Laminar Flow Of Incompressible Fluid Research Articles

Flow development region in the boundary layer: two-component molecular viscosity and partial slip

The subject of this study is the flow development region of laminar incompressible fluid flow in the boundary layer. This flow is an example where a direct application of the Navier-Stokes equations of gradient-free laminar incompressible fluid flow, in which the molecular viscosity is assumed to be a constant value independent of spatial coordinates, leads to a redefinition of the mathematical model. It is about the fluid boundary layer in the region of flow establishment in the motion problem of a semi-infinite plane, where the pressure gradient is zero. There is a situation when the number of equations is equal to three (two equations of momentum conservation and the equation of continuity), and the number of unknowns is equal to two - the number of the speed component. As a logical solution to the obtained inconsistency, it is proposed, as was already done for the problem of stationary motion of a plane and the problem of acceleration of a plane, to depart from the false statement about the constancy of molecular viscosity in the gradient-free boundary layer of an incompressible flow and consider molecular viscosity as a function of spatial coordinates. The need to consider the variable nature of molecular viscosity led to the discovery of another flaw in the Navier-Stokes theory. This non-trivial flaw was discovered during the application of the original numerical analytical method for solving the flow development region problem. The Navier-Stokes equations are supplemented by boundary conditions. The most important condition is the condition of fluid non-slipping on the surface of a solid body, which, by the way, does not follow any physical law. As a result, on the surface of a half-plane (or a moving body), the component of the velocity, which coincides with the direction of motion, has a constant value equal to the velocity of the body. It immediately follows from the continuity equation that the normal derivative of the normal component of the velocity must be equal to zero along the surface of the plane (body), since the longitudinal derivative of the velocity becomes zero. However, it is quite obvious that the velocity component normal to the surface of the plane (body) changes across the boundary layer in the region of current development, which indicates the presence of a normal gradient (both components) of the velocity. The conflict or contradiction is overcome by moving away from the generally accepted condition of non-slipping to the condition of partial non-slipping, or essentially the presence of sliding. Even with the sudden braking of any vehicle, the complete stop does not occur instantly, but after some finite time and distance, so in the case of the motion of a body in a stationary fluid, there is not an instant sticking, but a gradual one - from complete sliding, when a particle of liquid has just met a moving plane (body), to complete non-slipping at the end (and further) of the flow development region. Research methods. This work uses purely theoretical methods based on the use of calculus of variations, laws of physics, and ideas from everyday life. Conclusions. An improved model of a viscous Newtonian fluid in the area of flow development in the boundary layer was derived. On the basis of assumptions about the variable nature of the molecular viscosity, which already has two components, and the departure from the non-slipping condition, analytical solutions for both components of the velocity and both components of the molecular viscosity were obtained. A comparison of the obtained results with the results of other studies is presented.

Laminar Flow in a PEM Fuel Cell Cathode Channel

A Berman’s model for the laminar flow of incompressible fluid in a channel with permeable walls is extended for variable along the channel velocity of injection. The system of two–dimensional continuity and Navier–Stokes equations is reduced to a single ODE for the transversal velocity with coefficients depending on position along the channel. Numerical solution for the flow in the cathode channel of a PEM fuel cell is presented. The flow velocity profile across the channel is almost indistinguishable from the Poiseuille’s parabolic shape; however, the distribution of pressure gradient and longitudinal velocity differ quite significantly from the Berman’s result.

Viscous heating and cooling process in a mixed convection cavity with free-slip effect

Both no-slip and free-slip effects are enforced to the lid-driven cavity walls with Robin boundary condition. Viscous heating and cooling process happen when the shear force applied at the top lid is removed and the cavity is exposed to natural convection. This numerical study is performed for a two-dimensional laminar incompressible fluid flow with unsteadiness. The governing equations of time-dependent vorticity-stream function and thermal energy equation are solved using a finite difference method. Dissipation of viscous-heating of the cavity flow to the ambient is observed. The in-house Matlab® code is firstly verified with Ansys Fluent® finite element method at steady state. With no-slip boundary condition, temperature profile shows a reduction with acumination style at the beginning of the cooling process whereas a blunted profile is shown in free-slip boundary condition with temperature profile reduces dawdlingly. This is due to the movement of fluid is still able to create viscous heating at the outset of cooling process and this effect is even more pronounced at high Reynolds number when subject to free-slip boundary condition.

Temporal stability of multiple similarity solutions for porous channel flows with expanding or contracting walls

In this paper, the temporal stability of multiple similarity solutions (flow patterns) for the incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is analyzed. This work extends the recent results of similarity perturbations of Sun et al. [“Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls,” Appl. Math. Modell. 77, 738–755 (2020)] by examining the temporal stability with perturbations of general form (including similarity and nonsimilarity forms). Based on the linear stability theory, two-dimensional eigenvalue problems associated with the flow equations are formulated and numerically solved by a finite difference method on the staggered grids. The linear stability analysis reveals that the stability of the solutions is same with that under the perturbations of a similarity form within the range of the wall expansion ratio α. [−5≤α≤3 as in Sun et al., “Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls,” Appl. Math. Modell. 77, 738–755 (2020)]. Further, it is found that the expansion ratio α has a great influence on the stability of type I flows: in the case of wall contraction (α<0), the stability region of the cross-flow Reynolds number (R) increases as the contraction ratio (|α|) increases; in the case of wall expansion and 0<α≤1, the stability region increases as the expansion ratio (α) increases; in the case of 1≤α≤3, type I flows are stable for all R where they exist. The flows of other types (types II and III with −5≤α≤3 and type IV with α = 3) are always unstable. As a nonlinear stability analysis or a validation of the linear stability analysis, the original nonlinear two-dimensional time-dependent problem with an initial perturbation of general form over those flow patterns is solved directly. It is found that the stability with the nonlinear analysis is consistent with the linear stability analysis.

Numerical Analysis of Unsteady Laminar Flow past a Pentagonal Obstacle

The current article dispenses the numerical investigation of a two dimensional unsteady laminar flow of incompressible fluid passing a regular pentagonal obstacle in an open rectangular channel. The centre of attention of this work is the comparison of drag coefficients estimated for two distinct cases based on the orientation of face and corner of an obstacle against the flow direction. The numerical results shows that the corner – oriented obstacle bring about 42% larger value of drag coefficient at Re = 500 than face – oriented obstacle. The substantial growth in the expanse of vortex behind obstacle (presented as a function of fluid inertia 25 < Re < 500) is analyzed through the contours and streamline patterns of velocity field. The two eddies in the downstream become entirely unsymmetrical at Re = 500 for both the cases, whereas; the flow separation phenomena occurs a bit earlier in the face – oriented case at Re = 250. Two dimensional Pressure – Based – Segregated solver is employed to model the governing equations written in velocity and pressure fields. The numerical simulations of unsteady flow are presented for 50 seconds time frame with time step 0.01 by using one of the best available commercial based Computational Fluid Dynamics (CFD) software, ANSYS 15.0.

Using Fractal Calculus to Solve Fractal Navier–Stokes Equations, and Simulation of Laminar Static Mixing in COMSOL Multiphysics

Navier–Stokes equations describe the laminar flow of incompressible fluids. In most cases, one prefers to solve either these equations numerically, or the physical conditions of solving the problem are considered more straightforward than the real situation. In this paper, the Navier–Stokes equations are solved analytically and numerically for specific physical conditions. Using Fα-calculus, the fractal form of Navier–Stokes equations, which describes the laminar flow of incompressible fluids, has been solved analytically for two groups of general solutions. In the analytical section, for just “the single-phase fluid” analytical answers are obtained in a two-dimensional situation. However, in the numerical part, we simulate two fluids’ flow (liquid–liquid) in a three-dimensional case through several fractal structures and the sides of several fractal structures. Static mixers can be used to mix two fluids. These static mixers can be fractal in shape. The Sierpinski triangle, the Sierpinski carpet, and the circular fractal pattern have the static mixer’s role in our simulations. We apply these structures just in zero, first and second iterations. Using the COMSOL software, these equations for “fractal mixing” were solved numerically. For this purpose, fractal structures act as a barrier, and one can handle different types of their corresponding simulations. In COMSOL software, after the execution, we verify the defining model. We may present speed, pressure, and concentration distributions before and after passing fluids through or out of the fractal structure. The parameter for analyzing the quality of fractal mixing is the Coefficient of Variation (CoV).

Analysis of pressure characteristics under laminar and turbulent flow states inside the pilot stage of a deflection flapper servo-valve: Mathematical modeling with CFD study and experimental validation

Electro-hydraulic servo-valves are widely used components in the mechanical industry, aerospace and aerodynamic devices which precisely control the airplane or missile wings. Due to the small size and complex structure in the pilot stage of deflection flapper servo-valves, accurate mathematical models for the flow and pressure characteristics have always been very difficult to be built. In this paper, mathematical models for the pilot stage of deflection flapper servo-valve are investigated to overcome some gaps between the theoretical formulation and overall performance of the valve by considering different flow states. Here, a mathematical model of the velocity distribution at the flapper groove exit is established by using Schlichting velocity equations for in-compressible laminar fluid flow. Moreover, when the flow becomes turbulent, a mathematical model of pressure characteristics in the receiving ports is built on the basis of the assumption of the collision between the liquid and the jet as the impact of the jet on a moving block of fluid particles. To verify the analytical models for both laminar and turbulent flows, the pressure characteristics of the deflection flapper pilot stage are calculated and tested by using numerical simulation and experiment. Experimental verification of the theory is also presented. The computed numerical and analytical results show a good agreement with experimental data.

Numerical investigations of various aspects of plaque deposition through constricted artery

The pulsatile blood flow through constricted artery generates fluid mechanical forces on internal layer of artery, endothelium. These fluid mechanical factors affect endothelial lining from keeping artery healthy. In this paper, a series of numerical simulations of modeled bell shaped stenosed artery have been carried out for investigation of fluid mechanical factors of realistic pulsatile flow at the inlet of modeled stenosis with bell shaped geometry. The governing equations for two-dimensional unsteady laminar flow of incompressible fluid are solved by finite volume method followed by SIMPLER algorithm. The fluid mechanical factors, particularly wall pressure, streamline contour, peak wall shear stress, low wall shear stress and oscillatory shear index, having inferences to the arterial disease, are investigated by simulation results of different percentage of restrictions. All these parameters have a noticeable impact for the plaque deposition. The impacts of Reynolds number and Womersley number for both of mild stenosis and severe stenosis on arterial disease, atherosclerosis are also investigated by evaluating fractional flow reverse and oscillatory shear potential.

Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls

• We consider the temporal stability of multiple solutions for flows in a porous channel with moving walls for the first time. • We construct second order finite difference schemes for eigenproblems associated with the linear stability analysis . • We verify the linear stability results for the first time by solving the perturbed nonlinear time dependent equation. In this paper, a viscous, incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is considered. We present multiple symmetric steady-state solutions of this flow problem at several different expanding ratios, and use the linear stability theory to analyse the temporal stability for these solutions under symmetric, antisymmetric and general perturbations. We construct second order finite difference schemes for the eigenvalue problems with boundary conditions associated with those perturbations, and observe that most of these solutions which are stable under symmetric perturbations are unstable under antisymmetric perturbations. Furthermore, we verify the linear stability analysis results by directly solving the original perturbed nonlinear time dependent problem, and find that both stability results are consistent.

HEAT TRANSFER AND FRICTION CHARACTERISTICS OF AN ARRAY OF PERFORATED FINS UNDER LAMINAR FORCED CONVECTION

Three-dimensional incompressible laminar fluid flow and heat transfer of a heated array of circular perforated fins are examined numerically. The Navier–Stokes and energy equations are solved by a finite volume method using the SIMPLE algorithm. The second order upwind technique is employed to discretize the momentum and energy equations. Computations were performed for a range of Reynolds numbers 100 ≤Re ≤ 350. Thermal performance and effectiveness as well as friction coefficient of perforated and solid fins are determined for the optimum porosity. The results show that the average coefficient of friction reduces with increasing the Reynolds number and the number of perforations. The heat transfer rate increases with the porosity and the Reynolds number. It is found that perforated fin effectiveness decreases with the number of peroration at constant Reynolds numbers.

Adaptive topology optimization for incompressible laminar flow problems with mass flow constraints

This paper presents an adaptive no-slip boundary conditions scheme within the context of topology optimization of incompressible laminar fluid flow problems with mass flow constraints. The flow of fluid is described by the Incompressible Navier–Stokes (INS) equations and the density-based topology optimization methodology together with Heaviside projection is used to optimize the flow layout. Adjoint sensitivity analysis is used to compute the sensitivities of the objective and constraints which are used by a gradient-based optimizer to solve the problem. To accurately capture flow characteristics and improve computational efficiency of the forward analysis, no-slip nodal boundary conditions are adaptively added (and removed) as the design evolves, with these operations guided entirely by the formal sensitivity analysis. Several design examples are presented, including 2-D and 3-D design problems purposely constructed to challenge the proposed algorithm. Presented solutions are shown to achieve crisp (0–1) representations of topology satisfying required flow rates, with no pressure diffusion and flow seepage through the solid material phase. The results reveal the performance and efficiency of the proposed approach for use in such applications as the design of flow manifolds.

A fluid‐structure interaction approach with inherently fulfilled coupling conditions

AbstractIn the present contribution a fluid‐structure interaction (FSI) approach based on a mixed least‐squares finite element method (LSFEM) is introduced. The LSFEM is a variational approach for constructing finite element formulations, see e.g. [1] and [2]. In this work, the proposed FSI method is based on the formulation of mixed finite elements in terms of stresses and velocities for both the fluid and the solid regime. The conforming discretization of the stresses and velocities in the spaces H(div) and H1, respectively, leads to the inherent fulfillment of the coupling conditions of a FSI method. The idea of the LSFEM based FSI scheme is the solution of the full problem consisting of both domains in one triangulation at the same time. The general idea on this was firstly annotated in [8]. A numerical example considering an incompressible, linear elastic solid at small deformations (compare to [3]) together with an incompressible laminar fluid flow demonstrates the applicability of the LSFEM‐FSI approach.

Mathematical model of viscous petroleum fluid for consequential migration in porous solid

Viscous fluid transport or crude natural petroleum migration in a form of the laminar flow of incompressible fluids and hydrocarbon classification are principal to study the mass flow rate according to Darcy’s law and Hagen-Poiseuille law. Natural petroleum classification is concerned for its migration from a reservoir to underground surface. Its chemical aspect can be categorized into hydrocarbon as paraffinic, intermediate paraffinic, aromatic, and asphaltic classes, in the ratio of 5:10:4:1, respectively. Its property signifies two types of paraffinic and intermediate classes. The volumetric flow rates implicate to petroleum migration which refers to hydrocarbon classifications. These indicate possibility of migration with the mass flow of hydrocarbon class. Correlation of their volumetric flow rates can appraise the configuration of the volumetric flow rates as a function of mass of flow rates with three classifications: typical crude oil, heavy crude oil, bitumen. Crude petroleum classes are bitumen to heavy crude oil, and can be anticipated a progress of the migration zones of heavy crude oil, heavy crude oil-bitumen, and bitumen zone. Consequently, assessment results expose that the slow volumetric flow rates associate to petroleum migration, pressure depths, and crude petroleum classification.

Numerical study of the fluid-dynamic loading on pipes conveying fluid with a laminar velocity profile

The paper deals with the fluid-dynamic forces of laminar fluid flow in long vibrating pipes as well as in long curved pipes at rest. It focuses on the contribution of the fluid-dynamic force related to the centrifugal acceleration. The study is based on the results of a CFD numerical model of an incompressible laminar fluid flow in the pipe, which is deflected in a bending, beam-type mode. The results are presented in terms of the centrifugal correction factor, which is defined as the ratio between the actual centrifugal-related term of the observed fluid-dynamic force and the solution of the one-dimensional model. The estimated values of the respective correction factor are presented for a range of vibration frequencies, lengths of the pipe and amplitudes of the pipe deflection. The results show that all the listed parameters affect the centrifugal correction factor. The resulting values are also compared and assessed with respect to some accessible analytical solutions. The influence of the predicted values of the centrifugal correction factor on the onset of static and dynamic instabilities of pipes conveying fluid is also discussed.

Temperature Dependent Variable Properties on Mixed Convective Unsteady MHD laminar Incompressible fluid Flow with Heat Transfer and Viscous Dissipation

In the present paper we investigated that the numerical solution of mixed convective laminar boundary layer flow with uniform transverse magnetic field for a vertical porous plate. The fluid properties are considered the nonlinear functions of temperature as μ = μ∞(T / T∞)A and κ = κ∞(T / T∞)C For flue gases the values of the exponents A and C are taken as A=0.7, and C=0.83. Also time dependent suction velocity is considered. The governing equations are reduced into ordinary differential equations by introducing the suitable similarity variables and then solved numerically by using sixth ordered Runge-Kutta method and Nachtsheim Swigert iteration technique. The numerical results of our physical interest are shown graphically and tabular form.

Effects of wall waviness and temperature variations on the fluid flow and natural convection in an inclined corrugated channel

A parametric study involving the effects of some combinations of parameters, in particular, different combinations of the Rayleigh number, amplitude, temperature, and inclination angle of a two-dimensional long wavy-walled channel on a laminar incompressible fluid flow and natural convection within the channel is performed. The considered channel has an undulated wall as one side of the channel, and a parallel flat wall at a differentially different temperature as its counterpart. The channel sustains variable inclination angle, variable wavy wall amplitude, and variable temperature-difference between its two walls. A perturbation technique in terms of the small waviness of the undulated wall is performed to obtain a set of non-linear ordinary differential equations for the main flow and its perturbations. Solving this set of equations determines the streamline and temperature profiles for the imposed varying parameters. The results reveal that the fluid velocity along the channel axis increases with increasing the Rayleigh number, but decreases with increasing the channel inclination angle. The results also show that eddies appeared due to the steep undulations intensify as the temperature-difference between the two channel walls increases. The veracity of the present work is demonstrated through comparing the obtained results with those available in the literature.

Benchmark tests based on the Couette viscometer—I: Laminar flow of incompressible fluids with inertia effects and thermomechanical coupling

The Couette viscometer is a well-known problem of fluid mechanics, well-suited for the verification of numerical methods. The aim of this work is to extend the classical steady state mechanical solution obtained in fluid mechanics, both to strongly-coupled thermomechanical problems in the case of laminar and incompressible fluid flows, and to solid-type nonlinear behaviours. Extended solutions will allow for the verification of new formulations of a mixed P1+/P1 finite element developed both in fluid and solid mechanics, within a temperature/velocity/pressure formulation coupled with an implicit (backward) Euler algorithm in time. In the present Part I, we address the case of the laminar flow of incompressible fluids with inertia effects and thermomechanical coupling. The verification performed on the reference solutions developed clearly evidence the good behaviour of the fluid finite element. The extension to solid-type nonlinear behaviours for strongly-coupled thermomechanical problems will be the subject of Part II.

Benchmark tests based on the Couette viscometer—II: Thermo-elasto-plastic solid behaviour in small and large strains

The Couette viscometer is a well-known problem of fluid mechanics, well-suited for the verification of numerical methods. The aim of this work is to extend the classical steady state mechanical solution obtained in fluid mechanics and to use the extended solutions to assess new finite elements. Part I was devoted to the case of laminar flow of incompressible fluids with inertia effects and thermomechanical coupling. The present Part II focuses on solid-type nonlinear behaviours; we address the cases of elastic–plastic and thermo-elastic–plastic von Mises materials, both in small and large strains. The extended solutions permit to assess a new formulation of a mixed P1+/P1 finite element in solid mechanics, in a temperature/velocity/pressure formulation coupled with an implicit (backward) Euler algorithm in time. The verification evidences a good behaviour of the solid finite element.

A numerical study on the effects of 2d structured sinusoidal elements on fluid flow and heat transfer at microscale

Better understanding of laminar flow at microscale level is gaining importance with recent interest in microfluidics devices. The surface roughness has been acknowledged to affect the laminar flow, and this feature is the focus of the current work to evaluate its potential in heat transfer enhancement. Based on various roughness characterization schemes, the effect of structured roughness elements on incompressible laminar fluid flow is analyzed and the hydrodynamic and thermal characteristics of minichannels and microchannels are studied in the presence of roughness elements using CFD software, FLUENT. Structured roughness elements following a sinusoidal pattern are generated on two opposed rectangular channel walls with a variable gap. A detailed study is performed to understand the effects of roughness height, roughness pitch, and channel separation on pressure drop and heat transfer coefficient. As expected, the structured roughness elements on channel walls result in pressure drop and heat transfer enhancements as compared to smooth channels due to the combined effects of area increase and flow modification. The current numerical scheme is validated with the experimental data and can be used for design and estimation of transport processes in the presence of roughness features.

Meshless local Petrov–Galerkin method-higher Reynolds numbers fluid flow applications

The meshless local Petrov–Galerkin (MLPG) primitive variable based method is extended to analyze the incompressible laminar fluid flow within or over some different two-dimensional geometries. Although still in laminar regions, the Reynolds numbers considered in this study are in the ranges for which, in the literature, the MLPG primitive variable based method has never produced stable solutions and comparable results with those of the conventional methods. The considered test problems include, a steady lid-driven cavity flow with Reynolds numbers up to and including 10,000, a flow over a backward-facing step at 800 Reynolds number, and a transient fluid flow past a circular cylinder with Reynolds numbers up to and including 200. The present method solves the incompressible Navier–Stokes (N–S) equations in terms of the primitive variables using the characteristic-based split (CBS) scheme for discretization. The weighting function in the weak formulation of the governing equations is taken as unity, and the field variables are approximated using the moving least square (MLS) interpolation. For validation purposes, the obtained results are compared with those of the conventional numerical methods. The agreements of the compared results reveal a step forward towards further applications of the MLPG primitive variable based approach.