Homogeneous Shear Flow Research Articles

Mass and momentum balance during particle migration in the pressure-driven flow of frictional non-Brownian suspensions

The transient shear-induced particle migration of frictional non-Brownian suspensions is studied using particle-resolved simulations. The numerical method – the fictitious domain method – is well suited to heterogeneous flows thanks to a frame-invariant formulation of the subgrid (lubrication) corrections that does not involve the ambient flow (Orsi et al., J. Comput. Phys., vol. 474, 2023, 111823). The paper aims to give an accurate quantitative picture of the mass and momentum balance during the flow. The various assumptions and local constitutive laws that together form the suspension balance model (SBM) are thoroughly examined. To this purpose, the various quantities of interest are locally averaged in space and time, and their profile across the channel is extensively studied, with specific attention to the time evolution of the different contributions, either hydrodynamic in nature or from contact interactions, to the shear and normal stresses. The latter, together with the velocity gradient in the wall-normal direction and the volume fraction profile, yield the local constitutive laws, which are compared with their counterpart obtained in homogeneous shear flow. A fair agreement is observed except in a layering area at the boundaries and at the very centre of the channel. In addition, the main assumption of the SBM, i.e. the local relation between the hydrodynamic force on the particles and the particle flux, is meticulously investigated. The hydrodynamic force is found to be mainly a drag, except in the lower range of the probed volume fractions, where a non-drag contribution is observed.

Transient behavior and steady-state rheology of dense frictional suspensions in pressure-driven channel flow

AbstractResults from particle-resolved Direct numerical simulations are presented for dense suspensions of frictional non-colloidal spheres in viscous pressure-driven channel flow. The bulk solid volume fraction varies between $$\phi _b=0.2$$ ϕ b = 0.2 and 0.6, and the Coulomb friction coefficient is either $$\mu _c = 0$$ μ c = 0 or 0.5. The main objectives are to unravel the influence of (1) $$\phi _b$$ ϕ b and $$\mu _c$$ μ c on the flow development time and of (2) heterogeneous shear on the steady-state suspension rheology. Starting from an initially homogeneous distribution, the particles show shear-induced migration toward the core until equilibrium is reached. The flow development time decays exponentially with increasing $$\phi _b/\Phi _R$$ ϕ b / Φ R , where $$\Phi _R$$ Φ R is a friction-dependent reference bulk concentration beyond which particle contacts cause a rapid increase in the particle stress. The steady-state rheology is studied by means of the ‘viscous’ and ‘frictional’ rheology frameworks. Excluding the central core and wall regions, the data for the local relative suspension viscosity collapse onto a single curve as function of the normalized local concentration $${\bar{\phi }}/\phi _m$$ ϕ ¯ / ϕ m , where $$\phi _m$$ ϕ m is the friction-dependent maximum flowable packing fraction. The frictional rheology shows ‘subyielding’ at low viscous number $$I_v$$ I v in the core region, where the macroscopic friction coefficient $$\mu $$ μ drops below the minimal value found for homogeneous shear flows. A modified frictional rheology model is presented that captures subyielding. Finally, a model is presented for $${\bar{\phi }}/\phi _{mp}$$ ϕ ¯ / ϕ mp as function of $$I_v$$ I v , where $$\phi _{mp}$$ ϕ mp is a modified maximum flowable packing fraction. It captures both ‘overcompaction’ in the core beyond $$\phi _{m}$$ ϕ m at high $$\phi _b$$ ϕ b and maximum core concentrations below $$\phi _m$$ ϕ m at lower $$\phi _b$$ ϕ b .

The motion of three-dimensional fractal aggregates in homogeneous shear flow

A model for the motion of aggregates in low Reynolds number flow has been established by utilizing the Stokesian dynamics and applying the quaternion as global attitude parameter. The model has been validated by the theorical solution of sphere doublet in shear flow, the simple helical aggregates, and the experiment on the settling of some specific objects in still water. The motion of fractal aggregates has been further studied, aiming to get better understanding of the dynamic behavior of fine-grained sediment flocs in shear flow. The fractal aggregates have been generated using the diffusion-limited aggregation model, which has similar fractal dimension as natural flocs. The results illustrate that fractal aggregates undergo a complex rotation in shear flow, which exhibit a bi-periodic characteristic. The motion of a particle within the fractal aggregate shows three-dimensional trajectory in a simple shear flow, affected by its initial orientation. The major rotation period is approximately 4π/γ̇, which corresponds to the rotation period of a sphere with shear strength γ̇. The deviation decreases with the increase in the size of the fractal aggregate.

Simulation of fiber-induced melt pressure fluctuations within large scale polymer composite deposition beads

The process-structure-property relationship in Large Area Additive Manufacturing (LAAM) technology is an ongoing area of research as the inherent microstructural properties (chiefly fibers and voids) affect the performance of printed parts. Unfortunately, we currently lack adequate understanding of micro void nucleation and evolution during the LAAM and fused deposition modelling (FDM) processes. Modeling of the polymer melt flow during the extrusion process is important in understanding the underlying microstructural formation and associated properties of the print, that determines the part performance in service conditions. In this paper we compute fiber-induced local pressure fluctuations which may promote void formation in the bead’s microstructure. On a macro-scale, we determine flow fields of a purely viscous, Newtonian planar polymer deposition flow through a LAAM nozzle which are utilized on a micro-scale model where we simulate the evolution of a single ellipsoidal fiber along streamlines obtained from the macro-model. On the micro-scale, we determine instantaneous values of the translational and rotational velocities of the rigid ellipsoidal fiber that satisfies a balance of hydrodynamic forces and couples on the fiber’s surface based on a Newton Raphson algorithm and we track the fiber’s motion along the flow path via an explicit numerical iterative algorithm. Model verification is achieved by benchmarking results with solutions from well-known Jeffery’s equation of motion of a particle in homogeneous simple shear flow. We account for rotary diffusivity due to short-range fiber-fiber interaction in the FEA simulation by determining an effective fluid domain size representative of the interaction coefficient of the melt flow through a correlation analysis that yields an equivalent steady state orientation based on the Advani-Tucker equation. We also consider different possible motions of the fiber along individual LAAM flow paths from a given set of random initial fiber conditions to determine pressure bounds on the fiber surface along each streamline. For improved computational efficiency, calculations are carried out with respect to the fiber’s local coordinate axes to overcome the rigor of adaptive remeshing during the quasi-transient analysis. Results show low pressure extremes near the fiber’s surface which varies across the printed bead as well as through its thickness. Discussion is provided to gain insight into the effect of low-pressure extremes on micro void formation, particularly at the nozzle exit and during die swell/expansion.

Reynolds-stress root modeling based on a statistical theory

A new methodology of turbulence modeling is proposed by combining a statistical theory [T. Ariki, Phys. Fluids 31, 065104 (2019)] and the Reynolds-stress-root method [T. Ariki, Phys. Rev. E 92, 053010 (2015)], aiming at realizing practical turbulence model of wider applicability with the help of theoretical support. The resultant model integrates, at the same time, the following five features: coordinate covariance, realizability condition, near-wall behavior, history effect, and streamline curvature effect, which are all key ingredients to build up better turbulence model mimicking realistic behaviors. Numerical assessments of the model are conducted for homogeneous shear flow, channel flow, flow in a rotating pipe, and flow between concentric annuli, all of which show reasonable agreement with direct numerical simulations and experiments.

A compressible turbulence model for the dissipation rate

In this work, the ability of a Reynolds stress model to compute turbulent homogeneous shear flow with significant compressibility effects is discussed. Several studies of compressible turbulent flows carried out in the past years have shown that the pressure strain correlation is mainly responsible for the strong changes in the magnitude of the Reynolds stress anisotropies. Two recent compressible models of this term are considered in conjunction with the standard model of the dissipation rate of the turbulent kinetic energy to predict compressible homogeneous flow highly sheared are tested. It is found that deficiencies appear in the calculations even if the pressure strain model is improved by compressibility corrections. Consistent with earlier studies, this deficiency is attributed to the use of the incompressible model for turbulent dissipation. However, a compressibility correction of this equation model uncovers the main focus of the present study. This correction makes the standard coefficients of this equation depend on the turbulent gradient Mach and Mach numbers. The proposed model is tested for low and strong compressibility cases from the DNS results of Sarkar. A comparison of the proposed model predictions with the DNS results shows good qualitative agreement. Therefore, compressibility correction for the incompressible model of the turbulent dissipation rate is found to be an important issue for the compressible homogeneous turbulent shear flow.

Turbulence in Two-Phase Flows with Macro-, Micro- and Nanoparticles: A Review

Turbulent flows are nonstationary in nature. Since the turbulent fluctuations of most flow parameters satisfy a symmetric Gaussian distribution, the turbulent characteristics have the property of symmetry in the statistical meaning. A widespread simplest model of turbulent flows is the model of “symmetric” turbulence, namely, homogeneous isotropic turbulence (HIT). The presence of particles with non-uniform distribution of their concentration in the turbulent flow, even under HIT conditions, can lead to redistribution of different components of fluctuation velocities of the carrier gas, i.e., to the appearance of asymmetry. The subject of the review is turbulent flows of gas with solid particles. Particular attention is paid to the problem of the back influence of particles on carrier gas characteristics (first of all, on the turbulent kinetic energy). A review of the results of experimental and computational-theoretical studies of the effect of the presence of the dispersed phase in the form of particles on the parameters of the turbulent flow of the carrier gas phase has been carried out. The main physical mechanisms and dimensionless criteria determining the direction and magnitude of the impact of particles of different inertia on the carrier gas phase turbulence energy are described and analyzed. The peculiarities of the influence of particles on the turbulence energy of the gas for different classes of flows: homogeneous isotropic turbulence, homogeneous shear flow, and wall turbulence in a pipe (channel) have been considered. For the near-wall flow in the pipe, it is shown that the turbulizing effect of extremely low-inertia particles of sub-micrometer size (nanoparticles) is replaced by the laminarizing effect of low-inertia particles of micrometer size (microparticles), and then again it is replaced by turbulizing due to additional generation of turbulence in the wakes of large particles of millimeter size (macroparticles). The review is intended to some extent to fill in the currently existing gap associated with the absence of dimensionless criteria (or complexes of physical parameters) responsible for the direction (attenuation or enhancement) of turbulence modification, and the value of this change. Possible directions for further researches are given in the conclusion of the review.

Wind turbine wake simulation with explicit algebraic Reynolds stress modeling

Abstract. Reynolds-averaged Navier–Stokes (RANS) simulations of wind turbine wakes are usually conducted with two-equation turbulence models based on the Boussinesq hypothesis; these are simple and robust but lack the capability of predicting various turbulence phenomena. Using the explicit algebraic Reynolds stress model (EARSM) of Wallin and Johansson (2000) can alleviate some of these deficiencies while still being numerically robust and only slightly more computationally expensive than the traditional two-equation models. The model implementation is verified with the homogeneous shear flow, half-channel flow, and square duct flow cases, and subsequently full three-dimensional wake simulations are run and analyzed. The results are compared with reference large-eddy simulation (LES) data, which show that the EARSM especially improves the prediction of turbulence anisotropy and turbulence intensity but that it also predicts less Gaussian wake profile shapes.

A Compressible Turbulence Model for Pressure—Strain

This work focuses on the performance and validation of compressible turbulence models for the pressure-strain correlation. Considering the Launder Reece and Rodi (LRR) incompressible model for the pressure-strain correlation, Adumitroaie et al., Huang et al., and Marzougui et al., used different modeling approaches to develop turbulence models, taking into account compressibility effects for this term. Two numerical coefficients are dependent on the turbulent Mach number, and all of the remaining coefficients conserve the same values as in the original LRR model. The models do not correctly predict the compressible turbulence at a high-speed shear flow. So, the revision of these models is the major aim of this study. In the present work, the compressible model for the pressure-strain correlation developed by Khlifi−Lili, involving the turbulent Mach number, the gradient, and the convective Mach numbers, is used to modify the linear mean shear strain and the slow terms of the previous models. The models are tested in two compressible turbulent flows: homogeneous shear flow and the newly developed plane mixing layers. The predicted results of the proposed modifications of the Adumitroaie et al., Huang et al., and Marzougui et al., models and of its universal versions are compared with direct numerical simulation (DNS) and experiment data. The results show that the important parameters of compressibility in homogeneous shear flow and in the mixing layers are well predicted by the proposal models.

A Comparative Study for Compressible Turbulence Models

The incompressible models for the pressure-strain correlation are unable to correctly predict the turbulence flows evolving with significant compressibility. Huang and Fu use a damping function of the turbulent Mach number to modify two numerical coefficients of the incompressible model for the pressure strain developed by Launder, Reece and Rodi. This model predicts the spreading rate and the shear stress behavior in compressible turbulent mixing well. However, the model does not show the well-known compressibility effects on the compressible homogenous shear flow. In the present work, the model of Huang-Fu is revised, all resulting model coefficients become dependent on the turbulent Mach number, the gradient Mach number and the convective Mach number. The proposed model is tested in different compressible turbulent homogeneous shear flow and mixing layers cases. In general, the predicted results from the proposed model are in an acceptable agreement with DNS and experiment data.

Shear-driven Hall-magnetohydrodynamic dynamos

Nonlinear Hall-magnetohydrodynamic dynamos associated with coherent structures in subcritical shear flows are investigated by using unstable invariant solutions. The dynamo solution found has a relatively simple structure, but it captures the features of the typical nonlinear structures seen in simulations, such as current sheets. As is well known, the Hall effect destroys the symmetry of the magnetohydrodynamic equations and thus modifies the structure of the current sheet and mean field of the solution. Depending on the strength of the Hall effect, the generation of the magnetic field changes in a complex manner. However, a too strong Hall effect always acts to suppress the magnetic field generation. The hydrodynamic/magnetic Reynolds number dependence of the critical ion skin depth at which the dynamos start to feel the Hall effect is of interest from an astrophysical point of view. An important consequence of the matched asymptotic expansion analysis of the solution is that the higher the Reynolds number, the smaller the Hall current affects the flow. We also briefly discuss how the above results for a relatively simple shear flow can be extended to more general flows such as infinite homogeneous shear flows and boundary layer flows. The analysis of the latter flows suggests that interestingly a strong induction of the generated magnetic field might occur when there is a background shear layer.

Return to isotropy of homogeneous shear-released turbulence

In homogeneous shear flow, turbulence exhibits anisotropic properties affecting all scales of motion at finite Reynolds numbers. Upon releasing the mean shear, the anisotropy characterizing the velocity field decays over a large eddy turnover time. The decay of the anisotropy of the vorticity field, however involves a range of time-scales, from the short (Kolmogorov) time scale, up to the large eddy turnover time.

An Explicit Algebraic Closure for Passive Scalar-Flux: Applications in Channel Flows at a Wide Range of Reynolds Numbers

In this paper, we propose an algebraic model for turbulent scalar-flux vector that stems from tensor representation theory. The resulting closure contains direct dependence on mean velocity gradients and quadratic products of the Reynolds stress tensor. Model coefficients are determined from Direct Numerical Simulations (DNS) data of homogeneous shear flows subjected to arbitrary mean scalar gradient orientations, while a correction function was applied at one model coefficient based on a turbulent channel flow case. Model performance is evaluated in Poiseuille and Couette flows at several Reynolds numbers for Pr=0.7, along with a case at a higher Prandtl number (Pr=7.0) that typically occurs in water–boundary interaction applications. Overall, the proposed model provides promising results for wide near-wall interaction applications. To put the performance of the proposed model into context, we compare with Younis algebraic model, which is known to provide reasonable predictions for several engineering flows.

Application of the exact regularized point particle method (ERPP) to bubble laden turbulent shear flows in the two-way coupling regime

Direct numerical simulations of a bubbly laden homogeneous shear flow have been carried out using the exact regularized point particle method as the inter-phase momentum coupling approach. The aim of this study consists in addressing the modulation of shear turbulence and the bubble clustering geometry in the presence of different inter-phase momentum coupling conditions. Suspensions with different combinations of the void fraction and Kolmogorov-based Stokes number, in the dilute regime, have been addressed. Bubbles suppress the turbulent kinetic energy and turbulent dissipation as well. Turbulent modulation occurs via the direct change of the Reynolds shear stress. In fact, the bubble energy source is proved to be negligible in the scale-by-scale turbulent energy budget. The bubble clustering, in agreement with the literature, occurs in the form of thin elongated structures. The clusters are aligned with the principal strain direction of the mean flow, as usual in shear flows. The bubble clustering and turbulent modification are strictly related: both increase with the Stokes number and are independent of the void fraction. The data show that the turbulent modification is disadvantaged when the bubble distribution is homogeneous (i.e., small Stokes number). Finally, the small-scale bubble clustering is slightly reduced by two-way coupling effects even though the clustering anisotropy still persists at small scales as it occurs for inertial particles.

Modelling phase transition in granular materials: From discontinuum to continuum

This work focuses on the behaviour of granular materials under unsteady, simple shear conditions and, in particular, on from solid- to fluid-like phase transition. The authors introduce a theoretical model, based on continuum mechanics, able of predicting the mechanical behaviour of granular media under both quasi-static and dynamic conditions. The model assumes a parallel scheme where confining and shear stresses are computed as the sum of two contributions: the quasi-static and the collisional one. The quasi-static contribution is obtained by employing an elastic–plastic model including the critical state concept, while the collisional one is derived from the kinetic theory of granular gases.In order to test the model under unsteady conditions, DEM numerical simulations of time evolving homogeneous shear flows have been performed by considering an assembly of frictional, deformable spheres, under constant volume conditions. Simulations have been performed by systematically changing both void ratio and shear rate. The comparison between theoretical model predictions and DEM results is done in terms of time evolution of stresses and granular temperature. Suitable initial conditions are imposed to reproduce both solid to fluid (liquefaction) and fluid to solid (solidification) phase transitions.

Flow and rheology of frictional elongated grains

The rheology of a 3-dimensional granular system consisting of frictional elongated particles was investigated by means of discrete element model calculations. A homogenous shear flow of frictional spherocyliders was simulated, and a number of rheological quantities were calculated. In the framework of the μ(I) rheology, the effective friction was found to be a non-monotonic function of the aspect ratio for interparticle friction coefficient μp ≲ 0.4, while it was an increasing function for larger μp. We reveal the microscopic origin of this peculiar non-monotonic behaviour. We show the non-trivial dependence of the velocity fluctuations on the dissipation regime, and trace back the behaviour of the normal stress differences to particle-level quantities.

Acceleration of small heavy particles in homogeneous shear flow: direct numerical simulation and stochastic modelling of under-resolved intermittent turbulence

The acceleration of inertial particles in a homogeneous shear flow may vigorously respond to the intense flow structures induced by the mean shear. In this study, by direct numerical simulation (DNS) of particle-laden shear flow, we observe the statistical properties of those accelerations, and then we assessed the recently proposed simulation approach in which the effect of intermittency on residual scales is linked directly with coarsely resolved flow turbulence. First, we focused on the acceleration statistics of fluid particles in a homogeneous shear flow. Consistent with previous findings in homogenous isotropic turbulence, the norm and the direction of fluid acceleration in shear flow are shown to be conditioned by the dynamics of intermittent long-lived vortical structures. The averaged acceleration norm of the fluid particle exhibits a pseudo-cyclic behaviour, which is a signature of the periodic action of the largest confined vortices against the mean shear. The long correlation time of the acceleration norm, of the order of the integral time, the high level of fluctuations of the acceleration norm (very close to the magnitude of the mean norm of acceleration) and the log-normality in its statistical distribution reflect the impact of intense flow structures in shear flow. The presence of these zones results also in highly non-Gaussian statistics of the acceleration of a fluid particle and its velocity increments at small time lags. Contrary to the acceleration norm, the acceleration direction of a fluid particle is observed to be short, of the order of the Kolmogorov time, and to be statistically independent of the acceleration norm. The short-time correlation of the acceleration direction is attributed usually to effects of centripetal forces in intense vorticity filaments. We suggest that, in homogeneous shear turbulence, there may be a supplementary effect on the acceleration direction: the vortex, stretched by the mean shear, may exert the preferential direction of fluid particle acceleration. As evidence, our DNS shows that fluid particles are accelerated preferentially in the direction of longitudinal vortical tubes, effectively stretched by the imposed mean shear. Concerning simulations with heavy point-wise particles, it is shown that, when the inertia of a particle is not high, its acceleration closely follows all the aforementioned properties of the fluid particle acceleration. Particularly, an inertial particle is also entrained by accelerating motion in the direction of the effectively stretched vortical tubes. Although with increasing Stokes number of the inertial particle, the effects of strong intermittency in the flow are filtered, it is shown that the alignment of the particle acceleration direction with vortical tubes is amplified – particles with higher inertia respond to solicitations of stronger vortical structures. An alternative to the Maxey (J. Fluid Mech., vol. 174, 1987, pp. 441–465) preferential sweeping mechanism is discussed in this paper. When the shear turbulence is under-resolved, we employed the large-eddy simulation (LES) equations with a forcing term on the smallest resolved scales in order to simulate stochastically the effects of the dynamics on the residual scales. The forcing term is expressed with two independent stochastic processes, one for its norm and another for its direction. While the norm of acceleration is modelled using Pope’s log-normal process with the integral time for correlation, its direction is modelled in the framework of an Ornstein–Uhlenbeck process on the unit sphere. Consistently with our DNS, the latter process contains two presumed times: the homogeneous strain rate is specified as typical time of relaxation towards the direction of resolved vorticity, and the Kolmogorov time is presumed as a typical time of the diffusion process on the unit sphere. The high efficiency of this approach is demonstrated in prediction of the small-scale dynamics observed in DNS, even in the case when the shear length scale is not resolved by LES.

Evolution of wet agglomerates inside inertial shear flow of dry granular materials.

We use particle dynamics simulations to investigate the evolution of a wet agglomerate inside homogeneous shear flows of dry particles. The agglomerate is modeled by introducing approximate analytical expressions of capillary and viscous forces between particles in addition to frictional contacts. During shear flow, the agglomerate may elongate, break, or be eroded by loss of its capillary bonds and primary particles. By systematically varying the shear rate and surface tension of the binding liquid, we characterize the rates of these dispersion modes. All the rates increase with increasing inertial number of the flow and decreasing cohesion index of the agglomerate. We show that the data points for each mode collapse on a master curve for a dimensionless scaling parameter that combines the inertial number and the cohesion index. The erosion rate vanishes below a cutoff value of the scaling parameter. This leads to a power-law borderline between the vanishing erosion states and erosion states in the phase space defined by the inertial number and the cohesion index.

Time Correlation Functions of Equilibrium and Nonequilibrium Langevin Dynamics: Derivations and Numerics Using Random Numbers

We study the time correlation functions of coupled linear Langevin dynamics with and without inertial effects, both analytically and numerically. The model equation represents the physical behavior of a harmonic oscillator in two or three dimensions in the presence of friction, additive noise, and an external field with both rotational and deformational components. This simple model plays pivotal roles in understanding more complicated processes. The analytical solution presented serves as a test of numerical integration schemes, and its derivation is presented in a fashion that allows it to be repeated directly in a classroom. While the results in the absence of fields (equilibrium) or confinement (free particle) are omnipresent in the literature, we write down, apparently for the first time, the full nonequilibrium results that may correspond, e.g., to a Hookean dumbbell embedded in a macroscopically homogeneous shear or mixed flow field. We demonstrate how the inertial results reduce to their noninertial counterparts in the nontrivial limit of vanishing mass. While the results are derived using basic integrations over Dirac delta distributions, we also provide alternative approaches involving (i) Fourier transforms, which seem advantageous only if the measured quantities also reside in Fourier space, and (ii) a Fokker--Planck equation and the moments of the probability distribution. The results, verified by numerical experiments, provide additional means of measuring the performance of numerical methods for such systems. It should be emphasized that this article provides specific details regarding the derivations of the time correlation functions as well as the implementations of various numerical methods, so that it can serve as a standalone piece for lessons in the framework of Itô stochastic differential equations and calculus.

Higher-order realizable algebraic Reynolds stress modeling based on the square root tensor

In this study, realizable algebraic Reynolds stress modeling based on the square root tensor [Phys. Rev. E \textbf{92}, 053010 (2015)] is further developed for extending its applicability to more complex flows. In conventional methods, it was difficult to construct an algebraic Reynolds stress model satisfying the realizability conditions when the model involves higher-order nonlinear terms on the mean velocity gradient. Such higher-order nonlinear terms are required to predict turbulent flows with three-dimensional mean velocity. The present modeling based on the square root tensor enables us to make the model always satisfy the realizability conditions, even when it involves higher-order nonlinearity. To construct a realizable algebraic Reynolds stress model applicable to turbulent flows with three-dimensional mean velocity, a quartic-nonlinear eddy-viscosity model is proposed. The performance of the model is numerically verified in a turbulent channel flow, a homogeneous turbulent shear flow, and an axially rotating turbulent pipe flow. The present model gives a good result in each turbulent flow. Note that the mean swirl flow in an axially rotating turbulent pipe flow is reproduced because the present model involves cubic nonlinearity. Such a higher-order realizable algebraic Reynolds stress model, involving quartic nonlinearity on the mean velocity, is expected to be useful in numerically stable predictions of turbulent flows with three-dimensional mean velocity.