Laminar Boundary Layer Equations Research Articles

ADVECTION AND DEPOSITION OF MICRODROPLETS IN STAGNATION POINT FLOW

We investigate trajectories of microscale evaporating droplets in a stagnation point flow near a wall of a respiratory airway. The configuration is motivated by the problem of advection and deposition of microscale droplets of respiratory fluids in human airways during transmission of infectious diseases, such as tuberculosis and COVID-19. Laminar boundary layer equations are solved to describe the airflow while the equations of motion of the droplet include contributions from gravity, aerodynamic drag, and Saffman force. Evaporation is accounted for at both the droplet surface and the wall of the respiratory airway and is shown to delay droplet deposition as compared to the predictions of isothermal models. Evaporation at the airway wall has a stronger effect on droplet trajectories than evaporation at the droplet surface, leading to droplets being advected away by the flow and thus avoiding deposition in the stagnation point flow region.

Compressible modification of boundary-layer momentum thickness localized method in transition model

In current transition models, Menter’s correlation between the maximum vorticity Reynolds number Revmax in the boundary layer and momentum thickness Reynolds number Reθ has been widely used in the localization of the boundary layer length scale. However, the existing correlations have problems in correctly reflecting compressibility effects and only relying on local variables. In this paper, the similarity solutions of laminar compressible boundary layer equations are adopted to study the relation between Revmax and Reθ. Through extracting the local Mach number Malocal and the ratio of wall temperature to the local temperature Tw/Tlocal right at the position where Revmax happens, the influences of the Mach number effect and wall heat transfer effect are characterized, meanwhile a new correlation only relying on local variables between Revmax and Reθ is proposed. For plat-plate boundary layers, the relative error is tiny within a large Mach number and wall-to-adiabatic temperature ratio range and the error caused by the streamwise pressure gradient is not of great concern for general pressure gradients. Furthermore, numerical results over common geometries, including wedges, sharp cones and blunt cones, demonstrate that the proposed correlation could give much better predictions of the relation between Revmax and Reθ than previous correlations. At last, the present correlation is applied to the γ-Reθ transition model as an application presentation.

MHD and Heat Generation Along a Vertical Flat Plate with Variable Viscosity and Viscous Dissipation: A Numerical Solution

Numerous researchers have looked into the importance of natural convection in the context of engineering, and this topic has been extensively studied. This study investigates the nature of heat generation and viscous dissipation in MHD natural convection flow with changing viscosity. Laminar flow and boundary layer equations with unstable boundaries in two dimensions are the subject of this article. The fundamental governing equation is turned into a dimensionless governing equation by using the necessary variables. The Crank Nicolson scheme is an efficient implicit finite difference approach for numerical computations of these equations. Heat generation and viscosity dissipation owing to MHD and changing viscosity were explored in this work. Several effects of various parameters are demonstrated in this study, and they are compared to the velocity and temperature profiles, skin friction, and local heat transfer coefficients of other researchers. Compare the present numerical results to the work outcomes that were previously released. It also compares the number of works available to the number of works published previously. The results are given in both figures and tables for various values of related physical parameters.

Mathematical Formulation of Bubble Formation after Compressible Boundary Layer Separation: Preliminary Numerical Results

Laminar boundary layer (BL), under adverse pressure gradient, can separate. The separated shear layer reattaches to form a laminar separation bubble. Such bubbles are usually observed on gas turbine blades, on low Reynolds number wings and close to the leading edges of airfoils. Presence of bubbles has a weakening effect on the performance of a fluid device. The understanding of the prevailing mechanism of the separation bubble and ways to control it are essential for the efficient design of these devices. This is due to the significance of drag reduction in these various aerodynamic devices, such as gas turbines, re-entry space vehicles and airfoils. This study introduces a two-dimensional mathematical formulation of bubble formation after flow separation. The laminar BL equations with appropriate boundary conditions are dimensionalized using the Falkner-Skan transformation. Additionally, using the Keller-box method, the nonlinear system of partial differential equations (PDEs) is numerically solved. This study presents preliminary numerical results of bubble formation in low Mach numbers. These results reveal that after separation, a laminar bubble is formed in all studied cases, for Mach numbers, M = 0.2, 0.33 and 1.0. The flow after separation reverses close to the wall and finally reattaches downstream, in a new location. As the Mach number increases, this effect is more intense. After reattachment, the BL is again established in a lower energy level and the velocity field is substantially reduced, for all cases.

The MHD Newtonian hybrid nanofluid flow and mass transfer analysis due to super-linear stretching sheet embedded in porous medium

The steady magnetohydrodynamics (MHD) incompressible hybrid nanofluid flow and mass transfer due to porous stretching surface with quadratic velocity is investigated in the presence of mass transpiration and chemical reaction. The basic laminar boundary layer equations for momentum and mass transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The mass equation in the presence of chemical reaction is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation. The mass transfer is analyzed for two different boundary conditions of concentration field that are prescribed surface concentration (PSC) and prescribed mass flux (PMF). The asymptotic solution of concentration filed for large Schmidt number is analyzed using Wentzel-Kramer-Brillouin (WKB) method. The parameters influence the flow are suction/injection, superlinear stretching parameter, porosity, magnetic parameter, hybrid nanofluid terms, Brinkman ratio and the effect of these are analysed using graphs.

A CFD Tutorial in Julia: Introduction to Laminar Boundary-Layer Theory

Numerical simulations of laminar boundary-layer equations are used to investigate the origins of skin-friction drag, flow separation, and aerodynamic heating concepts in advanced undergraduate- and graduate-level fluid dynamics/aerodynamics courses. A boundary-layer is a thin layer of fluid near a solid surface, and viscous effects dominate it. Students must understand the modeling of flow physics and implement numerical methods to conduct successful simulations. Writing computer codes to solve equations numerically is a critical part of the simulation process. Julia is a new programming language that is designed to combine performance and productivity. It is dynamic and fast. However, it is crucial to understand the capabilities of a new programming language before attempting to use it in a new project. In this paper, fundamental flow problems such as Blasius, Hiemenz, Homann, and Falkner-Skan flow equations are derived from scratch and numerically solved using the Julia language. We used the finite difference scheme to discretize the governing equations, employed the Thomas algorithm to solve the resulting linear system, and compared the results with the published data. In addition, we released the Julia codes in GitHub to shorten the learning curve for new users and discussed the advantages of Julia over other programming languages. We found that the Julia language has significant advantages in productivity over other coding languages. Interested readers may access the Julia codes on our GitHub page.

Numerical research on airfoil transition delay by alternative current dielectric barrier discharge actuation

A Dielectric Barrier Discharge (DBD) plasma actuator can create a body force which locally accelerates the base flow leading to an attenuation of broadband disturbance to delay the transition. In this study, numerical simulation on an NLF0416 airfoil is conducted to investigate transition delay and drag reduction by a DBD plasma actuator. To simulate plasma’s effect more accurately, boundary-layer data is acquired from Reynolds Averaged Navier Stocks (RANS) equations instead of laminar boundary layer equations, although RANS equations need a much finer boundary-layer grid, and the linear stability analysis method is used to analyze the boundary layer and get the transition point. In this study, the influences of different actuation intensities and positions are investigated, and results show that if the actuation intensity is stronger and the actuation position is closer to the base transition point, more drag reduction can be obtained. However, the efficiency of plasma transition delay is really low. For example, when the actuation voltage is 16 kV, the actuation frequency is 1 kHz, and the main Mach number is 0.1, the saved power due to drag reduction is about 5.09 W, but the power consumed is about 32.61 W, and the efficiency is just 15.6%.

On non-Oberbeck–Boussinesq effects in Rayleigh–Bénard convection of air for large temperature differences

We present direct numerical simulations of non-Oberbeck–Boussinesq (NOB) Rayleigh–Benard (RB) convection due to large temperature differences in two-dimensional (2-D) and three-dimensional (3-D) cells. Perfect air is chosen as the operating fluid and the Prandtl number ( ) is fixed to 0.71 for the reference state . In the present system, we consider large temperature differences ranging from 60 K to 240 K, and relatively strong NOB effects are induced at moderate Rayleigh numbers ( ) in the range . The large temperature difference also induces the turbulence system with large density variation. Due to top-down symmetry breaking under NOB conditions, an increase of the centre temperature is found compared to the arithmetic mean temperature of the top and bottom plates, and the shift of is strongly dependent on Rayleigh number and temperature differential . The NOB effects on the Nusselt number ( ) are quite small ( ). The power-law scalings of versus are robust against NOB effects, even for the extremely large temperature difference 240 K, which has never been reached in previous experiments and simulations. The Reynolds numbers , as well as the scalings of versus , are also insensitive to NOB effects. It is noteworthy that the influence of NOB effects on and in 3-D RB flow are weaker than its 2-D counterpart. Furthermore, the extended laminar boundary layer (BL) equations are developed based on the low-Mach-number Navier–Stokes equations, which qualitatively predicts the NOB effects on velocity profiles. Direct numerical simulation results indicate that the top and bottom thermal BLs can compensate each other much better than the velocity BLs under NOB conditions, which contribute to the robustness of .

Analytical Study of Laminar Boundary Layers near Blunted Bodies

In high-speed flows, blunt body elements having an irregular shape due to which gas dynamic parameters undergo significant changes are, as a rule, the most thermally loaded parts. In this respect, a quick evaluation of the thermal load on blunt bodies is important. Laminar boundary-layer equations given in special coordinates in a constant axisymmetric flow of a compressible perfect gas are considered. The adhesion condition is accepted as a boundary condition on the wall and it is assumed that the speed and temperature on the boundary correspond to the values of the external flow. The Pohlhausen method introduces the concepts of the displacement thickness and momentum thickness, finds relations between these values and the boundary-layer thickness, and derives a differential equation to determine the boundary-layer form parameter such that all other characteristics of the boundary layer are determined via it. The Pohlhausen method is modified in order to simplify the calculation by excluding the differential equations from it. Similarly to the velocity, a special function including the enthalpy and dimensionless kinetic parameter to be determined is introduced as a biquadratic polynomial. The boundary conditions on the wall and on the border of the boundary layer are used to determine the polynomial coefficients. The kinetic parameter is defined in different ways for bodies of various shapes. We provide the results of applying the proposed method for the calculation of heat flux that have been analyzed numerically (in various papers) in complete systems of Navier-Stokes and Prandtl equations. The comparison of the results shows the efficiency of the proposed method.

Accelerated laminar flow near the heated permeable surface

The paper reports on numerical simulation of an accelerated laminar boundary layer over the heated permeable wall. Xenon was used as a working gas. The finite difference method was applied to solve the system of laminar boundary layer equations with variable gas properties. It has been shown that the gas injection in the mentioned conditions increases maximal velocity within the boundary layer and decreases skin-friction coefficient and thermal Stanton number. On the contrary, gas suction decreases an overshoot phenomenon down to its vanishing and increases skin-friction coefficient and thermal Stanton number up to the value at asymptotic suction. The paper considers a separation of thermal boundary layer at gas injection. The proposed conventional point of this separation is situated farther from the leading edge than the separation point of the dynamic boundary layer.

A numerical method for solution of the discharge coefficients in externally pressurized gas bearings with inherent orifice restrictors

To study the discharge coefficients of aerostatic bearings, a new numerical method which combines the method of “separation of variables” for solution of laminar boundary-layer equations and analytical solution of Reynolds equation is proposed. The discharge coefficients are suitable for solution of Reynolds equation. The influences of flow and geometry parameters on discharge coefficients are investigated and the results indicate that there exists parameters insensitive and sensitive regions in discharge coefficient analysis. Further research shows that flow and geometry parameters affect discharge coefficient by influencing the pressure ratio (pr/ps) and discharge coefficient tends to be constant when pressure ratio is approximately less than 0.6, which is the reason that there exists the parameters insensitive and sensitive regions.

Numerical research of pressure depression in aerostatic thrust bearing with inherent orifice

Abstract The undesirable pressure depression in aerostatic bearing will cause the loss of aerostatic bearing load capacity. Therefore, the method of “separation of variables” (MSV) for solution of laminar boundary-layer equations is utilized to investigate the pressure depression. The influences of supply pressure, orifice diameter, film thickness and pressure ratio on the pressure depression are studied. The results manifest that pressure depression is weakened with decreasing supply pressure and film thickness and increasing orifice diameter. The increase of pressure ratio which is determined by the flow and geometry parameters will weaken the inertial effect and then results in weakening the pressure depression. What's more, when pressure ratio is larger than the critical value 0.9409, the pressure depression will disappear.

Effect of gas properties on accelerated laminar boundary layer over a heated wall

The paper reports on numerical investigation of an accelerated laminar boundary layer over a heated wall at Prandtl numbers from 0.23 to 2.39. Properties of three gases were simulated: air, helium-xenon mixture and superheated water vapor. The finite difference method was applied to solve a system of laminar boundary layer equations with variable gas properties. It has been shown that the influence of Prandtl number, viscosity, heat conductivity and heat capacity on the flow at studied conditions is secondary relative to gas density, which is the determinant factor for the overshoot magnitude and other flow parameters. Nevertheless, at insignificant differences in density distributions the overall decrease of Prandtl number and decrease of viscosity and heat capacity and increase of heat conductivity separately lead to an increase of maximal velocity within the boundary layer.

Magnetohydrodynamics flow of a nanofluid driven by a stretching/shrinking sheet with suction.

The present paper investigates the effect of a mathematical model describing the aforementioned process in which the ambient nanofluid in the presence of suction/injection and magnetic field are taken into consideration. The flow is induced by an infinite elastic sheet which is stretched along its own plane. The stretching/shrinking of the sheet is assumed to be proportional to the distance from the slit. The governing equations are reduced to a nonlinear ordinary differential equation by means of similarity transformation. The consequential nonlinear equation is solved analytically. Consequences show that the flow field can be divided into a near-field region and a far-field region. Suction on the surface plays an important role in the flow development in the near-field whereas the far-field is responsible mainly by stretching. The electromagnetic effect plays exactly the same role as the MHD, which is to reduce the horizontal flow resulting from stretching. It is shown that the behavior of the fluid flow changes with the change of the nanoparticles type. The present study throws light on the analytical solution of a class of laminar boundary layer equations arising in the stretching/shrinking sheet problem.

Lagrangian analysis of the laminar flat plate boundary layer

The flow properties at the leading edge of a flat plate represent a singularity to the Blasius laminar boundary layer equations; by applying the Lagrangian approach, the leading edge velocity profiles of the laminar boundary layer over a flat plate are studied. Experimental observations as well as the theoretical analysis show an exact Gaussian distribution curve as the original starting profile of the laminar flow. Comparisons between the Blasius solution and the Gaussian curve solution are carried out providing a new insight into the physics of the laminar flow.

Adaptive viscous–inviscid interaction method for analysis of airfoils in ground effect

The behavior of an airfoil in ground proximity is significantly different from out of ground. Therefore, understanding the effect of proximity to ground can help determine its aerodynamic characteristics, which are important design parameters. In this paper, an accurate viscous–inviscid interacting method is investigated as an alternative method to computing heavy Navier–Stokes solution for utilizing in the surface effect zone. The inviscid part is modeled by a 2D panel method, and the viscous part is modeled by solving the integral form of the laminar and turbulent boundary layer equations. Validation of the incompressible 2D version of the written code (so-called WIGPMBL) is carried out against experimental and other numerical codes for different airfoil geometries at various Reynolds numbers. Various aspects of aerodynamic characteristics such as pressure distribution, lift and drag coefficient, transition and separation points of symmetric and asymmetric airfoils have been validated. Consequently, the assessments have verified that the present method has good accuracy for predicting the effect of surface proximity.

Predictor homotopy analysis method (PHAM) for nano boundary layer flows with nonlinear Navier boundary condition: Existence of four solutions

In the present work, the classical laminar boundary layer equation of the flow away from the origin past a wedge with the no-slip boundary condition replaced by a nonlinear Navier boundary condition is considered. This boundary condition contains an arbitrary index parameter, denoted by n > 0, which appears in the differential equation to be solved. Predictor homotopy analysis method (PHAM) is applied to this problem and more, it is proved corresponding to the value n = 1/3, there exist four solutions. Furthermore, these solutions are approximated by analytical series solution using PHAM for further physical interpretations.

Existence results for nano boundary layer flows with nonlinear Navier boundary condition

The standard no slip boundary condition of classical fluid mechanics is no longer valid at the micro- and nano-scale and should be replaced by a boundary condition that allows some degree of tangential slip. In the present work, the classical laminar boundary layer equation of the flow away from the origin past a wedge with the no-slip boundary condition replaced by a nonlinear Navier boundary condition is revisited. This boundary condition includes an arbitrary index parameter, denoted by n > 0 , which appears in the coefficients of the differential equation to be solved. It is proved corresponding to the value n = 1 3 , there are exactly three situations for the problem: (i) there is no solution; (ii) there exist two solutions; (iii) there exist four solutions. Furthermore, the exact analytical solution of the problem is given in terms of parabolic cylinder functions for further physical interpretations.

Natural Convection Induced by a Heated Vertical Plate Embedded in a Porous Medium with Transpiration: Local Thermal Non-equilibrium Similarity Solutions

This paper is concerned with the thermal non-equilibrium free convection boundary layer, which is induced by a vertical heated plate embedded in a saturated porous medium. The effect of suction or injection on the free convection boundary layer is also studied. The plate is assumed to have a linear temperature distribution, which yields a boundary layer of constant thickness. On assuming Darcy flow, similarity solutions are obtained for governing the steady laminar boundary layer equations. The reduced Nusselt numbers for both the solid and fluid phases are calculated for a wide range of parameters, and compared with asymptotic analyses.

Analytical Solution of Laminar Boundary Layer Equations Over A Flat Plate by Homotopy Perturbation Method

In this paper, the problem of an incompressible viscous flow over a flat plate is presented. The Homotopy perturbation method (HPM) is employed to solve the well-known Blasius nonlinear differential equation. It has been tried to use a special technique by which one be able to obtain solutions that are very close to the exact solution of the equation. The obtained results have been compared with the exact solution of Blasius equation and another results obtained in previous works so that the high accuracy of results are clear. Also it is found that this method is powerful mathematical tool and can be applied to a large class of linear and nonlinear problems in different fields of science and engineering especially some fluid mechanics equations.