Blasius Boundary Layer Research Articles

A Hydraulic Study of Trickle Irrigation.(Dept.C)

Polyethylene pipes are widely used in the trickle irrigation system. The main objective of this research work is to determine friction factors for this type of pipes. A field study was carried out on two trickle irrigation laterals each of which was about 240 m in length. One line had 18.5 mm inside diameter, the other consisted of two parts, the first was 21.5 mm inside diameter and the other was 17 mm inside diameter. The measured friction factor was compared by the corresponding friction factors given by conventional friction formulae. Blasius equation has to be modified to cope with the measured factors, and predicted equations are presented. The relationship between the discharge of the orifice type emitter and the pressure exerted on it, is given.

A variable step-size fourth-derivative hybrid block strategy for integrating third-order IVPs, with applications

ABSTRACT In this paper, an efficient fourth-derivative two-step hybrid block strategy (FDTHBS) to get the approximate solution of a third-order IVP with applications to problems in Fluid Dynamics and Engineering is constructed. The mathematical derivation of the proposed FDTHBS is based on the interpolation and collocation of the exact solution and its derivatives at the selected equidistant grid and off-grid points. The theoretical characteristics of the proposed method are analysed. An embedding-like procedure is considered and executed in variable step-size mode to get better performance of the newly developed strategy. Some test problems, including the well-known Blasius equation and three different types of non-linear thin-film flow problems, are integrated numerically to ascertain the superior impact of our developed error estimation and control strategy. It is worth concluding that the proposed technique is not only efficient in term of CPU time, but also minimizes errors and support the analytical results.

A boundary shape function iterative method for solving nonlinear singular boundary value problems

In this paper, a novel iterative algorithm is developed to solve second-order nonlinear singular boundary value problem, whose solution exactly satisfies the Robin boundary conditions specified on the boundaries of a unit interval. The boundary shape function is designed such that the boundary conditions can be fulfilled automatically, which renders a new algorithm with the solution playing the role of a boundary shape function. When the free function is viewed as a new variable, the original singular boundary value problem can be properly transformed to an initial value problem. For the new variable the initial values are given, whereas two unknown terminal values are determined iteratively by integrating the transformed ordinary differential equation to obtain the new terminal values until they are convergent. As a consequence, very accurate solutions for the nonlinear singular boundary value problems can be obtained through a few iterations. The present method is different from the traditional shooting method, which needs to guess initial values and solve nonlinear algebraic equations to approximate the missing initial values. As practical applications of the present method, we solve the Blasius equation for describing the boundary layer behavior of fluid flow over a flat plate, where the Crocco transformation is employed to transform the third-order differential equation to a second-order singular differential equation. We also solve a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity.

Continuous adjoint complement to the Blasius equation

This manuscript is concerned with a continuous adjoint complement to two-dimensional, incompressible, first-order boundary-layer equations for a flat plate boundary layer. The text is structured into three parts. The first part demonstrates that the adjoint complement can be derived in two ways, following either a first simplify then derive or a first derive and then simplify strategy. The simplification step comprises the classical boundary-layer (BL) approximation, and the derivation step transfers the primal flow equation into a companion adjoint equation. The second part of the paper comprises the analyses of the coupled primal/adjoint BL framework. This leads to similarity parameters, which turn the partial-differential-equation (PDE) problem into a boundary value problem described by a set of ordinary-differential-equations (ODEs) and support the formulation of an adjoint complement to the classical Blasius equation. Opposite to the primal Blasius equation, its adjoint complement consists of two ODEs, which can be simplified depending on the treatment of advection. It is shown that the advective fluxes, which are frequently debated in the literature, vanish for the investigated self-similar BL flows. Differences between the primal and the adjoint Blasius framework are discussed against numerical solutions, and analytical expressions are derived for the adjoint BL thickness, wall shear stress, and subordinated skin friction and drag coefficients. The analysis also provides an analytical expression for the shape sensitivity to shear driven drag objectives. The third part assesses the predictive agreement between the different Blasius solutions and numerical results for Navier–Stokes simulations of a flat plate BL at Reynolds numbers between 103≤ReL≤105. It is seen that the reversal of the inlet and outlet locations and the direction of the flow, inherent to the adjoint formulation of convective kinematics, poses a challenge when investigating real finite length (finite Re-number) flat plate boundary layer problems. Efforts to bypass related issues are discussed.

Exploring Blasius and Falkner-Skan Equations with Python

The solution of Blasius Equation with various numerical methods is reviewed. The Falkner-Skan equation is also solved with these methods. The application methods range from classical Euler, Runge-Kutta, to Artificial Intelligence Machine Learning methods. The code for each method is available for verification in python scripting language.

Numerical solution for Falkner-Skan flow of hybrid nanofluid with porosity effect

ABSTRACT Hybrid nanofluid is known to improve heat transfer performance, and its advantages have led to relatively reasonable expectations for their applications. This research considered a moving wedge, namely the Falkner-Skan model, which is well-known in the aerodynamic field. Hybrid nanofluid has been chosen where the dispersion of alumina and copper nanoparticles with water as the base fluid is considered in the unsteady mixed convection flow over moving wedge. By using similarity transformations, the governing equations are converted into ordinary differential equations and then numerically solved using MATLAB bvp4c solver. The increasing values of porosity parameter caused the velocity of hybrid nanofluid to increase. The results also indicated that, the effect of porosity parameter improved the values of skin friction coefficient but decrease the value of Nusselt number.

Homotopy Simulation of Dissipative Micropolar Flow and Heat Transfer from a Two-Dimensional Body with Heat Sink Effect

Non-Newtonian flow from a wedge constitutes a fundamental problem in chemical<br /> engineering systems and is relevant to processing of polymers, coating systems, etc. Motivated by such applications, the homotopy analysis method (HAM) was employed to<br /> obtain semi-analytical solutions for thermal convection boundary layer flow of incompressible micropolar fluid from a two-dimensional body (wedge). Viscous dissipation<br /> and heat sink effects were included. The non-dimensional boundary value problem<br /> emerges as a system of nonlinear coupled ordinary differential equations, by virtue of<br /> suitable coordinate transformations. The so-called Falkner-Skan flow cases are elaborated. Validation of the HAM solutions was achieved with earlier simpler models, as well as with a Nakamura finite difference method for the general model. The micropolar model employed simulates certain polymeric solutions quite accurately, and features rotary motions of micro-elements. Primary and secondary shear stress, wall couple stress, Nusselt number, microrotation velocity, and temperature were computed for the effect of<br /> vortex viscosity parameter (micropolar rheological), Eckert number (viscous dissipation),<br /> Falkner-Skan (pressure gradient) parameter, micro-inertia density, and heat sink parameter. The special cases of Blasius and stagnation flow were also addressed. It was observed from the study that the temperature and thermal boundary layer thickness are both suppressed with increasing wedge parameter and wall heat sink effect, which is beneficial to temperature regulation in polymer coating dynamics. Further, strong reverse spin was generated in the microrotation with increasing vortex viscosity, which resulted in<br /> increase in angular momentum boundary layer thickness. Also, both primary and secondary skin friction components were reduced with increasing wedge parameter. Nusselt number was also enhanced substantially with greater wedge parameter.

Linear to turbulent Görtler instability transition

We present results from a highly resolved large-eddy simulation of a freely developing Blasius profile over a concave boundary in a large spanwise domain. Due to the large initial Reynolds and Görtler numbers (Reθ,0 = 1175, Gθ,0 = 75), we observe the onset of two dominant wavelengths: the first dominating in the linear/transition region, λ1, and the second dominating in the turbulent region, λ2. Extending previous linear stability analysis (LSA) to higher Görtler numbers and non-dimensional wavenumbers, both dominant wavelengths of the Görtler instability correspond to predictions of LSA, the latter comparable to laminar theory by replacing the kinematic viscosity with the turbulent viscosity in the definition of the Görtler number. The predicted spatial modes compare well with the computed profiles for both λ1 and λ2. The skin friction coefficient Cf is found heterogeneous in the spanwise direction according to the emerging wavelengths λ1 and λ2 of the Görtler instability. We report a smooth increase in Cf from the theoretical predictions of a laminar boundary layer to those for a turbulent boundary layer over a flat plate. The values only slightly overshoot these predictions in the domain of existence of the second dominant wavelength λ2, very different from that reported at lower Reynolds numbers.

Significance of nonlinear stratification in convective Falkner-Skan flow of Jeffrey fluid near the stagnation point

The study of Falkner-Skan flows has promoted more interests that recently become more significant because of their wide applications in the field of engineering and in many industrial processes. Owing to this fact, the current attempt investigates the features of mixed convection on Jeffrey fluid flow under the influence of stagnation point. Modeling and analysis are carried out in the presence of Falkner-Skan wedge phenomenon. Nonlinear stratification and viscous dissipation phenomena are incorporated to scrutinize the heat transportation. The governing equations are made non-dimensional with the implementation of suitable variables and convergent analytical solutions are acquired via homotopic technique. Results for velocity and temperature fields are analyzed through pertinent parameters. Nusselt number and skin friction expressions at the wedge surface are examined graphically. The analysis shows that temperature field declines for higher stratification parameter while it depicts reverse trend for increased Eckert number. Moreover, heat transfer rate intensifies in case of larger Prandtle number.

A numerical study on the natural transition locations in the flat-plate boundary layers on superhydrophobic surfaces

In this paper, the natural transition locations in the flat-plate boundary layers on the superhydrophobic surfaces are studied by numerical methods. The laminar flow field in the whole stream-wise computational domain is obtained by solving the Blasius equation with the slip-velocity boundary condition on the wall. The boundary layer on the superhydrophobic surface becomes thinner than that on the ordinary surface. The linear instability analysis is performed on the laminar boundary layer, and the eN method is employed to predict the transition location. The two-dimensional (2D) Tollmien–Schlichting (T–S) waves are still more unstable than the three-dimensional (3D) ones on the superhydrophobic surfaces, so only the 2D waves are taken into consideration to predict transition. As the slip length becomes longer, the critical location of flow instability moves further downstream, and the unstable zone becomes smaller. Therefore, the superhydrophobic surfaces have the effect of delaying the natural transition and that the delay effect becomes stronger as the slip length becomes longer. The higher oncoming flow velocity leads to higher frequencies of the unstable T–S waves and the larger unstable zone. As the oncoming flow velocity rises, the transition location on the superhydrophobic surface moves once upstream and then downstream. Consequently, there is a “dangerous” oncoming flow velocity corresponding to the transition location, which is the closest to the lead edge. Furthermore, the transition delay effect of the superhydrophobic surface becomes stronger with the increase in the oncoming flow velocity.

Modal and nonmodal stability of a stably stratified boundary layer flow

We investigate the modal and nonmodal linear stability of a stably stratified Blasius boundary-layer flow, composed of a velocity and a thermal boundary layer. The nonmodal analysis, based on optimization of a weighted sum of the kinetic and potential energies, provides oblique optimal structures close to the wall at short target times and spanwise-homogeneous rolls at the freestream for long target times. The latter disappear when variation of stratification strength with height is accurately accounted for in the norm definition, underlining the importance of the choice of a meaningful norm for thermal boundary layer optimization.

Receptivity to Freestream Periodic Vorticity Disturbance on a Flat Plate with an Elliptic Leading Edge

The receptivity of the Blasius boundary layer over a semi-infinite flat plate with an elliptic leading edge and of aspect ratio five was investigated using a direct numerical solution of two-dimensional Navier-Stokes equations. The result of the computation where the slip condition is applied to the fluctuating component of velocity at the wall surface is compared with that of an ordinary computation using a nonslip condition. Another numerical experiment is performed where no vorticity fluctuation is supplied from a freestream while prerecorded values of vorticities at the wall in response to the passage of convecting fluctuations are used as the wall vorticity boundary condition. It is shown that vorticity fluctuations in the boundary layer can be classified according to their wavelengths. Waves with longer wavelengths originate from the freestream, whereas waves with shorter wavelengths close to T-S waves originate from the surface of the plate. In another numerical experiment, the slip boundary condition against the fluctuation component of vorticity is applied to the limited area of the wall surface. The aim of the study is to determine the part of the elliptic leading edge or flat plate that induces vorticity fluctuations, thereby resulting in the creation of T-S waves. The numerical results show that the contribution of vorticity fluctuations originating from the juncture is the most crucial, whereas the vorticities supplied in the elliptic leading-edge surface negatively affect the amplitude of vorticity fluctuations inside the boundary layer. And, the stagnation section did not show positive contribution.

Surface temperature effects of solar panels of fixed-wing drones on drag reduction and energy consumption

In this paper, the thermal effects of solar panels are investigated experimentally and computationally on the efficiency of an Unmanned Air Vehicle (UAV) in laminar and turbulent flows. At first, the impact of temperature on output power and efficiency of an eFlex 30 Wp solar panel is studied. Then, the surface temperature and output voltage of two different types of solar panels, a flexible and a solid panel, are measured under a heat lamp. The heat lamp provides the radiation and raises the temperature of the solar panels. A thermal camera and laser thermometer are used to measure the surface temperature of the solar panels. Considering a tilt-rotor UAV as a case study, an energy balance is modeled for the wing of UAV, which is assumed as a flat plate. Applying the Blasius boundary layer for laminar flow and 1/5 power law for turbulent flow, it is shown that there is skin friction drag changes on the top surface of the solar panel due to its dark blue color. In order to validate the results of the proposed model, a thermal-fluid study is carried out on the NACA 2412 airfoil through COMSOL to see whether changing the surface temperature on the solar panel relates to skin drag reduction. The results indicate that an increase in the surface temperature of the solar panel will decrease the output power and efficiency to a maximum of 8%; while this increase in temperature reduces drag by up to 10% in laminar flow. This research shows that despite the reduction of efficiency and generated power by solar panels with increasing the surface temperature on top of a UAV, the aerodynamic efficiency can be improved with drag reduction in laminar flow.

Three-dimensional simulation and experimental validation of solar air heater having sinusoidal rib

ABSTRACT A new geometry of roughness in the sinusoidal form has been investigated computationally using Computational Fluid Dynamics (CFD) software for examining the influence of roughness on heat transfer and fluid friction in solar air heater and results obtained have been compared with experimental results. The simulated results for smooth solar air heater are compared with Dittus-Boelter equation for heat transfer and Blasius equation for fluid friction. The deviation of simulated results are found to be 7.55% and 16.91%, respectively. In order to validate the simulated results experimental test rig was fabricated and experiments were carried out for the same set of parameters used in the simulation. The experimental results indicate that for the roughened absorber plate having a sinusoidal rib, the variation in heat transfer is within 9% from that of simulated results and for fluid friction, deviation is within 4%. The optimal value of thermohydraulic performance (THP) is obtained 2.05 for roughness pitch (P) = 10 mm and Reynolds number (Re) = 4000, where enhancement ratio of Nusselt number (Nur/Nus) and friction factor (ƒr/ƒs) are 2.48 and 1.76, respectively. The experimental value of THP is found to be within 6.77% of the value indicated by CFD analysis. The results are also compared with the triangular and square rib and found that the optimal THP of the proposed rib is 20.06% and 33.31% more.

Finite element simulation of bioconvection Falkner–Skan flow of a Maxwell nanofluid fluid along with activation energy over a wedge

A description of heat transportation in Maxwell nano-fluids mixed with self-motile thermophile microorganisms over a wedged wall is presented in this article. The porous and thermally convective boundary of the wedge undergoes a sudden movement. The physical mechanism is influenced by an invariant magnetic field. A transformed formulation is established in ordinary differential form by involving similarity functions. A robust coding based on finite element analysis is developed in Matlab script. The convergence and accuracy of the solution are tested against reliable criteria. The interest in computational effort centered on the formation of boundary layer patterns for microorganism distribution, fluid temperature, the volume fraction of nano-inclusions, and fluid velocity when influential parameters are varied. The larger Hartmann number Ha, unsteady parameter A, Deborah number β, and wedge parameter m made the flow along the wall faster and produced thinning of the boundary layer. The higher values of Hartmann number, mixed convection, buoyancy ratio parameter, thermophoresis parameter Nt, Brownian motion parameter Nb, wedge parameter m, radiation parameter Rd, and Biot number have raised the fluid temperature. The local heat transfer rate reduces against Nt and it is higher for stretching wedge and smaller for the shrinking wedge. An efficient heat transfer in macro-tech processes may utilize the procedure and findings of this study.

Numerical solution of a generalized Falkner–Skan flow of a FENE-P fluid

The purpose of this article is to introduce a novel strategy for the numerical solution of a viscous boundary layer flow past a flat plate at a non-zero pressure gradient for a viscoelastic fluid governed by the FENE-P model. The stream function of this problem obeys a generalized Falkner–Skan equation. The fundamental method is based on embedding an integral operator, expressed in terms of Green's function of the corresponding linear differential operator, into well-known fixed-point iterative procedures, including Picard's and Mann's. Estimates of the skin friction coefficient are computed. The numerical solutions obtained from the proposed method are compared with the analytical and numerical solutions that exist in the literature. The convergence of the iterative scheme is proved via manipulation of the contraction principle. Numerical experiments and calculations ascertain that the introduced scheme handles the governing equation very efficiently, provides rapid convergence, and yields highly accurate results.

Solution of Blasius Equation Concerning with Mohand Transform

In this article, we present solution of Blasius differential equation with condition at infinity and convert the series solution into rational function by using Pad $$\check{\rm e}$$ s approximation. A new method is introduced, called Adomian Mohand transform method, which is a combination of Adomian decomposition method and new integral transform specifically Mohand transform, for handling a differential equation of mixing layer that arises in viscous incompressible fluid. It offered not only the numerical values, but also the power series close-form solutions.

Effects of Non-uniform Suction, Heat Generation/Absorption and Chemical Reaction with Activation Energy on MHD Falkner-Skan Flow of Tangent Hyperbolic Nanofluid over a Stretching/Shrinking Eedge

In the present investigation, the magnetohydrodynamic Falkner-Skan flow of tangent hyperbolic nanofluids over a stretching/shrinking wedge with variable suction, internal heat generation/absorption and chemical reaction with activation energy have been scrutinized. Nanofluid model is composed of “Brownian motion’’ and “Thermophoresis’’. Transformed non-dimensional coupled non-linear equations are solved by adopting the fourth-order R-K method along with the shooting technique. A comprehensive analysis of nanofluid velocity, the relative temperature, and its concentration profiles has been addressed. The major outcomes of the current study include that augmentation in the Weissenberg parameter, Hartmann number along with suction impede fluid flow and the shrinkage of the related boundary layer while internal heating develops an ascending thermal boundary layer for static and moving (stretching/shrinking) wedge. An increment in reaction rate undermines the nanoparticle concentration while that of activation energy exhibits a reverse trend.

Actuator and sensor placement for closed-loop control of convective instabilities

This work deals with the characterization of the closed-loop control performance aiming at the delay of transition. We focus on convective wavepackets, typical of the initial stages of transition to turbulence, starting with the linearized Kuramoto–Sivashinsky equation as a model problem representative of the transitional 2D boundary layer; its simplified structure and reduced order provide a manageable framework for the study of fundamental concepts involving the control of linear wavepackets. The characterization is then extended to the 2D Blasius boundary layer. The objective of this study is to explore how the sensor–actuator placement affects the optimal control problem, formulated using linear quadratic Gaussian (LQG) regulators. This is carried out by evaluating errors of the optimal estimator at positions where control gains are significant, through a proposed metric, labelled as $$\gamma $$ . Results show, in quantitative manner, why some choices of sensor–actuator placement are more effective than others for flow control: good (respectively, bad) closed-loop performance is obtained when estimation errors are low (respectively, high) in the regions with significant gains in the full-state-feedback problem. Unsatisfactory performance is further understood as dominant estimation error modes that overlap spatially with control gains, which shows directions for improvement of a given set-up by moving sensors or actuators. The proposed metric and analysis explain most trends in closed-loop performance as a function of sensor and actuator position, obtained for the model problem and for the 2D Blasius boundary layer. The spatial characterization of the $$\gamma $$ -metric provides thus a valuable and intuitive tool for the problem of sensor–actuator placement, targeting here transition delay but possibly extending to other amplifier-type flows.

A new friction factor formula for single phase liquid flow through geothermal pipelines

Proper selection of pumps has been an important issue for energy efficiency in geothermal energy. In a research conducted among machines that consume electricity, pumps come first with 25 %. Accurate estimation of pressure losses is a key point in order to select pump systems. Hot water from the geothermal area is carried by pipeline systems. In this study, extensive experimental study is conducted at Izmir Katip Celebi University Flow Loop using pure water for fluid temperature from 25 °C to 60 °C, for four various relative roughness values, and for Reynolds Number ranges from 6.2 × 104 to 7.8 × 105. This study has mainly two objectives: First is to identify if the absolute pipe roughness values of commercially available pipes vary according to the pipe diameter. The second objective is to investigate fluid temperature effects on frictional pressure losses in rough pipes and to present a new friction factor correlation for using in geothermal pipelines, considering the fluid temperature effects. Experimental study shows that absolute pipe roughness varies substantially for pipe diameters. The proposed friction factor has a good agreement with explicit formulas of Blasius (1913) and Taler (2016) equations for smooth pipes and Colebrook (1939) equation for rough pipes, as well as experimental data at various temperature values.Additionally, frictional pressure losses are accurately estimated using new friction factor equation. Results show that both relative roughness and temperature have a significant influence on the friction factor for turbulent flow through rough pipes. For this reason, pipe roughness should be determined correctly and the effects of fluid temperature on pressure losses should be considered. This may influence frictional head losses per unit length and, consequently, the predicted pumping requirements in geothermal pipelines.