Body Force Term Research Articles

Cilia-assisted flow of viscoelastic fluid in a divergent channel under porosity effects.

Cilia-driven laminar flow of an incompressible viscoelastic fluid in a divergent channel has been conducted numerically using the BVP4C technique. The non-Newtonian Jeffrey rheological model is utilized to characterize the fluid. The flow equations are formulated in a curvilinear coordinate system, and the porosity effects are simulated with a body force term in the Navier-Stokes equation. The flow equations are transformed into a wave frame from a fixed frame of reference using a linear mathematical relationship. A biological approximation of creeping phenomena and the long-wavelength assumption is used in the flow analysis. The flow analysis is carried out by using a complex (wavy) propulsion of cilia beating. The two-dimensional flow is controlled by physical parameters-Darcy's number, curvature parameter, viscoelastic parameter, phase difference, cilia length, and divergent parameter. They also examined the ciliated pumping and bolus trapping in their flow analysis. The boundary layer phenomena in the velocity profile are noticed under more significant porosity and time relaxation effects. The bolus circulations are reduced for a larger porosity medium and larger numeric values of the time relaxation parameter.

Homogenized Flux-Body Force Treatment of Compressible Viscous Porous Wall Boundary Conditions

A homogenization approach is proposed for the treatment of porous wall boundary conditions in the computation of compressible viscous flows. Like any other homogenization approach, it eliminates the need for pore-resolved fluid meshes, and therefore enables practical flow simulations in computational fluid domains with porous wall boundaries. Unlike alternative approaches, however, it does not require prescribing a mass flow rate and does not introduce in the computational model a heuristic discharge coefficient. Instead, it models the inviscid flux through a porous wall surrounded by the flow as a weighted average of the inviscid flux at an impermeable surface and that through pores. It also introduces a body force term in the governing equations to account for friction loss along the pore boundaries. The source term depends on the thickness of the porous wall and the concept of an equivalent single pore. The feasibility of the latter concept is demonstrated using low-speed permeability test data for the fabric of the Mars Science Laboratory parachute canopy. The overall homogenization approach is illustrated with a series of supersonic flow computations through the same fabric and verified using supersonic pore-resolved numerical simulations.

Investigation of solution differences on natural convection in a tall cavity simulated by using the Boussinesq approximation and the real air density equation

Solution differences of natural convection in a tall cavity filled with air in which the density of the body force term are approximated by using the Boussinesq approximation (BA) and by using the real air density equation (RE) are numerically investigated. The main parameters for the investigation are Nusselt numbers and numbers of cells in the multi-cellular flow. The investigation is carried out for the Rayleigh number of 104, and the cavity aspect ratio equal to 60. The temperature differences of the vertical cavity walls are varied from 1 K to 200 K. These conditions are appropriate to form the multi-cellular flow in the tall cavity obviously, and close to the boundary of the laminar and turbulent flows. Consequently, the differences in both Nusselt numbers and numbers of the cells in the multi-cellular flow can be easily observed. The finite volume method with the Lagrange interpolating polynomial (LIP) scheme are selected to discretize the governing equations of the flow, and the Semi-implicit method for pressure linked equations (SIMPLE) algorithm is used to determine the pressure field of the flow. The maximum difference of the average Nusselt numbers of natural convection in the tall cavity simulated by using BA and RE is 0.436 % for the range of the investigation. This indicates that the difference of heat transfer solutions of natural convection in the tall cavity simulated by using BA and RE is insignificant for practical applications, but there is an obvious difference on the numbers of the cells in the multi-cellular flow, when the temperature difference of the vertical cavity walls is greater than or equal to 120 K.

A Simple Immersed Boundary Method for Modeling Forced Convection Heat Transfer

Abstract As nonbody-conforming numerical methods using simple Cartesian mesh, immersed boundary methods have become increasingly popular in modeling fluid–solid interaction. They usually do this by adding a body force term in the momentum equation. The magnitude and direction of this body force ensure that the boundary condition on the solid–fluid interface are satisfied without invoking complicated body-conforming numerical methods to impose the boundary condition. A similar path has been followed to model forced convection heat transfer by adding a source term in the energy equation. The added source term will ensure that thermal boundary conditions on the solid–fluid interface are imposed without invoking a boundary conforming mesh. These approaches were developed to handle the Dirichlet boundary condition (constant wall temperature). Few of them deal with the Neumann boundary condition (constant wall heat flux). This paper presents a simple new immersed boundary method. It can deal with the Dirichlet boundary condition, Neumann boundary condition, and conjugated heat transfer by adding an energy source or sink term in the energy conservation equation. The presented approach is validated against the analytical solutions and a very good match is achieved.

Compressible plane turbulent wakes under pressure gradients evolving in a constant area section

Development of statistically two-dimensional (2-D) turbulent wakes under pressure gradients is a common feature of many industrial and aerodynamic flows. The usual set-up is to generate a wake, and study its development through a passage with a variable area; if the downstream area decreases (increases) a favourable pressure gradient or FPG (adverse pressure gradient or APG) is imposed. In applications such as in turbomachinery, however, the wakes develop in a periodic constant area passage in the stator–rotor gap and with an imposed pressure gradient. To study these flows, here, we develop a canonical set-up for this new kind of wake evolution in FPG and APG of different strengths by placing a 2-D flat plate normal to the flow in a periodic constant area passage and a fixed inflow mass flux. Employing compressible direct numerical simulations, we impose pressure gradients through a ramped body force term to the momentum and total energy equations while the wake is allowed to develop spatially in a region of fixed width. The resultant mean velocity statistics, wake width, energy budgets and entropy generation rates are scrutinised to assess the effect of the pressure gradients, and where possible, the similarities and differences to the conventional case of variable area pressure gradients are discussed. The results show that the effect of a constant area pressure gradient on flow statistics is non-trivial, resulting from significant density changes. The pressure gradients also have an effect on the different energy budgets, which produces a gain for FPG and loss for APG in the mean kinetic energy. Consequently, the entropy generation rate diminishes and augments for the FPG and APG, respectively, compared to the zero pressure gradient. Finally, the effect of different passage heights ( ) relative to the wake half-width ( ) is studied using large eddy simulations. We find that wake width and hence the spreading, depends primarily on the wake–wake interaction for small and pressure gradients for larger , and this has implications for the design of turbomachinery.

Multi-Degree of Freedom Propeller Force Models Based on a Neural Network and Regression

Accurate and efficient prediction of the forces on a propeller is critical for analyzing a maneuvering vessel with numerical methods. CFD methods like RANS, LES, or DES can accurately predict the propeller forces, but are computationally expensive due to the need for added mesh discretization around the propeller as well as the requisite small time-step size. One way of mitigating the expense of modeling a maneuvering vessel with CFD is to apply the propeller force as a body force term in the Navier–Stokes equations and to apply the force to the equations of motion. The applied propeller force should be determined with minimal expense and good accuracy. This paper examines and compares nonlinear regression and neural network predictions of the thrust, torque, and side force of a propeller both in open water and in the behind condition. The methods are trained and tested with RANS CFD simulations. The neural network approach is shown to be more accurate and requires less training data than the regression technique.

Lattice Boltzmann modeling of fluid-particle interaction based on a two-phase mixture representation.

In this work, we derive a lattice Boltzmann model for fluid-particle interaction by considering the system as a two-phase mixture. A partially saturated type scheme is achieved rigorously without any viscosity-dependent weight parameter. The scheme is of second-order accuracy in both space and time including the body-force term. Moreover, we devise a scheme suitable for the scenario where two or more particles intersect a single computational cell, typically occurring for particles in contact or close to contact. Good performance is found when the present scheme is validated against three classic problems, namely the flow past a stationary cylinder, a cylindrical particle settling in a channel under gravity, and the flow around two impacting cylinders.

Body force model for simulating airflow through dynamically oscillating louvers

Louvers are primarily used for directional airflow delivery in air conditioning systems. To model the airflow passing through a louver, a detailed computational fluid dynamics (CFD) analysis can be performed by resolving all the geometrical features of the louver (we call this the Geometrically Resolved [GR] model). However, in the case of physical movement of louvers dynamic meshing is required, leading to high computational effort. In this paper, we propose a simple airflow model with a body force in the louver region (we call this the Body Force [BF] model), which can replicate similar downstream flow while obviating the necessity to resolve the louver geometry. In order to achieve this, a body force term of appropriate magnitude is introduced in a particular region in the vicinity of louver. The BF term specification region was identified based on the louver dimensions and using inputs from the GR model simulations. The time-dependent physical movement of louvers was achieved by specifying a time-varying body force term. The results obtained from the BF model and the GR model were in good agreement. This suggests that a simple BF model is an useful alternative to the GR model to simulate airflow through louvers.

Stability of an Axisymmetric Liquid Metal Flow Driven by a Multi-Pole Rotating Magnetic Field

The stability of an electrically conducting fluid flow in a cylinder driven by a multi-pole rotating magnetic field is numerically studied. A time-averaged Lorentz force term including the electric potential is derived on the condition that the skin effect can be neglected and then it is incorporated into the Navier-Stokes equation as a body force term. The axisymmetric velocity profile of the basic flow for the case of an infinitely long cylinder depends on the number of pole-pairs and the Hartmann number. A set of linearized disturbance equations to obtain a neutral state was successfully solved using the highly simplified marker and cell (HSMAC) method together with a Newton–Raphson method. For various cases of the basic flow, depending on both the number of pole-pairs and the Hartmann number, the corresponding critical rotational Reynolds numbers for the onset of secondary flow were obtained instead of using the conventional magnetic Taylor number. The linear stability analyses reveal that the critical Reynolds number takes its minimum at a certain value of the Hartmann number. On the other hand, the velocity profile for cases of a finite length cylinder having a no-slip condition at the flat walls generates the Bödewadt boundary layers and such flows need to be computed including the non-linear terms of the Navier-Stokes equation.

Plasma Actuator–Assisted Rocket Nozzle for Improved Launcher Performance

The need to avoid unstable separations and side loads during low-altitude operations imposes a strict limit on the nozzle expansion ratio of launchers propulsion systems. Remarkable performance losses are consequently induced. An innovative concept is here presented that overcomes this limit. Plasma actuators are proposed as a means to stabilize separation in highly overexpanded conventional bell-shaped nozzles. A larger expansion ratio can thus be adopted in boosters and first-stage engines, and their vacuum specific impulse is increased. The concept is here assessed by means of Reynolds-averaged Navier–Stokes simulations. The effects of the plasma actuators are described by average body force and heat source terms. The potentialities of this concept, adaptable to any bell-shaped nozzle, are assessed evaluating the possible payload gain for a representative case. The results show that the proposed concept allows for suitable payload growth and engine flexibility.

Verification and improvement of dynamic motion model in MARS for marine reactor thermal-hydraulic analysis under ocean condition

Abstract Unlike land-based nuclear power plants, a marine or floating reactor is affected by external forces due to ocean conditions. These external forces can cause additional accelerations and affect each system and equipment of the marine reactor. Therefore, in designing a marine reactor and evaluating its performance and stability, a thermal hydraulic safety analysis code is necessary to consider the thermal hydrodynamic effects of ship motion. MARS, which is a reactor system analysis code, includes a dynamic motion model that can simulate the thermal-hydraulic phenomena under three-dimensional motion by calculating the body force term included in the momentum equation. In this study, it was verified that the dynamic motion model can simulate fluid motion with reasonable accuracy using conceptual problems. In addition, two modifications were made to the dynamic motion model; first, a user-supplied table to simulate a realistic ship motion was implemented, and second, the flow regime map determination algorithm was improved by calculating the volume inclination information at every time step if the dynamic motion model was activated. With these modifications, MARS could simulate the thermal-hydraulic phenomena under ocean motion more realistically.

Time-Periodic Electro-Osmotic Flow With Nonuniform Surface Charges

Mixing in a microfluidic device is a major challenge due to creeping flow, which is a significant roadblock for development of lab-on-a-chip device. In this study, an analytical model is presented to study the fluid flow behavior in a microfluidic mixer using time-periodic electro-osmotic flow. To facilitate mixing through microvortices, nonuniform surface charge condition is considered. A generalized analytical solution is obtained for the time-periodic electro-osmotic flow using a stream function technique. The electro-osmotic body force term is accounted as a slip boundary condition on the channel wall, which is a function of time and space. To demonstrate the applicability of the analytical model, two different surface conditions are considered: sinusoidal and step change in zeta potential along the channel surface. Depending on the zeta potential distribution, we obtained diverse flow patterns and vortices. The flow circulation and its structures depend on channel size, charge distribution, and the applied electric field frequency. Our results indicate that the sinusoidal zeta potential distribution provides elliptical shaped vortices, whereas the step change zeta potential provides rectangular shaped vortices. This analytical model is expected to aid in the effective micromixer design.

Comprehensive investigation of solid and porous fins influence on natural convection in an inclined rectangular enclosure

A numerical analysis has been performed to investigate two-dimensional natural convection inside a differentially heated inclined rectangular enclosure with several porous fins attached to the hot wall. Various parameters were studied including Rayleigh number (103 ≤ Ra ≤ 106), Darcy number. (10−8 ≤ Da ≤ 10−2), relative thermal conductivity ratio of solid phase to fluid (1 ≤ Ke ≤ 100), cavity inclination angle (0 ≤ φ ≤ π/3), cavity aspect ratio (AR = 2:1 and 3:1), fin number (0 ≤ N ≤ 4), dimensionless fin position (0.25 ≤ S ≤ 0.75) and dimensionless fin length (0.25 ≤ L ≤ 0.75). Left vertical wall is maintained at a higher temperature Th, while the right vertical wall is kept at lower temperature Tc and horizontal walls are insulated. Cavity is filled with an incompressible Newtonian fluid with Prandtl number of 0.71. Fins with length of lf are attached to the hot vertical wall. Due to a small temperature difference between the hot and cold vertical walls, all of the thermo-physical properties, except the density variation in the body force term, which is modeled using the Boussinesq approximation, are assumed constant. Governing steady state equations for mass, momentum and energy within the porous fin is based on volume-averaging method and the finite volume code based on SIMPLE method was utilized in the present investigation to solve the governing equations. The investigated results illustrated that using several conductive porous fins, increase the average Nusselt number within the cavity up to 41% compared to enclosures with solid fins and up to 20% compared to cavities without any fins. Furthermore, it was found that increasing heat transfer using more porous fins highly depends on the relative thermal conductivity ratio of solid phase to fluid and rising the thermal conductivity ratio from 10 to 100, results in 17% of heat transfer enhancement. It was found that Da = 10−5 is a limit for positive performance of the porous fins, below which no heat transfer enhancement occurs compared to cavities with solid fins. Optimum porous fin length was found to be a decreasing function of Rayleigh number.

Lattice Boltzmann method in hydrodynamics and thermophysics

The variants of two- and three-dimensional lattice Boltzmann methods (LBM) are described for computer simulations of fluids with liquid-vapor phase transitions and heat transfer. Several improvements of LBM are described (correct implementation of the body force term, “pseudoforces” for the method of “passive scalar” for internal energy and implementation of a latent heat of evaporation). The formation of rivulets in a flow along an inclined flat surface, the fall of a droplet onto a rigid nonwettable plane and the process of spinodal decomposition of initially uniform fluid are modelled.

Turbulent molten pool analysis of tandem GMA automotive steel sheet welding

In this research, a three-dimensional turbulent weld pool simulation technique of tandem gas tungsten arc welding (GMAW) in a lap joint fillet is described and compared to a conventional laminar flow simulation. Basically, four governing equations are adopted for continuity, namely, continuity, Navier-Stokes, energy, and the popular volume of fluid equations. According to the basic theory of an arc weld pool, all known characteristics such as the arc heat input, arc pressure, electromagnetic force, and Marangoni flow are applied to the analysis model as a body force term or boundary conditions. A conventional arc weld pool analysis usually adopts a laminar flow; however, its results have shown a weld bead shape that is quite different from an experiment of a tandem GMA lap joint fillet welding of an automotive steel sheet. Therefore, the authors suggest the use of a k-εturbulent analysis model and show that its results coincide with the experiment results.

A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force

This paper determines a class of exact solutions for plane steady motion of incompressible fluids of variable viscosity with body force term in the Navier-Stokes equations. The class consists of stream function  characterized by equation , in polar coordinates ,  where ,  and  are continuously differentiable functions, derivative of  is non-zero but double derivative of  is zero. We find exact solutions, for a suitable component of body force, considering two cases based on velocity profile. The first case fixes both the functions ,  and provides viscosity as function of temperature. Where as the second case fixes the function , leaves  arbitrary and provides viscosity and temperature for the arbitrary function . In both the cases, we can create infinite set of expressions for streamlines, viscosity function, generalized energy function and temperature distribution in the presence of body force. Â

A class of new exact solutions of the system of PDE for the plane motion of viscous incompressible fluids in the presence of body force

The purpose of this paper is to indicate a class of exact solutions of the system of partial differential equations governing the steady, plane motion of incompressible fluid of variable viscosity with body force term to the right-hand side of Navier-Stokes equations. The class consists of the stream function characterized by the equation  in polar coordinates  and  where  and  are continuously differentiable functions and the function  is such that  where a non-zero constant is  and overhead prime represents derivative with respect to . When  or  we show exact solutions for given one component of the body force for both the cases when the function  is arbitrary and when it is not. For the arbitrary function case,  appears in the coefficient of a linear second order ordinary differential equation showing a large numbers of solutions of this equation. This in turn establishes an infinite set of exact solutions to the problem concerned however; we show three examples of such exact solutions. The alternate case fixes  and provides viscosity as derivative of temperature function for  and . Anyhow, we find an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution.

Review: Electric field driven pumping in microfluidic device.

Pumping of fluids with precise control is one of the key components in a microfluidic device. The electric field has been used as one of the most popular and efficient nonmechanical pumping mechanism to transport fluids in microchannels from the very early stage of microfluidic technology development. This review presents fundamental physics and theories of the different microscale phenomena that arise due to the application of an electric field in fluids, which can be applied for pumping of fluids in microdevices. Specific mechanisms considered in this report are electroosmosis, AC electroosmosis, AC electrothermal, induced charge electroosmosis, traveling wave dielectrophoresis, and liquid dielectrophoresis. Each phenomenon is discussed systematically with theoretical rigor and role of relevant key parameters are identified for pumping in microdevices. We specifically discussed the electric field driven body force term for each phenomenon using generalized Maxwell stress tensor as well as simplified effective dipole moment based method. Both experimental and theoretical works by several researchers are highlighted in this article for each electric field driven pumping mechanism. The detailed understanding of these phenomena and relevant key parameters are critical for better utilization, modulation, and selection of appropriate phenomenon for efficient pumping in a specific microfluidic application.

Lattice Boltzmann simulations of flow and heat transfer from a permeable triangular cylinder under the influence of aiding buoyancy

Mesoscopic numerical simulations have been carried out to learn about the flow and heat transfer characteristics of a 2-D permeable triangular cylinder, aligned at two different orientations under the influence of aiding buoyancy. Objective of this study is to investigate the effects of Darcy number and forebody shape on the hydrodynamic and thermal behaviour of the porous cylinder, under forced convection (i.e.Ri=0) and aiding buoyancy conditions (Ri = 0.5 and 1) for a Prandtl number value of 0.71 (air). The ranges of Reynolds number (Re) and Darcy number (Da) considered in this study are 1⩽Re⩽40 and 10-6⩽Da⩽10-2, respectively. Lattice Boltzmann method with two distribution functions is employed to perform the numerical experiments. Alongwith BGK collision operator, a body force term with viscous and inertial effects of the porous medium is employed at the porous zone. Detailed results are exhibited in the form of wake length, drag coefficient, streamlines, isotherm contours, heat transfer enhancement ratio and mean Nusselt number. Furthermore, a comparative investigation of drag coefficient and mean Nusselt number of permeable triangular cylinder (apex and side facing flow) with that of the square cylinder is carried out at Da=10-6 for different buoyancy levels. Under aiding buoyancy condition (i.e.Ri>0), the side facing triangular cylinder experiences less drag force than the apex facing for all values of Da. A significant thermal dissipation is observed for increasing values of non-dimensional permeability (or Da) and Richardson number. Furthermore, simple expressions for mean Nusselt number, valid for the range of parameters embraced in the present study, are also provided. The appropriate selection of non-dimensional permeability under different buoyancy conditions is important while applying porous media modeling technique in diverse fields of engineering.

Estimation of turbulent channel flow at based on the wall measurement using a simple sequential approach

The unsteady flow estimation problem of wall-bounded turbulence, numerically benchmarked by Chevalier et al. (J. Fluid Mech., vol. 552, 2006, pp. 167–187), is re-tackled with simple approaches. A turbulent channel flow at $Re_{\unicode[STIX]{x1D70F}}=100$ with periodic boundary conditions is reconstructed with linear stochastic estimation only based on the wall measurement, i.e. the wall shear stress in the streamwise and spanwise directions as well as the wall pressure over the entire wavenumber space. The results reveal that instantaneous information on the wall governs the success of the estimation in the vicinity of the wall ($y^{+}\lesssim 20$). The degrees of agreement are equivalent to those reported by Chevalier et al. using the extended Kalman filter as well as the ensemble Kalman filter performed in this study. This suggests that the instantaneous information on the wall dictates the reconstruction rather than the prediction step in these state observers solving the dynamical system. Subsequently, we feed the velocity components given by the linear stochastic estimation via the body-force term into the Navier–Stokes system: such an observer slightly improves the estimation in the log layer, indicating a small benefit of involving a dynamical system but over-suppression of turbulent motions beyond the viscous sublayer due to their low correlation with the wall measurement. Errors in the estimation grow in the buffer layer and prevent further reconstruction toward the centreline even if we relax the feedback forcing and let the flow evolve nonlinearly through the observer. We also discuss the flow components truly reconstructible from the wall measurement, which has limited degrees of freedom and poor correlation across wavenumbers.