A mixing-length model for magnetohydrodynamic flows in channels and ducts with wall-parallel magnetic field

A spanwise magnetic field leads to turbulent drag reduction in channel flow of a conducting liquid due to the selective Joule damping of certain flow structures. This effect can be captured by a simple modification of Prandtl's classical mixing-length idea. The mixing length over which a turbulent fluctuation loses its momentum is not only constrained geometrically but also by magnetic damping. We therefore introduce a magnetic damping length that is proportional to friction velocity and the Joule damping time. The limitation of mixing length is implemented by using the harmonic mean between wall distance and this damping length. By combining this ansatz with the van-Driest model for turbulent stresses in channel flow we obtain a satisfactory prediction for the mean velocity distribution in magnetohydrodynamic channel flow with spanwise field for different Reynolds and Hartmann numbers. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation and analysis of turbulent MHD channel flow with a streamwise magnetic field

We perform large-eddy simulations of turbulent MHD channel flow with a streamwise magnetic field using a pseudo spectral method. The streamwise magnetic field leads to turbulent drag reduction due to the selective Joule damping of certain flow structures. Near the walls, the turbulent mean velocity profile retains the logarithmic layer but the von Karman constant decreases with increasing magnetic field strength. In the outer region, the flow is characterized by persistent streaky structures of large streamwise extent, which lead to a rather flat mean velocity profile. In addition, the streamwise velocity fluctuations develop a pronounced second peak upon increasing the magnetic induction as well as a second logarithmic layer that increases in steepness. We find that Prandtl's classical mixing-length model with a variable Kármán constant can describe the modified logarithmic layer reasonably accurately in a wide range of Reynolds and Hartmann numbers. However, the flow modification near the center of the channel is not properly captured by this approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A theoretical model for Reynolds-stress and dissipation-rate budgets in near-wall region

A 3-D wave model for the turbulent coherent structures in near-wall region is proposed. The transport nature of the Reynolds stresses and dissipation rate of the turbulence kinetic energy are shown via computation based on the theoretical model. The mean velocity profile is also computed by using the same theoretical model. The theoretical results are in good agreement with those found from DNS, indicating that the theoretical model proposed can correctly describe the physical mechanism of turbulence in near wall region and it thus possibly opens a new way for turbulence modeling in this region.     

Asymptotic study of turbulent Poiseuille flow with wall transpiration

An incompressible, pressure–driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary–value problem for a second–order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. A limiting transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one–parameter family of curves. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High-Reynolds-number effects on turbulent scalings in compressible channel flow

The effect of the Reynolds number in a supersonic isothermal channel flow is studied using a direct numerical simulation (DNS). The bulk Mach number based on the wall temperature is 1.5, and the bulk Reynolds number is increased up to Re_τ ≈︁ 1000. The use of van Driest velocity transformation in the presence of heated walls has been questioned due to the poor accuracy at low Reynolds number. For this reason alternative transformations of the velocity profile and turbulence statistics have been proposed, as, for instance, semi-local scalings. We show that the van Driest transformation recovers its accuracy as the Reynolds number is increased. The Reynolds stresses collapse on the incompressible ones, when properly scaled with density, and very good agreement with the incompressible stresses is found in the outer layer. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fluid flow in an inhomogeneous granular medium

The seepage of a compressible fluid in an inhomogeneous undeformable granular medium is investigated. It is assumed that the fluid flow in a porous space is described by the Navier–Stokes equations. It is shown that, in the case of an inhomogeneous velocity field, a tensor of additional effective stresses occurs in connection with the transfer of fluid particles in a transverse direction when flow occurs around the granules of the medium in a longitudinal direction. Using the fundamental propositions of Reynolds’ averaging theory and Prandtl's mixing path, the structure of the effective viscosity coefficient is determined and hypotheses are formulated which enable it to be assumed to be independent of the flow velocity. It is established by comparison with experimental data that the effective viscosity coefficient can exceed the viscosity coefficient of the flowing fluid by an order of magnitude. The equations of average motion are obtained, which in the case of an incompressible fluid have the form of the Navier–Stokes equations with body forces proportional to the velocity. It is established that, in addition to the well-known dimensionless flow numbers, there is a new number which characterizes the ratio of the Darcy porous drag forces to the effective viscosity forces. The proposed equations are extended to the case of the flow of an aerated fluid. The components of the angular momentum vector are used as the required functions instead of the components of the velocity vector. This enables a solving system of equations to be obtained, which, apart from the notation, is identical with the similar equations for the case of an incompressible fluid. The solution of a new problem of the fluid flow in a plane channel with permeable walls is presented using three models: Darcy's law for an incompressible and aerated fluid, and also of an aerated fluid taking the effective viscosity into account. It is established that, for the same pressure drop, the maximum flow rate corresponds to Darcy's law. Compressibility leads to its reduction, but by simultaneously taking into account the compressibility and the effective viscosity one obtains minimum values of the flow rate. The effective viscosity and aeration of the fluid has a considerable effect on the flow parameters.     

Turbulent Poiseuille flow with near-critical wall transpiration

An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress with the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary-value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. The case of near-critical transpiration, when the shear stress at the injection wall vanishes, is considered. It is shown that the maximum point on the mean velocity profile lies in a thin sublayer near the suction wall in this case. A formula for the position of the maximum point as a function of the transpiration factor is obtained. The mean velocity profiles near the suction wall are calculated. A friction law for Poiseuille flow with near-critical transpiration is found, which makes it possible to describe the relation between the shear stress at the wall, the Reynolds number, and the transpiration velocity by a single function of one variable. Direct numerical simulation of the flow for some transpiration factors is performed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of constriction height on flow separation in a two-dimensional channel

We studied numerically the effect of the constriction height on viscous flow separation past a two-dimensional channel with locally symmetric constrictions. A numerically stable scheme in primitive variables (velocity and pressure) for the solution of two-dimensional incompressible time-dependent Navier–Stokes equations is employed using finite-difference approximation in staggered grid. The wall shear stresses at different heights of the constriction are computed and presented graphically. It is noticed that the maximum stress and the length of the recirculating region associated with two shear layers of the constriction increase with the increase of the area reduction of the constriction. The critical Reynolds number for symmetry breaking bifurcation for the 50%, 60% and 70% area reduction are obtained numerically. The flow field separates after the symmetry breaking bifurcation and the symmetry of the flow depends on the Reynolds number and the height of the constriction.     

Numerical calculation of turbulent corner flows

Numerical predictions are presented of the hydrodynamic characteristics of developing and fully-developed turbulent flow in a square duct. The turbulent stresses in the plane of the cross-section, gradients of which cause the familiar secondary flows, are approximated by gradients in the axial mean velocity. Two distinct approximations are investigated, one of which specifies some of the model ‘constants’ as functions of the gradient of the length scale to account for wall effects. The stresses in the axial momentum equation are calculated from an eddy viscosity deduced from the K-W model of turbulence, K being the turbulence energy and W, a measure of the time-mean-square-vorticity fluctuations. The approximation incorporating wall effects generally performs better than the other when compared with fully-developed flow-data. This same approximation also compares favourably with data for developing flow and predictions based on K-? models in the literature.     

Influence of Mass Loading on Particle-Laden Turbulent Pipe Flow

A CFD code in the framework of OpenFOAM was validated for simulations of particle-laden pipe and channel flows at low to intermediate mass loadings. The code is based on an Eulerian two-fluid approach with Reynolds-averaged conservation equations, including turbulence modeling and four-way coupling. Pipe flow simulations of particles in air against gravity were conducted at Reynolds numbers up to 50000. The particle mass loading was varied and its effect on the mean velocities and turbulent fluctuations of the two phases was studied. Special attention was paid to the influence of mass loading on the centerline velocity and the wall shear velocity of the fluid phase for various flow parameters and particle properties. Empirical correlations were established between these two quantities and the flow Reynolds number, particle Reynolds number, Stokes number and particle to fluid density ratio for a range of particle mass loadings. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Self-similar turbulent boundary layer in pressure gradient

Self-similar flows in a turbulent boundary layer when the free-stream velocity is specified as a power function of longitudinal coordinate are investigated. The self-similar formulation not only simplifies solving of the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. It is shown that for the class of flows under consideration that depend on three governing parameters the dimensionless mixing length is a function of the normalised distance from the wall and the exponent in the law specifying the free-stream velocity distribution in the outer region and a universal function of local Reynolds number in the wall region, the latter corollary being true even when the skin friction vanishes. In calculations this function is set to be independent of pressure gradient, which gives the results very close to experimental data. There exist four different self-similar flow regimes. Each regime is related to its similarity parameter, one of which is the well-known Clauser equilibrium parameter and the other three are established for the first time. In case of adverse pressure gradient when the exponent lies within certain limits, which depend on Reynolds number, the problem has two solutions with different values of the boundary layer thickness and skin friction, which points out the possibility of hysteresis in near-separating flow. Separation occurs not at the minimum value of the exponent that corresponds to the strongest adverse pressure gradient but at a higher one whose dependence on Reynolds number is calculated in the paper. The results of the theory are in good agreement with experimental data. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional numerical simulation of nonisotropic thermal density flow in a strongly curved open channel

The results from a 3D nonisotropic algebraic stress/flux turbulence model are presented to investigate the structure of thermal density flow and the temperature distribution in a strongly curved open channel (180° bend). The numerically simulated results show that (i) several secondary flows take place at the bend cross-section 90° of the curved open channel, the feature which is not found for the isothermal flows and thermal density flow in a straight channel, and (ii) the thermocline in a curved channel is thicker than that in a straight channel due to the secondary flows-induced strong mixing process taking place in the former. Such features may be ascribed to the complex interaction of the buoyant force, the centrifugal force and the Reynolds stresses taking place only in curved channels. The simulated results are in good agreement with available experimental data, which indicates that the developed model can be applied for predicting the motion of the nonisotropic thermal density flow in the curved open channel.     

Computation of dilute two-phase flow in a pump

This paper is a report on a joint project between academia and industry which is concerned with computation of dilute two-phase flow through a pump in turbulent condition. The flow field for the continuous phase is computed using the Reynolds averaged Navier–Stokes equations together with mixing length turbulence modeling. The dispersed phase is treated using the Lagrangian approach by tracking it's trajectory along which the information is passed. It is found that the bubbles and small solid particles flow out of the chamber (between the rotating impeller and the casing wall) with the conveying fluid. The solid particles of relatively bigger sizes accumulate at the low pressure zones near the cashing wall or the rotating shaft.     

Asymptotic behaviour of commutation errors and the divergence of the Reynolds stress tensor near the wall in the turbulent channel flow

The derivation of the space averaged Navier–Stokes equations for the large eddy simulation (LES) of turbulent incompressible flows introduces two groups of terms which do not depend only on the space averaged flow field variables: the divergence of the Reynolds stress tensor and commutation errors. Whereas the former is studied intensively in the literature, the latter terms are usually neglected. This note studies the asymptotic behaviour of these terms for the turbulent channel flow at a wall in the case that the commutation errors arise from the application of a non‐uniform box filter. To perform analytical calculations, the unknown flow field is modelled by a wall law (Reichardt law and 1/αth power law) for the mean velocity profile and highly oscillating functions model the turbulent fluctuations. The asymptotics show that near the wall, the commutation errors are at least as important as the divergence of the Reynolds stress tensor. Copyright © 2006 John Wiley & Sons, Ltd.     

Heat transfer at high energy devices with prescribed cooling flow

We study the heat transfer from a high‐energy electric device into a surrounding cooling flow. We analyse several simplifications of the model to allow an easier numerical treatment. First, the flow variables velocity and pressure are assumed to be independent from the temperature which allows a reduction to Prandtl's boundary layer model and leads to a coupled nonlinear transmission problem for the temperature distribution. Second, a further simplification using a Kirchhoff transform leads to a coupled Laplace equation with nonlinear boundary conditions. We analyse existence and uniqueness of both the continuous and discrete systems. Finally, we provide some numerical results for a simple two‐dimensional model problem. Copyright © 2006 John Wiley & Sons, Ltd.     

Micro diffuser flow modeling for cold gas propulsion systems

The use of highly diluted and hypersonic gas flow is in the scope of application of cold gas thrusters for space applications. Satellites and small spacecrafts are navigated to their orbital trajectory with these nozzles. Inside these propulsion systems high density gradients are dominating the efficiency and the thrust steering behavior of the propulsion systems. Micro flows in the transient regime between free molecular flow and continuous flow are not able to be computed with trustworthy results by using a continuous model with no-slip boundary conditions. Therefore boundary slip-velocity models are used for modeling the reduced wall shear stress. Molecular shear stresses decrease the molecular mean velocity near the wall. With a Knudsen number depending slip-velocity model the effective shear stress is computed by the mean gradient of the velocity profile near the wall. In the present study a trans-sonic nozzle flow is computed by using a calibrated velocity slip model what depends on the Knudsen number. The Knudsen numbers are lower the Kn=1 at the nozzle neck of the propulsion system. The results are compared with simulation results of a uniform channel flow and computations of the corresponding no-slip approach. The differences in the hypersonic region are following discussed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of physiological pulsatile flow in a model arterial stenosis using large-eddy and direct numerical simulations

Physiological pulsatile flow in a 3D model of arterial stenosis is investigated by using large eddy simulation (LES) technique. The computational domain chosen is a simple channel with a biological type stenosis formed eccentrically on the top wall. The physiological pulsation is generated at the inlet using the first harmonic of the Fourier series of pressure pulse. In LES, the large scale flows are resolved fully while the unresolved subgrid scale (SGS) motions are modelled using a localized dynamic model. Due to the narrowing of artery the pulsatile flow becomes transition-to-turbulent in the downstream region of the stenosis, where a high level of turbulent fluctuations is achieved, and some detailed information about the nature of these fluctuations are revealed through the investigation of the turbulent energy spectra. Transition-to-turbulent of the pulsatile flow in the post stenosis is examined through the various numerical results such as velocity, streamlines, velocity vectors, vortices, wall pressure and shear stresses, turbulent kinetic energy, and pressure gradient. A comparison of the LES results with the coarse DNS are given for the Reynolds number of 2000 in terms of the mean pressure, wall shear stress as well as the turbulent characteristics. The results show that the shear stress at the upper wall is low just prior to the centre of the stenosis, while it is maximum in the throat of the stenosis. But, at the immediate post stenotic region, the wall shear stress takes the oscillating form which is quite harmful to the blood cells and vessels. In addition, the pressure drops at the throat of the stenosis where the re-circulated flow region is created due to the adverse pressure gradient. The maximum turbulent kinetic energy is located at the post stenosis with the presence of the inertial sub-range region of slope −5/3.     

Thermodynamics / turbulence analogy modelling dissipating molecular gas flows for high Knudsen numbers

Modelling micro channel flows momentum and heat diffusion / convection are recent parameters modelling the molecule velocity distribution. Macroscopic models describe velocity and energy / enthalpie with integrals of mass increments. Using microscopic models motion and forces of a molecular flow have to be computed by models of physical properties, whose are described by statistical power moments of the molecule velocity. Therefore dilute flows have to be investigated in small channels with a mean free path length of molecules higher than the channel width of the the micro channel itself (λ₀≥H₀). Modelling this process by a continuous flow the boundary conditions have to be modified (e.g. [6]). The present model uses the statistical approximation of the molecule velocity distribution to simulate the behaviour of this discrete flow with a weighted averaged molecule velocity ∼ξ_i, its standard deviation σ and the characterisic molecule collision rate z. The number density N per volume V near one position is used for the weighting factor averaging method describing the mean molecule velocity. The present model is validated computing Poiseuille and Couette flows with different Knudsen numbers. Showing the advantages of the present model the simulation results are compared with simulation results of the wall–distance depending diffusivity model of Lockerby and Reese [4] and BGK results of a Lattice–Boltzmann simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Laws of the wall for velocity and temperature in supersonic turbulent boundary layers

Scaling laws for velocity and temperature profiles in the near-wall region of sub- and supersonic turbulent boundary layers have been developed, which allow us to represent velocity and temperature profiles in compressible gas stream in terms of those in an incompressible boundary layer. They are obtained as asymptotic expansions of the solutions to the Reynolds equations in a small parameter — the Mach number based on the friction velocity and gas enthalpy on the wall. The leading term of the expansion for velocity corresponds to known Van Driest's formula. However, the obtained solution contains additional terms of order unity, which explains the contradiction between Van Driest's formula and experimental data. The law of the wall for temperature, which has been formulated for the first time, has an analogous structure. Besides the von Kármán constant and the turbulent Prandtl number in the logarithmic region, known for incompressible flow, the obtained relations contain three new universal constants, which do not depend on gas molecular properties and the specific heat ratio. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The compression of a thin strip of material in a superplasticity state

The viscoplastic flow of a thin strip of material in a superplasticity state between rigid, converging parallel planes (an analogue of Prandtl's problem) is investigated. An analytical quadrature solution of the problem is constructed, asymptotically precise in the same sense as Prandtl's solution. Special cases are considered where the solution (including an approximate solution) is written out fully. The effects of superplasticity are determined.