Non-uniform Body Force Research Articles

Turbulence enhancement in body force opposed flows

Turbulence may be strongly influenced by near-wall nonuniform body forces. A widely encountered example is buoyancy acting in mixed convection flows. Here we use direct numerical simulations to present two key ideas: First we show that introducing opposing body forces whilst maintaining the pressure gradient does not cause key turbulence characteristics to change. It follows that the concept of “apparent Reynolds number” introduced by He [] to explain laminarization can now be extended to explain the opposite phenomena of turbulence enhancement by opposing body forces. This new understanding has allowed turbulent shear stress and skin friction to be easily predicted simply from the nonuniform body force profile and reference data despite the nonequilibrium nature of the flow. Second, we analyze the flow in the context of turbulence regeneration cycle and provide evidence that complements the recent study from Jiménez [], which suggests that streaks are more decoupled from the regeneration cycle than previously thought. Published by the American Physical Society 2024

Second-order computational homogenisation enhanced with non-uniform body forces for non-linear cellular materials and metamaterials

Although “classical” multi-scale methods can capture the behaviour of cellular, including lattice, materials, when considering lattices or metamaterial local instabilities, corresponding to a change of the micro-structure morphology, classical computational homogenisation methods fail. On the one hand, first order computational homogenisation, which considers a classical continuum at the macro-scale cannot capture localisation bands inherent to cell buckling propagation. On the other hand, second-order computational homogenisation, which considers a higher order continuum at the macro-scale, introduces a size effect with respect to the Representative Volume Element (RVE) size, which is problematic when the RVE has to consider several cells to recover periodicity during local instability. In this paper we reformulate in a finite-strain setting the second-order computational homogenisation using the idea of equivalent homogenised volume. From this equivalence, arises at the micro-scale a non-uniform body force that acts as a supplementary volume term over the RVE. In the presented method, this non-uniform body-force term arises from the equivalence of energy, i.e. the Hill–Mandel condition, between the micro- and macroscopic volumes and depends mainly on the relation between the micro-scale and macro-scale deformation gradient. We show by considering elastic and elasto-plastic metamaterials and cellular materials that this approach reduces the RVE size dependency on the homogenised response.

Relaminarized and recovered turbulence under nonuniform body forces

We show that wall-bounded turbulence can be relaminarized and reorganized by adding nonuniform streamwise body forces. A nonuniform body force can distort the parabolic mean velocity profile, thus altering turbulence production due to mean shear. In the quasi-laminar state, all Reynolds stress tensor components are fairly weak except for streamwise fluctuations far from the wall, indicating a collapse of the near-wall turbulence self-sustaining cycle. In the recovered turbulence regime, the Reynolds shear stress has a negative range in the bulk connected with a positive range near the wall corresponding to the M-shaped velocity profile.

Discrete effects on the forcing term for the lattice Boltzmann modeling of steady hydrodynamics

This work concerns with the clean inclusion of the forcing term in the lattice Boltzmann method (LBM) for the modeling of non-uniform body forces in steady hydrodynamics. The study is conducted for the two-relaxation-time (TRT) scheme. Here, we consider a simple, but yet sufficiently generic, flow configuration driven by a spatially varying body force based on which we derive the analytical solution of the forced LBM-TRT and compare two force strategies set by first- and second-order expansions. This procedure exactly establishes the macroscopic system satisfied by each force formulation at discrete level. The obtained theoretical results are further verified in two distinct benchmark channel flow problems. Overall, this study shows that the spatial discrete effects posed by the LBM modeling of the force term may come in through two sources. The first one is a defect inherent to LBM, arising from the non-local spatial discretization of the forcing term, and given by the discrete Laplacian of the body force. While it corrupts the discrete momentum balance with a non-linear viscosity dependent term in the single-relaxation-time schemes, this inconsistency is avoided with the TRT scheme, through its free-tunable relaxation degree of freedom Λ. The other error source is unique to the use of second-order force expansions, where its non-zero second-order velocity moment interferes with the discrete momentum balance through a spurious first-order derivative term, leading to several forms of inconsistency in the LBM steady solution. For simulations under the convective scaling it makes the LBM scheme a numerical representation of a differential system distinct from the physical one, whereas under the diffusive scaling it leads to viscosity-dependent numerical errors, which corrupt the otherwise consistent structure of TRT steady-state solutions. In contrast, the defects reported herein are absent with the first-order force expansion scheme operated within the scope of the LBM-TRT model for steady hydrodynamics.

Characteristic scales of Townsend's wall-attached eddies.

Townsend (The Structure of Turbulent Shear Flow, 1976, Cambridge University Press) proposed a structural model for the logarithmic layer (log layer) of wall turbulence at high Reynolds numbers, where the dominant momentum-carrying motions are organised into a multiscale population of eddies attached to the wall. In the attached-eddy framework, the relevant length and velocity scales of the wall-attached eddies are the friction velocity and the distance to the wall. In the present work, we hypothesise that the momentum-carrying eddies are controlled by the mean momentum flux and mean shear with no explicit reference to the distance to the wall and propose new characteristic velocity, length and time scales consistent with this argument. Our hypothesis is supported by direct numerical simulation of turbulent channel flows driven by non-uniform body forces and modified mean velocity profiles, where the resulting outer-layer flow structures are substantially altered to accommodate the new mean momentum transfer. The proposed scaling is further corroborated by simulations where the no-slip wall is replaced by a Robin boundary condition for the three velocity components, allowing for substantial wall-normal transpiration at all length scales. We show that the outer-layer one-point statistics and spectra of this channel with transpiration agree quantitatively with those of its wall-bounded counterpart. The results reveal that the wall-parallel no-slip condition is not required to recover classic wall-bounded turbulence far from the wall and, more importantly, neither is the impermeability condition at the wall.

Buoyancy induced turbulence modulation in pipe flow at supercritical pressure under cooling conditions

A numerical investigation of cooling of a fluid at supercritical pressure has been performed by means of direct numerical simulations. The simulations were conducted with a uniform heat flux imposed at the wall at an inlet bulk Reynolds number of 5400. The aim of this work is to understand the role of buoyancy in modulating the turbulence in a flow where properties are spatially varying. Heat transfer deterioration followed by recovery is observed in the downward flow while enhancement occurs in the upward flow as compared to forced convection. The decomposition of the skin friction factor and the Nusselt number was performed. The major effects on the skin friction factor were brought by the non-uniform body force due to the gravity. The turbulent parts equally influence the Nusselt number as well as the skin friction factor in supercritical flows. Quadrant analysis and its weighted joint probability density function were analyzed to understand the role of sweep (Q4) and ejection (Q2) events. During the heat transfer deterioration, sweep and ejection events are decreased greatly, triggering the reduction in turbulence. The recovery in turbulence is brought by the Q1 and Q3 (also known as outward and inward interaction) events, contrary to the conventional belief about turbulence generation. The turbulence anisotropy of the Reynolds stress tensor is investigated showing that the turbulence structure becomes rod-like during the deteriorated heat transfer regime in the downward flow and disc-like for the upward flow.

Laminarisation of flow at low Reynolds number due to streamwise body force

It is well established that when a turbulent flow is subjected to a non-uniform body force, the turbulence may be significantly suppressed in comparison with that of the flow of the same flow rate and hence the flow is said to be laminarised. This is the situation in buoyancy-aided mixed convection when severe heat transfer deterioration may occur. Here we report results of direct numerical simulations of flow with a linear or a step-change profile of body force. In contrast to the conventional view, we show that applying a body force to a turbulent flow while keeping the pressure force unchanged causes little changes to the key characteristics of the turbulence. In particular, the mixing characteristics of the turbulence represented by the turbulent viscosity remain largely unaffected. The so-called flow laminarisation due to a body force is in effect a reduction in the apparent Reynolds number of the flow, based on an apparent friction velocity associated with only the pressure force of the flow (i.e. excluding the contribution of the body force). The new understanding allows the level of the flow ‘laminarisation’ and when the full laminarisation occurs to be readily predicted. In terms of the near-wall turbulence structure, the numbers of ejections and sweeps are little influenced by the imposition of the body force, whereas the strength of each event may be enhanced if the coverage of the body force extends significantly away from the wall. The streamwise turbulent stress is usually increased in accordance with the observation of more and stronger elongated streaks, but the wall-normal and the circumferential turbulent stresses are largely unchanged.

Effect of disjoining pressure in a thin film equation with non-uniform forcing

We explore the effect of disjoining pressure on a thin film equation in the presence of a non-uniform body force, motivated by a model describing the reverse draining of a magnetic film. To this end, we use a combination of numerical investigations and analytical considerations. The disjoining pressure has a regularizing influence on the evolution of the system and appears to select a single steady-state solution for fixed height boundary conditions; this is in contrast with the existence of a continuum of locally attracting solutions that exist in the absence of disjoining pressure for the same boundary conditions. We numerically implement matched asymptotic expansions to construct equilibrium solutions and also investigate how they behave as the disjoining pressure is sent to zero. Finally, we consider the effect of the competition between forcing and disjoining pressure on the coarsening dynamics of the thin film for fixed contact angle boundary conditions.

A Spectral Theory Formulation for Elastostatics by Means of Tensor Spherical Harmonics

Consider a set of (N+1)-phase concentric spherical ensemble consisting of a core region encased by a sequence of nested spherical layers. Each phase is spherically isotropic and is functionally graded (FG) in the radial direction. Determination of the elastic fields when the outermost spherical surface is subjected to a nonuniform loading and the constituent phases are subjected to some prescribed nonuniform body force and eigenstrain fields is of interest. When the outermost layer is an unbounded medium with zero eigenstrain and body force fields, then an N-phase multi-inhomogeneous inclusion problem is realized. Based on higher-order spherical harmonics, presenting a three-dimensional strain formulation with a robust form of compatibility equations, a spectral theory of elasticity in the spherical coordinate system is developed. Application of the established spectral theory leads to the exact closed-form solution when the elastic moduli of each phase vary as power-law functions of radius.

Mathematical Design of an Electromagnetic Separation Sensor in Molten Aluminum

This paper describes an investigation of an electromagnetic separation approach for separating inclusions in molten aluminum. The separation is effected by a magnetic field generated by a direct current, resulting in a Lorentz force. On the basis of this magnetic separation method, we constructed a sensor probe and rigorously analyzed it from a mathematical point of view. Specifically, we calculated the magnetic field inside the probe and showed that a nonuniform body force density is established throughout the melt volume, resulting in a decrease in probe efficiency. Moreover, we analyzed the rotational motion of the melt under the influence of the spatially dependent body force, and reasoned that this rotational motion also adversely affects the probe performance. To overcome these two drawbacks, we propose a new probe design. The goal of the new design is to increase strength and uniformity of the body force distribution within the aluminum melt.

Stepwise linear regression particular integrals for uncoupled thermoelasticity with boundary elements

This paper presents a simple technique for the efficient treatment of non-uniform body forces in boundary elements without the need for domain discretization and additional surface integral evaluations. The distribution of the non-uniform forces in the domain is described using regression polynomials. Particular integrals corresponding to these polynomials are developed for the solution of the non-homogeneous differential equations of elasticity. Two-dimensional and axisymmetric boundary element results for steady state and transient thermoelastic deformations are obtained using this formulation. Regression polynomials based on only the boundary points provide good results. Improved results are obtained by including internal points in the regression analysis. The method allows the use of low order polynomials to model a general distribution of thermal effects in the domain. Numerical data are given to provide comparisons of low order versus high order regression polynomials, and boundary only data versus combined boundary and domain data based regression polynomials.

Treatment of body forces in axisymmetric boundary element design sensitivity formulation

This paper deals with the treatment of uniform and non-uniform body forces in the implicit-differentiation formulation for axisymmetric boundary element design sensitivity analysis. The particular integral concept is extended to obtain the sensitivities due to gravitational and centrifugal body forces of uniform type. For thermal body forces of non-uniform type, the sensitivities are obtained through the implicit-differentiation of a surface integral. The efficiency and the accuracy of the formulation are determined for a wide range of problems which include different axisymmetric geometries under centrifugal, gravitational and thermal body forces.

Vorticity Transfer and Production in Steady Inviscid Flow

General equations are derived for the rate of change of vorticity in compressible, inviscid flow into which fluid may be introduced, and in which nonuniform body forces may exist. Examples of vorticity transfer and change are given and analogies are drawn between these examples.