box. By varying the operational parameters of the randomly actuated synthetic jet array, we also find that we are able to control the turbulence levels, including integral length scales and dissipation rates, by changing the mean on-times in the jet algorithm.

Optimization of the wire electrode height and pitch for 3-D electrohydrodynamic enhanced water evaporation

In this study, numerical and experimental analyses were carried out to study the electrohydrodynamic (EHD) effect on the evaporation rate of a channel forced convection flow. Three-dimensional steady turbulent flow equations combined with Maxwell equations were solved. The optimization of the electrode height (H) and longitudinal pitch (SL) was investigated numerically along with an optimal strategy (SCGM, simplified conjugate-gradient method). The mass transfer gain per power consumption was taken as the objective function to be maximized. The results showed that the EHD effect on the evaporating rate increased with increases in applied voltage and electrode pitch (SL) and decreases in electrode height (H). For example, as air flow inlet velocity uin = 1 m/s and applied voltage V0 = 15 kV, the mass transfer enhancement was doubled for SL = 40–100 mm at H = 20 mm, while the mass transfer enhancement was 3.5 times greater for H = 30–15 mm at SL = 100 mm. In addition, the optimization analysis indicated that the mass transfer gain enhanced per Watt power consumption by 316.9–179.7% when combined with the optimal (H, SL) design ranging from V0 = 13 to 17 kV and uin = 1.0 m/s. The comparisons of numerical results and experimental data obtained satisfactory consistency within a discrepancy of 19%.