Assessment of turbulence modeling for gas flow in two-dimensional convergent–divergent rocket nozzle

In the present study, the turbulent gas flow dynamics in a two-dimensional convergent–divergent rocket nozzle is numerically predicted and the associated physical phenomena are investigated for various operating conditions. The nozzle is assumed to have impermeable and adiabatic walls with a flow straightener in the upstream side and is connected to a plenum surrounding the nozzle geometry and extended in the downstream direction. In this integrated component model, the inlet flow is assumed a two-dimensional, steady, compressible, turbulent and subsonic. The physics based mathematical model of the considered flow consists of conservation of mass, momentum and energy equations subject to appropriate boundary conditions as defined by the physical problem stated above. The system of the governing equations with turbulent effects is solved numerically using different turbulence models to demonstrate their numerical accuracy in predicting the characteristics of turbulent gas flow in such complex geometry. The performance of the different turbulence models adopted has been assessed by comparing the obtained results of the static wall pressure and the shock position with the available experimental and numerical data. The dimensionless shear stress at the nozzle wall and the separation point are also computed and the flow field is illustrated. The various implemented turbulence models have shown different behavior of the turbulent characteristics. However, the shear-stress transport (SST) k–ω model exhibits the best overall agreement with the experimental measurements. In general, the proposed numerical procedure applied in the present paper shows good capability in predicting the physical phenomena and the flow characteristics encountered in such kinds of complex turbulent flow.     

反映强流动曲率效应的非线性湍流模型

首先定性地分析了流线曲率效应对流场湍流结构的影响,然后以U型槽道流为典型算例,对多种湍流模型进行了评估．评估的模型包括:线性涡粘性模型,二阶和三阶非线性涡粘性模型,二阶显式代数应力模型和Reynolds应力模型．评估结果表明,性能良好的三阶非线性涡粘性模型,如黄于宁等人发展的HM模型以及CLS模型,可以较好地描述流线的曲率效应对湍流结构的影响,如凸曲率作用下内壁附近湍流强度的衰减和凹曲率作用下外壁附近湍流的增强,并且较好地确定了管道下游的分离点位置和分离泡长度,其预测的结果和实验符合较好,与Reynolds力模型的结果十分接近,因此可以较好地应用于具有曲率效应的工程湍流的计算．     

Numerical modeling of turbulence-induced interfacial instability in two-phase flow with moving interface

The present paper introduces a new interfacial marker-level set method (IMLS) which is coupled with the Reynolds averaged Navier–Stokes (RANS) equations to predict the turbulence-induced interfacial instability of two-phase flow with moving interface. The governing RANS equations for time-dependent, axisymmetric and incompressible two-phase flow are described in both phases and solved separately using the control volume approach on structured cell-centered collocated grids. The transition from one phase to another is performed through a consistent balance of kinematic and dynamic conditions on the interface separating the two phases. The topological changes of the interface are predicted by applying the level set approach. By fitting a number of interfacial markers on the intersection points of the computational grids with the interface, the interfacial stresses and consequently, the interfacial driving forces are easily estimated. Moreover, the normal interface velocity, calculated at the interfacial markers positions, can be extended to the higher dimensional level set function and used for the interface advection process. The performance of linear and non-linear two-equation k–ε turbulence models is investigated in the context of the considered two-phase flow impinging problem, where a turbulent gas jet impinging on a free liquid surface. The numerical results obtained are evaluated through the comparison with the available experimental and analytical data. The nonlinear turbulence model showed superiority in predicting the interface deformation resulting from turbulent normal stresses. However, both linear and nonlinear turbulence models showed a similar behavior in predicting the interface deformation due to turbulent tangential stresses. In general, the developed IMLS numerical method showed a remarkable capability in predicting the dynamics of the considered two-phase immiscible flow problems and therefore it can be applied to quite a number of interface stability problems.     

Performance of turbulence models for flows through rough pipes

The Reynolds-averaged Navier–Stokes (RANS) equations were solved along with turbulence models, namely k–ε, k–ω, Reynolds stress models (RSM), and filtered Navier–Stokes equations along with Large Eddy Simulation (LES) to study the fully-developed turbulent flows in circular pipes roughened by repeated square ribs with various spacings. Solutions of these flows were obtained using the commercial computational fluid dynamics (CFD) software Fluent. The numerical results were validated against experimental measurements and other numerical data published in literature. The performance of the turbulence models was compared and discussed. All the RANS models and LES model were observed to perform equally well in predicting the time-averaged flow statistics. However no instantaneous information can be obtained from the RANS results. Therefore, when a rough overview of the flow process in a pipe roughened by repeated ribs is needed, any one of the RANS models can be of value. On the other hand, the instantaneous as well as time-averaged flows could be studied with more insight using LES, albeit at a cost of CPU effort at least one order higher.     

Direct numerical simulation of turbulent non-Newtonian flow using OpenFOAM

Understanding transition and turbulence in the flow of shear-thinning non-Newtonian fluids remains substantially unresolved and additional research is required to develop better computational methods for wall-bounded turbulent flows of these fluids. Previous DNS studies of shear-thinning fluids mainly use purpose-built codes and simple geometries such as pipes and channels. However in practical application, the geometry of mixing vessels, pumps and other process equipment is far more complex, and more flexible computational methods are required. In this paper a general-purpose DNS approach for shear-thinning fluids is undertaken using the OpenFOAM CFD library. DNS of turbulent Newtonian and non-Newtonian flow in a pipe flow are conducted and the accuracy and efficiency of OpenFOAM are assessed against a validated high-order spectral element-Fourier DNS code – Semtex. The results show that OpenFOAM predicts the flow of shear-thinning fluids to be a little more transitional than the predictions from Semtex, with lower radial and azimuthal turbulence intensities and higher axial intensity. Despite this, the first and second order turbulence statistics differ by at most 16%, and usually much less. An assessment of the parallel scaling of OpenFOAM indicates that OpenFOAM scales very well for the CPUs from 8 to 512, but the intranode scalability is poor for less than 8CPUs. The present work shows that OpenFOAM can be used for DNS of shear-thinning fluids in the simple case of pipe flow, and suggests that more complex flows, where flow separation is often important, are likely to be simulated with accuracies that are acceptably good for engineering application.     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.     

Significance of plaque morphology in modifying flow characteristics in a diseased coronary artery: Numerical simulation using plaque measurements from intravascular ultrasound imaging

The paper deals with numerical investigation of the effect of plaque morphology on the flow characteristics in a diseased coronary artery using realistic plaque morphology. The morphological information of the lumen and the plaque is obtained from intravascular ultrasound imaging measurements of 42 patients performed at Cleveland Clinic Foundation, Ohio. For this data, study of Bhaganagar et al. (2010) [1] has revealed the stenosis for 42 patients can be categorized into four types – type I (peak-valley), type II (ascending), type III (descending), and type IV (diffuse). The aim of the present study is to isolate the effect of shape of the stenosis on the flow characteristics for a given degree of the stenosis. In this study, we conduct fluid dynamic simulations for the four stenosis types (type I–IV) and analyze the differences in the flow characteristics between these types. Finely refined tetrahedral mesh for the 3-D solid model of the artery with plaques has been generated. The 3-D steady flow simulations were performed using the turbulence (k–ε) model in a finite volume based computational fluid dynamics solver. The axial velocity, the radial velocity, turbulence kinetic energy and wall shear stress profiles of the plaque have been analyzed. From the axial and radial velocity profiles results the differences in the velocity patterns are significantly visible at proximal as well as distal to the throat, region of maximum stenosis. Turbulent kinetic energy and wall shear stress profiles have revealed significant differences in the vicinity of the plaque. Additional unsteady flow simulations have been performed to validate the hypothesis of the significance of plaque morphology in flow alterations in diseased coronary artery. The results revealed the importance of accounting for plaque morphology in addition to plaque height to accurately characterize the turbulent flow in a diseased coronary artery.     

Assessment of an Eulerian CFD model for prediction of dilute droplet dispersion in a turbulent jet

The turbulent dispersion of non-evaporating droplets in an axisymmetric round jet issuing from a nozzle is investigated both experimentally and theoretically. The experimental data set has a well-defined inlet boundary with low turbulence intensity at the nozzle exit, so that droplet dispersion is not affected by the transport of nozzle-generated fluctuating motion into the jet, and is influenced solely by turbulence in the gas phase produced in the shear layer of the jet. This data set is thus ideal for testing algebraic models of droplet fluctuating motion that assume local equilibrium with the turbulence in the gas phase. Moreover, the droplet flux measurements are sufficiently accurate that conservation of the total volume flow of the droplet phase has been demonstrated. A two-fluid turbulence modelling approach is adopted, which uses the k–ε turbulence model and a simple algebraic model that assumes local equilibrium to predict the fluid and droplet turbulent correlations, respectively. We have shown that the k–ε turbulence model lacks generality for predicting the spread of momentum in jets with and without a potential core. However, in general, the model predicts the radial dispersion of droplets in the considered turbulent jet with reasonable accuracy over a broad range of droplet sizes, once deficiencies in the k–ε turbulence model are taken into account.     

Numerical modelling of subcritical open channel flow using the K-ε turbulence model and the penalty function finite element technique

A numerical model has been developed that employs the penalty function finite element technique to solve the vertically averaged hydrodynamic and turbulence model equations for a water body using isoparametric elements. The full elliptic forms of the equations are solved, thereby allowing recirculating flows to be calculated. Alternative momentum dispersion and turbulence closure models are proposed and evaluated by comparing model predictions with experimental data for strongly curved subcritical open channel flow. The results of these simulations indicate that the depth-averaged two-equation k-ε turbulence model yields excellent agreement with experimental observations. In addition, it appears that neither the streamline curvature modification of the depth-averaged k-ε model, nor the momentum dispersion models based on the assumption of helicoidal flow in a curved channel, yield significant improvement in the present model predictions. Overall model predictions are found to be as good as those of a more complex and restricted three-dimensional model.     

Conservation laws of turbulence models

The conservation of mass, momentum, energy, helicity, and enstrophy in fluid flow are important because these quantities organize a flow, and characterize change in the flow's structure over time. In turbulent flow, conservation laws remain important in the inertial range of wave numbers, where viscous effects are negligible. It is in the inertial range where energy, helicity (3d), and enstrophy (2d) must be accurately cascaded for a turbulence model to be qualitatively correct. A first and necessary step for an accurate cascade is conservation; however, many turbulent flow simulations are based on turbulence models whose conservation properties are little explored and might be very different from those of the Navier-Stokes equations.We explore conservation laws and approximate conservation laws satisfied by LES turbulence models. For the Leray, Leray deconvolution, Bardina, and Nth order deconvolution models, we give exact or approximate laws for a model mass, momentum, energy, enstrophy and helicity. The possibility of cascades for model quantities is also discussed.     

Numerical study of turbulent diesel flow in a pipe with sudden expansion

Three two-equation models and a second-moment closure are implemented in the case of turbulent diesel flow in a pipe with sudden expansion. The chosen two-equation closures are: the standard k–ε, the RNG k–ε and the two-scale k–ε models. The performance of the models is investigated with regard to the non-equilibrium parameter η and the mean strain of the flow, S. Velocity and turbulence kinetic energy predictions of the different models are compared among themselves and with experimental data and are interpreted on the basis of the aforementioned quantities. The effect of more accurate near-wall modeling to the two-equation models is also investigated. The results of the study demonstrate the superiority of the second-moment closure in predicting the flow characteristics over the entire domain. From the two-equation models the RNG derived k–ε model also gave very good predictions, especially when non-equilibrium wall-functions were implemented. As far as η and S are concerned, only the closures with greater physical consistency, such as the two-scale k–ε model, give satisfactory results.     

Statistical modeling of compressible turbulent channel flow

Several questions that are relevant to turbulence modeling are addressed on the basis of recently obtained direct numerical simulation results of turbulent supersonic channel flow. In particular, this concerns the turbulence frequency production mechanism, wall damping effects on turbulence model parameters, and the relevance of compressibility effects. Limited support is found for usually applied models for the turbulence frequency production and wall damping effects. In contrast to that it is shown that turbulence frequency production mechanisms and wall damping effects may be explained very well on the basis of a frequency scaling that characterizes mean flow changes. The influence of compressibility is found to be relevant. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study of the effect of changing the cyclone cone length on the gas flow field

Models of six different cyclone separator designs have been investigated numerically to study the effect of changing cone length on the gas flow field. The cone length was changed in two ways, one by fixing the cone diameter and the other by fixing the cone angle. The Reynolds stress model was applied to predict the turbulence of gas flow. The model was validated by comparing the numerical results to published experimental data. The results show that, for the cyclone without cylindrical part, the pressure drop and the value of tangential and axial velocity components increase with increasing cone length with fixed cone angle until they reach maximum value. It was found that increasing the cone length with the same cone diameter increases the back flow and the lip leakage below the vortex finder. The back flow and the lip leakage phenomena was observed to be reduced by increasing cone length with fixed cone angle. The effect of contraction ratio D_x/D, which is defined as the ratio between the vortex finder diameter and the cone diameter, on the turbulent kinetic energy and the velocity components also was investigated. As the contraction ratio decreases, the maximum value of the turbulent kinetic energy and the tangential and axial velocity components increases. The pressure drop was fitted into a correlation as a function of dimensionless quantities (Re, D_x/D, and l_c/D) for each design, so that the designers could select the appropriate dimensions according to their design limitations.     

Numerical investigation of the effect of helix angle and leaf margin on the flow pattern and the performance of the axial flow cyclone separator

A two-stage turbulence model based on the RNG κ–ε model combined with the Reynolds stress model is developed in this paper to analyze the gas flow in an axial flow cyclone separator. Five representative simulation cases are obtained by changing the helix angle and leaf margins of the cyclone. The pressure field and velocity field of the five cases are simulated, and then the effects of helix angle and leaf margins on the internal flow field of the cyclone are analyzed. When the continuum fluid (air) flow is relatively convergent, the discrete particle phase is added into the continuous phase and the gas-solid two-phase flow is simulated. One-way coupling method is used to solve the two-phase flow and a stochastic trajectory model is implemented for simulation of the particle phase. Finally, the pressure drop and separation efficiency of one case are measured and compare quantitatively well with the numerical results, which validates the reliability and accuracy of the simulation method based on the two-stage turbulence model.     

Numerical modelling of aeroelastic behaviour of an airfoil in viscous incompressible flow

In this paper, the problem of the numerical approximation of a two-dimensional incompressible viscous fluid flow interacting with a flexible structure is considered. Due to high Reynolds numbers in the range 10⁴ − 10⁶ the turbulent character of the flow is considered and modelled with the aid of Reynolds equations coupled with the k − ω turbulence model. The structure motion is described by a system of ordinary differential equations for three degrees of freedom: vertical displacement, rotation and rotation of the aileron. The problem is discretized in space by the Galerkin Least-Squares stabilized finite element method and the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian method.     

Direct numerical simulation of turbulent non-Newtonian flow using a spectral element method

A spectral element—Fourier method (SEM) for Direct Numerical Simulation (DNS) of the turbulent flow of non-Newtonian fluids is described and the particular requirements for non-Newtonian rheology are discussed. The method is implemented in parallel using the MPI message passing kernel, and execution times scale somewhat less than linearly with the number of CPUs, however this is more than compensated by the improved simulation turn around times. The method is applied to the case of turbulent pipe flow, where simulation results for a shear-thinning (power law) fluid are compared to those of a yield stress (Herschel–Bulkley) fluid at the same generalised Reynolds number. It is seen that the yield stress significantly dampens turbulence intensities in the core of the flow where the quasi-laminar flow region there co-exists with a transitional wall zone. An additional simulation of the flow of blood in a channel is undertaken using a Carreau–Yasuda rheology model, and results compared to those of the one-equation Spalart-Allmaras RANS (Reynolds-Averaged Navier–Stokes) model. Agreement between the mean flow velocity profile predictions is seen to be good. Use of a DNS technique to study turbulence in non-Newtonian fluids shows great promise in understanding transition and turbulence in shear thinning, non-Newtonian flows.     

A geometric introduction to the two-loop renormalization group flow

The Ricci flow has been of fundamental importance in mathematics, most famously through its use as a tool for proving the Poincaré conjecture and Thurston’s geometrization conjecture. It has a parallel life in physics, arising as the first-order approximation of the renormalization group flow for the nonlinear sigma model of quantum field theory. There recently has been interest in the second-order approximation of this flow, called the RG-2 flow, which mathematically appears as a natural nonlinear deformation of the Ricci flow. A curvature flow arising from quantum field theory seems to us to capture the spirit of Yvonne Choquet-Bruhat’s extensive work in mathematical physics, and so in this commemorative article we give a geometric introduction to the RG-2 flow. A number of new results are presented as part of this narrative: short-time existence and uniqueness results in all dimensions if the sectional curvatures K _ij satisfy certain inequalities; the calculation of fixed points for n =  3 dimensions; a reformulation of constant curvature solutions in terms of the Lambert W function; a classification of the solutions that evolve only by homothety; an analogue for RG flow of the 2-dimensional Ricci flow solution known to mathematicians as the cigar soliton, and discussed in the physics literature as Witten’s black hole. We conclude with a list of open problems whose resolutions would substantially increase our understanding of the RG-2 flow both physically and mathematically.     

Turbulent flow around single concentric long capsule in a pipe

A numerical solution was developed for the equations governing the turbulent flow around single concentric long capsule in a pipe. First, a turbulence model was established for the concentric annulus between the capsule and the pipe to simulate the flow as axi-symmetric, two dimensional, steady flow without edge effect. Second, the same case was considered taking into account the edge effect. Finally, turbulence modelling was established to simulate the case as a three dimensional steady flow, with a view of investigating the validity of axi-symmetric flow assumption. Three different turbulence models were used: an algebraic model (Baldwin–Lomax model) and two types of two-equation models (k–ε and k–ω). Obtained results of pressure gradient along the capsule were compared with available experimental data to verify the used models. In addition, experimental data of the velocity profiles of other investigators were also used in this concern. The results predicted by the three different turbulence models were shown to agree well with the experimental data, though precision differed from one to another.     

Oscillations of a gas in an open-ended tube near resonance

The pressure as function of time was measured near resonance in different axial locations of an open-ended tube. Flow visualisation showed that transition to turbulence was not influenced by the strong disturbance of the open end, except in a region near the open end which had a length of about three particle displacements. The pressure readings were decomposed into the first, second and third harmonic and compared with two different theories. In one case, the linearized theory for the oscillating flow in a tube was fitted to the boundary conditions, the obvious one at the piston and a model at the open end. In the second case, the nonlinear theory of Chester [1] was used. Both theories assume a relation between pressure and velocity at the open end that contains two free constants. The constants were determined by comparing the amplitude of the first and the second harmonic ofone pressure measurement with the theoretical predictions. Once the constants are fixed, the pressurep(ωt, x/L) is completely determined. For weak nonlinear effects, the pressure is essentially determined by one constantα(=k ₂) and the second constantβ(=k ₁) loses its significance. For the range of parameters given there isα=0.825±0.015. A very good approximation of the pressure near resonance can therefore be calculated with the following simple boundary condition at the open end $$p_E = \frac{{4\alpha }}{{3\pi }}\rho \hat u_E u_E = 0.350 \rho \hat u_E u_E .$$ Both theories predict a resonance frequency slightly above the experimental one. Changing Levine and Schwingers [2], end correction from 0.6133R to 1R eliminates the discrepancy for all tube lengths. For the first harmonic the variation of the amplitude and the phase of the pressure signal withω andx is very well predicted by both theories. The nonlinear theory describes also the small second and third harmonics fairly well while the linear theory predicts only the correct order of magnitude of these higher harmonics. The constantα that determines the energy loss at the open end shows an apparent increase if the boundary layer on the tube wall becomes turbulent. This occurs for \(A = 2\hat u/\sqrt {v\omega } \geqq 550\) to 750 which is close to the value observed in a tube with a closed end.     

Turbulent Boundary Layer Under the Influence of Adverse Pressure Gradient

The paper deals with the experimental analysis of turbulent boundary layer at the flat plate for large value of Reynolds number equal Re_θ ≈︁ 3000. The adverse pressure gradient generated by curvature of the upper wall corresponded to the case of pressure variation in axial compressor. The fully developed structure of turbulence was achieved by proper triggering of the boundary layer. The mean and turbulent flow-fields were investigated with the use of hot-wire technique while mean and instantaneous pressure fields were examined with piezoelectric transducers. The scaling and turbulence structure of fully developed turbulent boundary layer under the influence of adverse pressure gradient revealed the more pronounced contribution of outer region to the downstream development of turbulent boundary layer. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)