Fluid Dyn Research Articles

Local similarity of velocity distributions in homogeneous isotropic turbulence

Succeeding to the previous paper [Tatsumi and Yoshimura, 2004. Inertial similarity of velocity distributions in homogeneous isotropic turbulence. Fluid Dyn. Res. 35, 123–158] which dealt with the inertial similarity of the velocity distributions of homogeneous isotropic turbulence, the local similarity of the velocity distributions is investigated. The equations for the one- and two-point velocity distributions are expressed in the local dimensionless variables based on the mean energy-dissipation rate ε ¯ and the kinetic viscosity ν and solved by making use of the cross-independence closure hypothesis [Tatsumi, 2001. Mathematical physics of turbulence. In: Kambe, T., et al. (Eds.), Geometry and Statistics of Turbulence. Kluwer Academic Publishers, Dordrecht, pp. 3–12]. The velocity distributions are obtained as continuous solutions in the local variables which coincide with the inertial normal distributions out of the local similarity range. The one-point velocity distribution, which was given by the normal distribution N1 with the parameter α 0 under the inertial similarity, is expressed in the local variables as the distribution N1 but with the local parameter α 0 * ( = α 0 / ν ) . The two-point velocity distribution is expressed in terms of the velocity-sum distribution and the velocity-difference distribution as before. The velocity-sum distribution, which was given by the normal distribution N2 with the parameter α 0 / 2 under the inertial similarity, is obtained as the normal distribution N3 with the local parameter α + * ( r * ) which changes with the local distance r * ( = | r * | ) between the two points. Since α + * ( r * ) tends to α 0 * / 2 , corresponding to α 0 / 2 of N2, for r * → ∞ and to α 0 * , corresponding to α 0 of N1, for r * → 0 , the distribution N3 satisfies the boundary conditions at both ends of the local similarity range. The velocity-difference distribution, which was given by the isotropic distribution N2 under the inertial similarity, becomes axi-symmetric with respect to r * in the local range. The lateral distribution is obtained as the one-dimensional normal distribution N4 which satisfies the boundary conditions at both ends of the local range. The longitudinal distribution is obtained in three different similarity forms in the local range. It is expressed as the intermediate normal distribution N5 with the parameter α - 0 * ( r * ) in the intermediate subrange, as the algebraic distribution A1 with the same α - 0 * ( r * ) in the inertial subrange, and as the slightly asymmetric algebraic distribution A2 in the viscous subrange. The physical concepts of these results are discussed in comparison with existing experimental and numerical results.

The critical Reynolds number of a laminar incompressible mixing layer from minimal composite theory

According to parallel-flow theory based on the Orr–Sommerfeld equation, a mixing layer is unstable at all Reynolds numbers. However this is untenable from energy considerations, which demand that there exist a non-zero Reynolds number below which disturbances cannot extract net energy from the mean flow. It is shown here that a linear stability analysis of similarity solutions of the plane mixing layer, including the effects of flow non-parallelism using the minimal composite theory and the properties of adjoints, following Govindarajan & Narasimha (Theor. Comput. Fluid Dyn. vol. 19, 2005, p. 229) resolves the issue by yielding a non-zero critical Reynolds number for co-flowing streams of any velocity ratio. The critical Reynolds number for the total disturbance kinetic energy is found to be close to 30 for all velocity ratios in the range from zero to unity.

Vortex filaments in MHD

Two theorems on the Riemannian geometrical constraints on vortex magnetic filaments acting as dynamos in (MHD) flows are presented. The use of Gauss–Mainard–Codazzi equations allows us to investigate in detail the influence of curvature and torsion of vortex filaments in the MHD dynamos. This application follows closely previous applications to Heisenberg spin equation to the investigations in magnetohydrostatics given by Schief (2003 Plasma Phys. J. 10 2677). The Lorentz forces on vortex filaments are computed and the ratios between the forces along different directions are obtained in terms of the ratio between the corresponding magnetic fields which also equals the ratio between the Frenet torsion and vortex line curvature. A similar relation between Lorentz forces, magnetic fields and twist, which is proportional to total torsion integral, has been obtained by Ricca (2005 Fluid Dyn. Res. 36 319) in the case of inflexional disequilibrium of magnetic flux tubes. This is due to the fact that the magnetic vortex lines are a limiting case of the magnetic flux tubes when the length of the tube is much greater than the radius of the tube. The magnetic helicity equation of the filament allows us again to determine the magnetic fields ratio from Frenet curvature and torsion of the vortex lines. Recently, Schekochihin et al (2001 Phys. Rev. E 65 016305) obtained a similar relation between the ratios of magnetic field components by using a detailed analysis of the statistics of curvature. However, in their work no reference is made to torsion or helical vortex filaments.

Riemannian geometry of twisted magnetic flux tubes in almost helical plasma flows

Riemannian geometry of curves applied recently by Ricca [Fluid Dyn. Res 36, 319 (2005)] in the case of inflectional disequilibrium of twisted magnetic flux tubes is used here to compute the magnetic helicity force-free field case. Here the application of Lorentz force-free to the magnetic flux tube in tokamaks allows one to obtain an equation that generalizes the cylindrical tokamak equation by a term that contains the curvature of the magnetic flux tube. Another example of the use of the magnetic flux tube is done by taking the electron magnetohydrodynamics (MHD) fluid model (EMHD) of plasma physics that allows one to compute the velocity of the fluid in helical and almost helical flows in terms of the Frenet torsion of thin magnetic flux tubes. The cases of straight and curved twisted tubes are examined. Second-order effects on the Frenet torsion arise on the poloidal component of the magnetic field, while curvature effects appear in the toroidal component. The magnetic fields are computed in terms of the penetration depth used in superconductors. The ratio between poloidal and toroidal components of the magnetic field depends on the torsion and curvature of the magnetic flux tube. It is shown that the rotation of the almost helical plasma flow contributes to the twist of the magnetic flux tube through the total Frenet torsion along the tube.

Capillary breakup of liquid threads: a singularity-free solution

The process of capillary breakup of a thread of Newtonian liquid is considered theoretically in the simplest case where the thread is surrounded by an inviscid, dynamically passive gas. The goal is to remove the singularities inherent in the known solutions to the problem obtained in the framework of the standard model and explain some puzzling qualitative features of the process observed in experiments. The analysis is based on the idea that, since the known solutions indicate that the rate at which fresh free-surface area is created tends to infinity as breakup is approached, one has that the surface tension, whose relaxation to equilibrium is always associated with a small but finite relaxation time, is bound to deviate from its equilibrium value in the process of breakup. This gives rise to a surface-tension gradient which starts to pull the liquid thread apart (the flow-induced Marangoni effect), whilst the role of the capillary pressure-driven squeezing of the liquid out of the neck diminishes as the surface tension in the minimal cross-section decreases. An earlier developed theory incorporating the interface formation process is applied without any ad hoc alterations and analysed in the framework of the slender-jet approximation. The resulting solution is singularity-free and allows one to describe some previously unexplained features of experiment by Kowalewski (1996, Fluid Dyn. Res., 17, 121).

An optimal scale separation for a dynamo experiment

Scale separation between the flow and the magnetic field is a common feature of natural dynamos. It has also been used in the Karlsruhe sodium experiment in which the scale of the magnetic field is roughly seven times larger than the scale of the flow [R. Stieglitz and U. Müller, “Experimental demonstration of the homogeneous two-scale dynamo,” Phys. Fluids 13, 561 (2001)]. Recently, Fauve and Pétrélis [Peyresq Lectures on Nonlinear Phenomena, edited by J. Sepulchre (World Scientific, Singapore, 2003), p. 1] have shown that the power needed to reach the dynamo threshold in a dynamo experiment increases with the scale separation in the limit of large scale separation. With a more elaborate method based on subharmonic solutions [F. Plunian and K.-H. Rädler, “Subharmonic dynamo action in the Roberts flow,” Geophys. Astrophys. Fluid Dyn. 96, 115 (2002)], we show, for the Roberts flow, the existence of an optimal scale separation for which this power is minimum. Previous results obtained by Tilgner [“A kinematic dynamo with a small scale velocity field,” Phys. Lett. A 226, 75 (1997)] with a completely different numerical method are also reconsidered here. Again, we find an optimal scale separation in terms of minimum power for dynamo action. In addition we find that this scale separation compares very well with the one derived from the subharmonic solutions method.

Incompatibility of the exponential scaling law for a zero pressure gradient boundary layer flow with Reynolds averaged turbulence models

Analyzing the multipoint-correlation equations for parallel turbulent shear flows and zero pressure gradient turbulent boundary layer flows with the help of Lie symmetry methods, a new exponential scaling law has been derived in the work by Oberlack [“A unified approach for symmetries in plane parallel turbulent shear flows,” J. Fluid Mech. 427, 299 (2001)]. In the frame work of Lie group methods these solutions are called invariant solutions. From experiments [B. Lindgren, J. M. Österlund, and A. Johansson, “Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressuere-gradient turbulent boundary layers,” J. Fluid Mech. 502, 127 (2004)] and direct numerical simulations [G. Khujadze and M. Oberlack, “DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow,” Theor. Comput. Fluid Dyn. 18, 5 (2004)] the exponential velocity profile was clearly validated in the mid-wake region of high Reynolds number flat-plate boundary layers. It was identified as an explicit analytic form of the velocity defect law. Implementing the latter invariant solution into various Reynolds stress models it was found that none of the investigated models are in accordance with the exponential velocity law.

Two-dimensional compressible multimaterial flow calculations in a unified coordinate system

We extend the unified coordinate system method introduced in [J Comput Phys 153 (1999) 596, Comput Fluid Dyn J 8 (2000) 495] for single-fluid computations to multimaterial fluid computations by incorporating a γ-model scheme [P.L. Roe, A new approach to computing discontinuous flow of several ideal gases, Technical Report, (Cranfield Institute of Technology) 1984, J Comput Phys 169 (2001) 594], and propose therefore a γ-model unified coordinate system method, in which interfaces of multifluids are described by contact discontinuities of a conservation equation, i.e., the γ-model that is added into the Euler equations, forming the extended Euler equations (of a single fluid). The Godunov/MUSCL method is then used to solve the extended Euler equations numerically. Numerical examples validate the scheme in avoiding disadvantages of excessive numerical diffusion across interfaces in the Eulerian description and of severe mesh deformation in the Lagrangian description which may result in numerical inaccuracy and even break down computations.

An adaptive multilevel wavelet collocation method for elliptic problems

An adaptive multilevel wavelet collocation method for solving multi-dimensional elliptic problems with localized structures is described. The method is based on multi-dimensional second generation wavelets, and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems [Int. J. Comp. Fluid Dyn. 17 (2003) 151]. Wavelet decomposition is used for grid adaptation and interpolation, while a hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid. In order to accelerate the convergence of the solver, an iterative procedure analogous to the multigrid algorithm is developed. The overall computational complexity of the solver is O ( N ) , where N is the number of adapted grid points. The accuracy and computational efficiency of the method are demonstrated for the solution of two- and three-dimensional elliptic test problems.

Implementation of a conditional moment closure for mixing sensitive reactions

An implementation of the conditional moment closure (CMC) of Bilger (Phy. Fluids A 5 (1993) 436) and Klimenko (Fluid Dyn. 25 (1990) 327) has been carried out in the commercial CFD software FLUENT. A new implementation method is suggested that uses the CMC-equations to calculate an unconditional reaction rate that is subsequently inserted in a Reynolds averaged species equation. The objectives are to ensure conservation of reacting species by elimination of the spurious term that usually contaminates a CMC solution and to obtain convergence on a coarser grid than is usually required in CMC. The implementation has been successfully carried out for two sets of well-known mixing sensitive reaction pairs (parallel competitive and competitive consecutive) that both have one instantaneous and one fast irreversible reaction. Calculations with different discretizations of the CMC-equations in mixture fraction space have been performed and compared to experiments, full/finite-mode PDF models and the interpolation model of Baldyga. It is shown that the suggested implementation can achieve sufficient numerical accuracy with a much coarser mixture fraction space discretization than standard CMC implementations, which allows for more efficient CFD calculations.

Statistical Mechanics of Turbulent Flows

Statistical Mechanics of Turbulent Flows

Numerical modeling of buoyancy-driven turbulent flows in enclosures

Modeling turbulent natural convection in enclosures with differentially heated vertical walls is numerically challenging, in particular, when low-Reynolds-number (low- Re) models are adopted. When the turbulence level in the core region of cavity is low, most low- Re models, particular those showing good performance for bypass transitional flows, tend to relaminarize the flow and, as a consequence, significantly underpredict the near-wall turbulence intensities and boundary-layer thickness. Another challenge associated with low-turbulence buoyancy-driven flows in enclosures is its inherent unsteadiness, which can pose convergence problems when a steady Reynolds-averaged Navier–Stokes (RANS) equation is solved. In the present study, an unsteady RANS approach in conjunction with the low- Re k– ϵ model of Lien and Leschziner [Int. J. Comput. Fluid Dyn. 12 (1999) 1] is initially adopted and the predicted flow field is found effectively relaminarized. To overcome this difficulty, likely caused by the low- Re functions in the ϵ-equation, the two-layer approach is attempted, in which ϵ is prescribed algebraically using the one-equation k– l model of Wolfshtein [Int. J. Heat Mass Transfer 12 (1969) 301]. The two-layer approach combined with a quadratic stress–strain relation gives overall the best performance in terms of mean velocities, temperature and turbulence quantities.

On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method

We present a general framework to design Godunov-type schemes for multidimensional ideal magnetohydrodynamic (MHD) systems, having the divergence-free relation and the related properties of the magnetic field B as built-in conditions. Our approach mostly relies on the constrained transport (CT) discretization technique for the magnetic field components, originally developed for the linear induction equation, which assures [ ∇· B ] num =0 and its preservation in time to within machine accuracy in a finite-volume setting. We show that the CT formalism, when fully exploited, can be used as a general guideline to design the reconstruction procedures of the B vector field, to adapt standard upwind procedures for the momentum and energy equations, avoiding the onset of numerical monopoles of O(1) size, and to formulate approximate Riemann solvers for the induction equation. This general framework will be named here upwind constrained transport (UCT). To demonstrate the versatility of our method, we apply it to a variety of schemes, which are finally validated numerically and compared: a novel implementation for the MHD case of the second-order Roe-type positive scheme by Liu and Lax [J. Comput. Fluid Dyn. 5 (1996) 133], and both the second- and third-order versions of a central-type MHD scheme presented by Londrillo and Del Zanna [Astrophys. J. 530 (2000) 508], where the basic UCT strategies have been first outlined.

Dynamics of turbulence strongly influenced by buoyancy

The dynamics of quasi-horizontal motions in a stably stratified fluid have been simulated for Froude numbers of order 1, so that the flows are strongly affected by the stable density stratification, and for a range of Reynolds numbers. It is found that the horizontal scales of the motion grow continuously in time. The vertical scales decrease and the vertical shearing increases with time, maintaining the Richardson number of order 1, as suggested by Lilly [J. Atmos. Sci. 40, 749 (1983)] and Babin et al. [Theor. Comput. Fluid Dyn. 9, 223 (1997)]. Small-scale instabilities and turbulent-like motions are observed to occur in the high shearing regions, while the larger-scale motions appear to evolve somewhat independently of the Reynolds number. The results suggest that the larger-scale, quasi-horizontal motions would be a continuous source of smaller-scale turbulence until the local Reynolds number drops below a critical value, which is estimated. Finally, a Froude number based upon a vertical differential scale and used in previous scaling arguments and theories is estimated in terms of other parameters.

Symmetry breaking of flow and heat transfer in multiple impinging jets

This paper focuses on the effect of nozzle arrangement on the heat transfer of multiple impinging jets. Two different geometrical arrangements with an equal number of nozzles are studied, a square set-up (3 × 3 regularly spaced jets) and a circular set-up (eight jets surrounding a central jet). The predictions with two turbulence models are compared, the standard k– ε model with wall functions and Durbin’s v 2 – f model [Theor. Comput. Fluid Dyn. 3 (1991) 1] with integration to the wall. Quite surprisingly we find for one of the arrangements––the square set-up––an asymmetric flow field, even though the geometry, as well as the initial and boundary conditions are perfectly symmetric. Both turbulence models predict this peculiar broken symmetry. The other arrangement––the circular set-up––behaves more in line of expectation: both turbulence models predict a flow field that reflects all the symmetries of the physical situation. Interestingly, the differences between the two arrangements are quite small, but the corresponding flow fields clearly exhibit marked differences. Although both turbulence models agree qualitatively quite well on the mean features of the flow field for the two different set-ups, there are significant quantitative differences between the models regarding the turbulent kinetic energy and heat transfer. In particular the peak values of both the kinetic energy and the Nusselt number are quite different. Finally, we make a detailed comparison between the resulting heat transfer of the different arrangements. In the circular set-up the individual jets have a similar heat transfer, except for the central jet which is stabilized by the outer jets. In the square set-up, the difference between the jets is much more pronounced: due to the aforementioned asymmetry, some jets impinge vigorously, with a high heat transfer at the expense of the other, deflected, jets.

Two-way coupling simulations of instabilities in a plane bubble plume

In the present study we aim at investigating the instabilities in a plane bubble plume by means of two-way coupling simulations. The continuous phase motion is obtained by direct numerical solution of the Navier–Stokes equations forced by the presence of the bubbles. The collective effects induced by the presence of the bubbles are modeled by a spatiotemporal distribution of momentum. Time evolution of the dispersed phase is solved by Lagrangian tracking of all the bubbles. In the present study, the motion of the carrying fluid is initiated and driven by the induced buoyancy of bubbles released from a source located in an initially quiescent fluid layer. A quantitative analysis of the flow transition is thus investigated for several plume widths and for various fluid viscosities over a range of Grashof numbers based on the injection conditions. An analogy is drawn with buoyant single-phase flows for the steady laminar region. Following the similarity formulation of Fujii [Int. J. Heat Mass Transfer 6, 597 (1963)] under boundary layer approximations for free thermal plumes, the velocity profiles can be collapsed to a single self-similar plot. Nevertheless, this analogy with single-phase flow shows some discrepancies in the description of the transition. Numerical data emphasize that the key parameter controlling the height of transition is the Grashof number, which is based on injection conditions of the dispersed phase. Our results concur with the recent experiments of Alam and Arakeri [J. Fluid Mech. 254, 363 (1993)]. Although the Grashof number also determines the transition in thermal plumes [Wakitani and Yosinobu, Fluid Dyn. Res. 2, 363 (1988)], the two-phase configuration is more unstable. These new results underline the important role played by the slip velocity of the bubbles in plume stability. Indeed, it tends to delay the plume transition when the slip velocity increases and approaches the buoyancy-induced velocity. This feature should also be related to the lack of diffusion by the bubble cloud in the Lagrangian transport of the density gradient.

Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields

The Hamiltonian structure of the inhomogeneous layer models for geophysical fluid dynamics devised by Ripa [Geophys. Astrophys. Fluid Dyn. 70, 85 (1993)] involves the same Poisson bracket as a Hamiltonian formulation of shallow water magnetohydrodynamics in velocity, height, and magnetic flux function variables. This Poisson bracket becomes the Lie–Poisson bracket for a semidirect product Lie algebra under a change of variables, giving a simple and direct proof of the Jacobi identity in place of Ripa’s long outline proof. The same bracket has appeared before in compressible and relativistic magnetohydrodynamics. The Hamiltonian is the integral of the three dimensional energy density for both the inhomogeneous layer and magnetohydrodynamic systems, which provides a compact derivation of Ripa’s models.

Development of a nonlinear near-wall turbulence model for turbulent flow and heat transfer

A new nonlinear near-wall turbulence model is developed on the basis of realizability constraints to predict turbulent flow and heat transfer in strongly nonequilibrium flows. The linear k– ε– f μ model of Park and Sung (Fluid Dyn. Res., 20 (1997) 97) is extended to a nonlinear formulation. The stress–strain relationship is derived from the Cayley–Hamilton theorem in a homogeneous flow. The ratio of production to dissipation ( P k/ ε) is employed to solve an algebraic equation of the strain dependent coefficients. A near-wall treatment is dealt with by reproducing the model coefficients from a modified strain variable. An improved explicit heat flux model is proposed with the aid of Cayley–Hamilton theorem, which includes the quadratic effects of flow deformations. The near-wall asymptotic behavior is incorporated by modifying the f λ function. Emphasis is placed on the model performance on the truncated strain terms. The model performance is shown to be generally satisfactory.

Fluid Dynamics: Theory, Computation, and Numerical Simulation

Fluid Dynamics: Theory, Computation, and Numerical Simulation

Erratum: Engineering Applications of Noncummutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups

Erratum: Engineering Applications of Noncummutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups