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Homotopy perturbation method (HPM) is one of the most popular semi-analytical methods to solve a nonlinear differential equation. However, in HPM, there is no strict rule for the choice of its linear operator, and its series solution may not always converges. In this study, firstly, we define the linear operator as an auxiliary linear operator ( L a ), in the frame of homotopy analysis method (HAM). Then we generalize this L a based on the auxiliary roots of L a = 0 . Finally, using the optimization technique (on minimization of the residual error) we determine the best-fitted optimal L a for a problem. By doing this we ensure and accelerate the convergence of our semi-analytical homotopy perturbation series solution. Thereby we rename the HPM as Optimal and Modified Homotopy Perturbation Method (OMHPM). We consider three strongly nonlinear differential equations of nonlinear dynamical phenomena associated with the fluid dynamics to certify our technique. The dependencies of the form of the optimal L a and the convergence of the solution (obtained by HPM, Optimal HAM and OMHPM) on the values of parameters (involved in the scale transformation), initial/boundary conditions and artificial controlling parameters (involved in optimal HAM) are explored here. It is reported that our OMHPM is highly accurate and efficient than HPM, optimal HAM and Domain decomposition optimal HAM. Moreover, OMHPM is simple and can be applied to directly to any singular/non-singular highly nonlinear ordinary differential equations without any decomposition, special/scale transformation, linearization, artificial controlling parameters and discretization. An attempt is made to apply our optimal auxiliary linear operator onto the optimal HAM for possible fastest convergence. [ABSTRACT FROM AUTHOR]

This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

Nonlinear Blasius equation of boundary layer plays a crucial role in different physics and engineering fields. For nearly a century, researchers have been trying to find a convergent and closed analytical solution of the Blasius equation. In this paper, an analytical self‐consistent method (ASCM) is presented to solve this equation. Then, the method was implemented in other forms of Blasius equation, that is, Sakiadis flow and shrinking sheet problem. The analytical solutions obtained show the success of ASCM and agree closely with the corresponding numerical results. The reliability and efficiency of the proposed method are demonstrated on the numerical experiments. [ABSTRACT FROM AUTHOR]

A new approach to the polynomial approximation of the laminar boundary layer on the surface of a flat plate on the basis of the Karman momentum integral with the use of additional optimum constraints is proposed. The polynomial coefficients of a solution of the problem on this layer were determined for the first time with the use of the system of zero boundary conditions for the surface of the plate and definite boundary conditions for the outer side of the boundary layer on it. Optimum polynomial solutions of the problem in the zero to twentieth approximations have been obtained. A solution of the problem obtained in the seventeenth approximation is almost identical to the high-accuracy numerical solution of the Blasius equation, obtained by B. D. Ganapol, with a maximum deviation of 6∙10–7. [ABSTRACT FROM AUTHOR]

A generalized critical analysis of the main known polynomial solutions obtained for the velocity profile of the fluid flow in the laminar boundary layer on a plane plate with the use of the integral Karman–Pohlhausen method has been performed. The main reasons for the low accuracy of these solutions were revealed and ways of its increasing were determined. Efficient schemes of calculating the parameters of the indicated boundary layer with minimum errors, in particular, a new trinomial polynomial, defining the velocity profile of the fluid flow in this layer, are proposed. A new solution of the problem on the boundary layer flow over a plane plate in the form of the fourth-degree (Sutton) polynomial gives an almost exact value of the friction stress of this flow with small errors in determining its displacement thickness (0.12%) and shape parameter (0.12%), and an analogous solution in the form of the seventhdegree (Mughal) polynomial provides a very high accuracy of approximation of the friction stress of the indicated flow with an almost zero error at negligible small errors in calculating its displacement thickness (0.04%) and shape parameter (0.03%) of this layer. [ABSTRACT FROM AUTHOR]

In this paper, the fractal derivative is employed to define the fractal Blasius equation in a microgravity space arising in fluid dynamics, and its fractal variational principle is successfully established by the fractal semi-inverse transform method. The approximate analytical solution of the fractal Blasius equation is obtained by the variational iteration method. Our results have theoretical significance and practical application value. The example shows the approximate approach is reliable and accurate. [ABSTRACT FROM AUTHOR]

The present study aims to investigate time-dependent two-dimensional Sakiadis ow of quiescent fluid. This ow is induced by stationary at plate through uniform free-stream (Blasius ow). The variable conductivity and viscosity ratio parameters as well as non-linear chemical reaction were taken into account in the mathematical equations. Similarity variables were employed in the governing transport expressions to convert them into the ordinary differential system. The transformed system is numerically computed using Runge-Kutta scheme and based on shooting criteria. The results regarding concentration, velocity, and temperature distributions were also studied through plots. Moreover, mass and heat transfer rates and friction factor were discussed in detail. According to the findings, the constraint of chemical reaction slowed down the rates of friction factors and heat transportation in the case of the Sakiadis-Blasius ow and enhanced the mass transportation rate in both cases. In addition, the rate of mass transportation was smaller in Sakiadis ow than that of Blasius ow. The obtained results of the rate of heat transfer were in excellent agreement with those from the literature. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the transverse magnetic field is unraveled on the development of steady flow regime for an incompressible fluid in the boundary layer limit of a semi-infinite vertical plate. The sensitivity of real fluids to changes in temperature suggests a variable thermal conductivity modeling approach. Using appropriate similarity variables, solutions to the governing nonlinear partial differential equations are obtained by numerical integration. The approach used here is based on using the shooting method together with the Runge-Kutta-Fehlberg integration scheme. Representative velocity and temperature profiles are presented at various values of the governing parameters. The skin-friction coefficient and the rate of heat transfer are also calculated for different parameter values. Pertinent results are displayed graphically and discussed. It is found that the heat transfer rate improves with an upsurge in a magnetic field but lessens with an elevation in the fluid thermal conductivity. [ABSTRACT FROM AUTHOR]

The objective of the present work is to propose an extended analytical wall function that is capable of predicting the bypass transition from laminar to turbulent flow. The algebraic γ transition model, the k−ω turbulence model and the analytical wall function are integrated together in this work to detect the transition onset and start the transition process. The present analytical wall function is validated with the experimental data, the Blasius solution and the law of the wall. With this analytical wall function, the transition onset in the skin friction coefficient is detected and the growth rate of transition is properly generated. The predicted mean velocity profiles are found to be in good agreement with the Blasius solution in the laminar flow, the experimental data in the transition zone and the law of the wall in the fully turbulent flow. [ABSTRACT FROM AUTHOR]

In this research, a numerical algorithm is employed to investigate the classical Blasius equation which is the governing equation of boundary layer problem. The base of this algorithm is on the development of RCW (Rahmanzadeh-Cai-White) method. In fact, in the current work, an attempt is made to solve the Blasius equation by using the sum of Taylor and Fourier series. While, in the most common numerical methods, the answer is considered only as a Taylor series. It should be noted that in these algorithms which use Taylor expansion, the values of the truncation error are considerable. However, adding the Fourier series to the Taylor series leads to reduce the amount of the truncation error. Nevertheless, the results of this research show the RCW method has the ability to achieve the accuracy of analytical solution. Moreover, it is well illustrated that the accuracy of RCW method is higher than the Runge-Kutta one.

In this paper, the Hydon method to determine discrete symmetries for a differential equation is employed to construct discrete symmetries for a family of ordinary, partial, and fractional differential equations. An application of those discrete symmetries to construct other solutions from known ones is illustrated. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate theoretically the magneto-thermomechanical interaction between a heated viscous incompressible ferrofluid and a cold wall in the presence of a spatially varying field. Similarity transformation is used to convert the governing non-linear boundary-layer equations into coupled non-linear ordinary differential equations. These equations are numerically solved using a discretization scheme using higher derivative method (HDM). The effects of governing parameters corresponding to various physical conditions are analyzed. Numerical results are obtained for distributions of velocity and temperature, the dimensionless wall skin friction and heat-transfer coefficients. The results indicate that two solution exists in some cases. A comparison with previous studies available in the literature has been done and we found an excellent agreement with it.

This paper presents an iterative technique based on homotopy analysis method for solving system of Volterra integro-differential equations. The technique provides us series solutions to the problems which are combined with the diagonal Padé approximants and Laplace transform to obtain closed-form solutions. The technique is effectively applied on system of linear and nonlinear Volterra integro-differential equations which eventually yield closed-form solutions of the problems and this technique is also extended to boundary value problem for the integro-differential equation related to Blasius problem. An interesting comparison of the present solution is made with solutions of other methods and it is observed that the results are in excellent agreement with other methods in the literature. [ABSTRACT FROM AUTHOR]

The Blasius equation is a well known third-order nonlinear ordinary differential equation, which arises in certain boundary layer problems in the fluid dynamics. In this paper we prove the existence and the uniqueness of the solution of a generalized Blasius equation using nonstandard analysis techniques. [ABSTRACT FROM AUTHOR]

In this work we applied a feed forward neural network to solve Blasius equation which is a third-order nonlinear differential equation. Blasius equation is a kind of boundary layer flow. We solved Blasius equation without reducing it into a system of first order equation. Numerical results are presented and a comparison according to some studies is made in the form of their results. Obtained results are found to be in good agreement with the given studies. [ABSTRACT FROM AUTHOR]

A new collocation method, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) collocation method, is given for solving some nonlinear boundary value problems in the semi-infinite domain, such as equations of the unsteady isothermal flow of a gas, the third grade fluid, the Blasius, and the field equation determining the vortex profile. The method reduces the solution of the problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of the method, the numerical results of the present method are compared with several numerical results.

The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semi-infinite flat plate. The definition of a non-iterative transformation method for the celebrated Blasius problem is due to Töpfer and dates more than a century ago. Here we define a non-iterative transformation method for Blasius equation with a moving wall, a slip flow condition or a surface gasification. The defined method allows us to deal with classes of problems in boundary layer theory that, depending on a parameter, admit multiple or no solutions. This approach is particularly convenient when the main interest is on the behaviour of the considered models with respect to the involved parameter. The obtained numerical results are found to be in good agreement with those available in literature. [ABSTRACT FROM AUTHOR]

The numerical solutions for the Blasius equation and for the Ostrach system where investigated. A combination of optimization procedure and Shooting Method where systematized in order to produce a powerful method for solving nonlinear systems of differential equations, namely Initial Value Problem Approximation by Sequential Parameter Optimization (IVASO). Using the IVASO method, it was shown to be possible and easy to obtain an accurate solution for the Blasius equation. It was also demonstrated that the Ostrach system can be solved by IVASO. Thissystem has a large sensibility to initial conditions, and that its solution for near zero (η ≈ 0) is strongly correlated with its solution far from the origin (η ≫ 0), and consequently, an accurate solution for the boundary layer demands a highly accurate solution of the initial values problem, this is similar to the butterfly effect usually studied in chaotic systems. [ABSTRACT FROM AUTHOR]

An investigation is made to study the Blasius flow of a nanofluid in the presence of homogeneous-heterogeneous chemical reactions. Here, the diffusion coefficients of the reactant and autocatalyst are considered to be in comparable sizes. The Buongiorno's mathematical model is applied in describing the behavior of nanofluids. Multiple solutions of the steady-state system of nonlinear ordinary differential equations are obtained. Results show that nanofluids significantly participate in the transport mechanism of the homogeneous-heterogeneous reactions, which play different roles in the procedures of homogeneous and heterogeneous reactions. [ABSTRACT FROM AUTHOR]

This paper explores the mechanisms underlying roughness induced transition (RIT) caused by discrete roughness elements (DREs) using immersed boundary direct numerical simulations. We show via favorable comparison between RIT in Blasius boundary layers and equivalent Couette flows that linear instability of the boundary layer profile does not play a significant role for the DREs considered (k < 0.6δ*, where k is the height of the DRE) and that k+ = uτk/ν is the dominant parameter (for a given shape of the DRE) which strongly affects the transition location. For a suitable range of k+, the flow evolution can be separated into four distinct stages: (i) generation of vortical disturbances at the roughness, (ii) a steady and spatial amplification of a three dimensional disturbance, (iii) the emergence and amplification of unsteady disturbances, and (iv) the emergence of chaotic behavior leading to a "turbulent wedge" (with a relatively high mean wall shear stress). Each of these stages is studied in detail. A mechanistic understanding of RIT is suggested which includes a new and fundamental understanding of the final stage. Novel results include the description of a mutual stretching mechanism leading to the near wall amplification of streamwise vorticity at the onset of stage IV, complementary interpretations of the lift up and the "modal instability" using a control volume formulation for different components of the enstrophy, and a demonstration of a passive RIT mitigation strategy using an "anti-roughness" element (i.e., a second downstream roughness element), which exploits this understanding of RIT mechanisms from the vorticity-based analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear boundary-layer equation of Falkner-Skan defined on a semi-infinite domain. An iterative finite difference (IFD) scheme is proposed to numerically solve such nonlinear ordinary differential equation. A computational iterative scheme is developed based on Newton-Kantorovich quasilinearization. At every iteration, the obtained linearized differential equation is numerically solved using the standard finite difference method. Numerical experiments show the accuracy and efficiency of the method compared to existing solvers. The computation is performed for different parameter values, including the special case of Blasius problem.

Highlights • PIV measurements of the K-type transition of natural convection boundary layers are presented. • The aligned ˄-shaped flow structures are confirmed by PIV measurements for the first time. • The appearance of the aligned ˄-shaped flow structures is an 'abrupt' process. • The development of the three-dimensionality in the boundary layer is a 'gradual' process. Abstract The K-type transition of a natural convection boundary layer of Ra = 9.8 × 109 is studied by PIV (Particle Image Velocimetry) measurements. To excite the transition, Tollmien-Schlitchting (TS) and oblique waves of the same frequency are introduced into the upstream boundary layer in the form of velocity perturbations. It is found that resonant interactions between the characteristic frequency of the natural convection boundary layer and the superimposed TS and oblique waves are present, which trigger the K-type transition of the boundary layer. The typical aligned Λ-shaped flow structures characterising the K-type transition present in the transition of Blasius boundary layers are also observed in the present natural convection boundary layers. They occur when the primary instability grows to a certain extent. The appearance of the spanwise mode characterised by the aligned Λ-shaped flow structures is found to be an 'abrupt' process, although the development of the three-dimensionality in the natural convection boundary layer is found to be a 'gradual' process. A peak in the typical profile of the Root-Mean-Square (RMS) of the amplitude of streamwise velocity is also noted around the transition point, beyond which distinct three-dimensional Λ-shaped flow structures are observed. The transition point has been determined consistently using different approaches. A Bicoherence analysis suggests that the interaction between the external excitation frequency and the characteristic frequency of the boundary layer is responsible for the production of new harmonic frequencies and resonance groups. The PIV measurements have been extended to a range of perturbation frequencies for the boundary layer, and the dependence of the K-type transition on the perturbation frequency is discussed. [ABSTRACT FROM AUTHOR]

Abstract Airway resistance describes the ratio between pressure drop and flow rate through the conducting respiratory airways. Analytical models of airway resistance for tracheobronchial airways have previously been developed and assessed without upper airways positioned upstream of the trachea. This work investigated pressure drop as a function of flow rate and gas properties for upper and central airway replicas of 10 child subjects, ages 4–8. Replica geometries were built based on computed tomography scan data and included airways from the nose through 3–5 distal branching airway generations. Pressure drop through the replicas was measured for constant inspiratory flows of air and heliox. For both the nose-throat and branching airways, the relationship between non-dimensional coefficient of friction, C F , with Reynolds number, Re , was found to resemble the turbulent Blasius equation for pipe flow, where C F ∝ R e - 0.25 . Additionally, pressure drop ratios between heliox and air were consistent with analytical predictions for turbulent flow. The presence of turbulence in the branching airways likely resulted from convection of turbulence produced upstream in the nose and throat. An airway resistance model based on the Blasius pipe friction correlation for turbulent flow was proposed for prediction of pressure drop through the branching bronchial airways downstream from the upper airway. [ABSTRACT FROM AUTHOR]

The flow over a porous laminated flat plate is investigated from a flow control perspective through experiments and computations. A square array of circular cylinders is used to model the porous lamination. We determine the velocities at the fluid-porous interface by solving the two-dimensional Navier-Stokes and the continuity equations using a staggered flow solver and using LDV in experiments. The control parameters for the porous region are porosity, ϕ and Reynolds number, Re, based on the diameter of the circular cylinders used to model the porous lamination. Computations are conducted for 0.4<ϕ<0.9 and 25, and the experiments are conducted for ϕ=0.65 and 0.8 at Re≈391,497 and 803. The permeability of the porous lamination is observed to induce a slip velocity at the interface, effectively making it a slip wall. The slip velocity is seen to be increasing functions of ϕ and Re. For higher porosities at higher Re, the slip velocity shows non-uniform and unsteady behavior and a breakdown Reynolds number is defined based on this characteristic. A map demarcating the two regimes of flow is drawn from the computational and experimental data. We observe that the boundary layer over the porous lamination is thinner than the Blasius boundary layer and the shear stress is higher at locations over the porous lamination. We note that the porous lamination helps maintain a favorable pressure gradient at the interface which delays separation. The suitable range of porosities for effective passive separation control is deduced from the results. [ABSTRACT FROM AUTHOR]

Streaks are a common feature of disturbed boundary-layer flows. They play a central role in transient growth mechanisms and are a building block of self-sustained structures. Most theoretical work has focused on streaks that are periodic in the spanwise direction, but in this work we consider a single spatially localised streak embedded into a Blasius boundary layer. For small streak amplitudes, we show the perturbation can be described in terms of a set of eigenmodes that correspond to an isolated streak/roll structure. These modes are new, and arise from a bi-global eigenvalue calculation; they decay algebraically downstream and may be viewed as the natural three-dimensional extension of the classical two-dimensional Libby & Fox (J. Fluid Mech., vol. 17 (3), 1963, pp. 433-449) solutions. Despite their bi-global nature, we show that a subset of these eigenmodes (including the slowest decaying) is fundamentally related to the solutions first presented by Luchini (J. Fluid Mech., vol. 327, 1996, pp. 101-116), as derived for spanwise-periodic disturbances (at small spanwise wavenumber). This surprising connection is made by an analysis of the far-field decay of the bi-global state. We also address the fully non-parallel downstream development of nonlinear streaks, confirming that the aforementioned eigenmodes are recovered as the streak/roll decays downstream. Encouraging comparisons are made with available experimental data. [ABSTRACT FROM AUTHOR]

The present work extends Kolmogorov’s micro-scales to a large family of viscoplastic fluids. The new micro-scales, combined with Gioia and Chakaborty’s (2006) friction phenomenology theory, lead to a unified framework for the description of the friction coefficient in turbulent flows. A resulting Blasius-type friction equation is tested against some available experimental data and shows good agreement over a significant range of Hedstrom and Reynolds numbers. The work also comments on the role of the new expression as a possible benchmark test for the convergence of DNS simulations. The formula also provides limits for the maximum drag reduction of viscoplastic flows. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-linear sequential differential equation with Caputo fractional derivative of Blasius type and we reduce the problem to the equivalent non-linear integral equation. We prove the complete continuity of the non-linear integral operator. The theorem on the existence of a solution of the problem for the Blasius equation of fractional order is also proved. [ABSTRACT FROM AUTHOR]

With the wide spread of drip irrigation, low-pressure subsurface drip system has come into notice as it can reduce the operation cost and improve water and fertilizer use efficiency, and it has become an emerging trend in drip technology development. A field experiment was carried out in Xinjiang to investigate the variation law of hydraulic performance and irrigation uniformity in subsurface drip system under low pressure. We chose lateral capillary from the head and end of the main branch to be the representatives, as well as the middle ones. The inlet pressure of branch pipe was set between 1.40 and 6.55 m; then we compared subsurface drip system with surface drip system under the same conditions in relation to flow rate, working pressure, and total head loss. The results indicate that: 1) The operation status of laterals can be monitored by working pressure and flow rate in the drip irrigation system. 2) When the emitter outflow is driven by the soil matric potential, the flowrate of underground laterals is larger than that of the surface laterals, and the coefficient of flow variation is increasing by 0.12 to 0.9, and it keeps increasing with the decrease of working pressure. 3) When comparing the working pressure of subsurface drip system with that of the surface drip system, 90% of subsurface drip laterals are less than that in the surface drip irrigation system, and the variation coefficient of working pressure is from ?10% to 0 with the probability of 70%. 4) The increasing coefficient considering local head loss of laterals (ICCLHLL) is calculated by using the total head loss, the Blasius formula, and the flow from multiple outlets, and the value of surface drip system is within the range of 1.32-5.94, while range between 1.37 and 2.18 for subsurface drip system, and the ICCLHLL increases with the decrease of working pressure. The flow rate of subsurface laterals keeps increasing under the influence of the soil matric potential, which leads to the decrease of ICCLHLL in surface drip laterals. 5) When decreasing working pressure, the irrigation uniformity of surface drip system will also decline. The soil matrix potential improves the irrigation uniformity of subsurface drip system. The deviation rates of working pressure and flow rate in the subsurface drip system are 0.62%-3.44% and 8.15%-22.4% lower than those in the surface drip system, respectively. The research can provide a scientific basis for design and management of low-pressure subsurface drip irrigation. [ABSTRACT FROM AUTHOR]

We consider the Crocco equation (the reduction of the Blasius equation). The use of this more simple equation for computation of the Blasius constant leads to some unexpected difficulties, which have been unexplained. We computed the asymptotic expansion of the solution to Crocco equation at its singularity. This expansion was unknown before. We describe the structure of the Riemann surface of the Crocco solution at the singularity. These results were used for construction of an effective numerical algorithm, which is based on analytical continuation, for computation of the Blasius constant with an arbitrary and guaranteed accuracy. We computed the Blasius constant with a 100 decimal places. [ABSTRACT FROM AUTHOR]

Large eddy simulation (LES) of a transitional, thermal Blasius flowfield at a sufficiently low Reynolds number along an isothermal, flat plate is presented herein. Prior experiments observed that boundary layer transition is induced by the presence of streamwise vortex instability caused by the complex interaction between thermal buoyancy and forced convection dynamics. The maximum Grashof and Reynolds numbers employed in the LES were approximately 1.05 × 10 11 and 1.18 × 10 5 , respectively. To further enhance the accuracy, computational efficiency, and numerical stability, the LES solved the low-Mach number compressible flow governing equations, which included fluctuating density effects and pressure-density decoupling. For the subgrid scale ( SGS ) closure, a locally dynamic Smagorinsky SGS model was implemented into the LES solver to enable the backscatter phenomenon intrinsic to transitional boundary layer flows. The LES accurately predicted the onset of streamwise vortex instability and the eventual three-dimensional vortex breakdown of the underlying mean flow into a fully developed turbulent boundary layer, when compared to previously measured data. In the developed turbulence region, quadrant analyses indicated Q 2 and Q 4 events dominated the contribution to the Reynolds shear stress in the near-wall region, whereas Q 1 and Q 3 events contributed considerably to the wall-normal turbulent heat flux. And, as a result of the instability within the conduction layer, the layer erupts and intermittently releases buoyant thermal plumes. These recurring intermittent events inside the conduction layer deflect and deform the flowfield quantities near the wall, resulting in a plurality of peculiar peaks and shear layers inside the boundary layer. Furthermore, these buoyant thermal plumes are the dominant turbulence production mechanism farther away from the wall in the downstream region of the developed turbulent boundary layer flow. Near the wall, however, forced convection turbulent flow effects were observed, in which the shear production term was the primary contributor to the generation of turbulent kinetic energy, and quasi-streamwise and horseshoe-like vortex structures were observed in the developed turbulence region. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear boundary-layer equation of Falkner-Skan defined on a semi-infinite domain. An iterative finite difference (IFD) scheme is proposed to numerically solve such nonlinear ordinary differential equation. A computational iterative scheme is developed based on Newton-Kantorovich quasilinearization. At every iteration, the obtained linearized differential equation is numerically solved using the standard finite difference method. Numerical experiments show the accuracy and efficiency of the method compared to existing solvers. The computation is performed for different parameter values, including the special case of Blasius problem. [ABSTRACT FROM AUTHOR]

The power law and logarithmic law among others are the most commonly well-known theories for equilibrium mean profile of dust concentration in a neutral atmospheric surface layer. However, these theories are based on the assumption of homogeneity in horizontal (streamwise) direction, thereby the horizontal advection term in the transport equation is neglected. In this study, with the help of a self-similar analytical solution of the two-dimensional steady-state dust transport equation (Chamecki and Meneveau, J Fluid Mech 683:1-26, 2011), we examine the applicabilities of the assumption which the one-dimensional theories are based on. In addition, we also propose an empirical fit of the theory, and good agreements with reference data have been obtained. The new approximate profile can be used as an alternative of the one-dimensional theories in practical applications. [ABSTRACT FROM AUTHOR]

A systematic survey and evaluation on the thermal-hydraulic performance of gas inside internally finned, twisted tape or swirl generator inserted, corrugated, and dimpled, totally 436 pipes is conducted in this work. The gases in the investigations involve air, nitrogen, exhaust gases, and helium. Prandtl number is around 0.6–0.7. It is found that in the Reynolds number of 2 × 10 3 to 100 × 10 3 , the ratios of Nusselt number over Dittus-Boelter equation for internal finned tubes are typically in the range of 1–6, tubes with twisted tape and other inserts are 1.5–6, corrugated tubes are 1–3 and dimpled tubes are 1–4, including compound enhancement techniques. The ratios of friction factor over Blasius equation is normally in the range of 1.5–14 for internally finned tubes, 2–200 for inserted twisted tapes and swirl generators, corrugated tubes is 1.5–10 and dimpled tubes is 1–8. The heat transfer enhancement ratios for gases are generally similar with liquid, while the friction factor increased ratios for gases are higher than that for liquids. The number of investigations on the tubes fitted with twisted tapes inserts, coil loops, and swirl generators are more than other three enhancement methods. The increment of pressure drop for twisted tape inserts are also the largest. By using performance evaluation plot, different enhancement techniques with the same reference are compared for their effectiveness. It indicates that the efficiency of pipes with different types of inserts for gases are mostly lower than internal finned, corrugated and dimpled tubes in this survey. [ABSTRACT FROM AUTHOR]

This paper concerns the study of direct numerical simulation data of awavepacket in laminar turbulent transition in a Blasius boundary layer. The decomposition of this wavepacket into a set of "modes" (a basis that spans an approximate solution space) can be achieved in a wide variety ofways. Two wellknown tools are the fast Fourier transform (FFT) and the proper orthogonal decomposition (POD). To synergize the strengths of both methods, a hybrid POD-FFT is pioneered, using the FFT as a tool for interpreting the POD modes. The POD-FFT automatically identifies well-known fundamental, subharmonic, and Klebanoff modes in the flow, even though it is blind to the underlying physics. Moreover, the POD-FFT further separates the subharmonic content of the wavepacket into three fairly distinct parts: a positively detuned mode resembling a Lambda-vortex, a Craik-type tuned mode, and a Herbert-type positive-negative detuned mode pair, in decreasing order of energy. This distinction is less widely recognized, but it provides a possible explanation for the slightly positively detuned subharmonic mode often observed in previous experiments and simulations. [ABSTRACT FROM AUTHOR]

In this work, we apply the reproducing kernel method for investigating Blasius equations with two different boundary conditions in semi-infinite domains. Convergence analysis of the reproducing kernel method is given. The numerical approximations are presented and compared with some other techniques, Howarth's numerical solution and Runge-Kutta Fehlberg method. [ABSTRACT FROM AUTHOR]

For about a century, people have been trying to seek for a globally convergent and closed analytical solution (CAS) of the Blasius Equation (BE). In this paper, we proposed a formally satisfied solution which could be parametrically expressed by two power series. Some analytical results of the laminar boundary layer of a flat plate, that were not analytically given in former studies, e.g. the thickness of the boundary layer and higher order derivatives, could be obtained based on the solution. Besides, the heat transfer in the laminar boundary layer of a flat plate with constant temperature could also be analytically formulated. Especially, the solution of the singular situation with Prandtl number Pr=0, which seems impossible to be analyzed in prior studies, could be given analytically. The method for finding the CAS of Blasius equation was also utilized in the problem of the boundary layer regulation throughwall injection and slip velocity on the wall surface. [ABSTRACT FROM AUTHOR]

This study investigates the impact of the magnetic field on nanofluids in the presence of heterogeneous and homogeneous reactions for Blasius flow with thermal radiations. A ferrofluid is used to study the effects of nanoparticles. The developed problems are solved numerically by shooting method. The impact of physical parameters on the flow characteristics is examined and discussed through graphs. It is noticed that the velocity and temperature profiles are increased with an increase in by adding nanoparticles in the fluid. Also it is observed from present study that concentration of the fluid is decreases function by increasing the strength of homogeneous and heterogeneous reaction. [ABSTRACT FROM AUTHOR]

The central aim of the paper is to apply and assess a nonlinear optimal control strategy to suppress bypass transition, due to bimodal interactions [T. A. Zaki and P. A. Durbin, "Mode interaction and the bypass route to transition," J. Fluid Mech. 531, 85 (2005)] in a zero-pressure-gradient boundary layer. To this end, a Lagrange variational formulation is employed that results in a set of adjoint equations. The optimal wall actuation (blowing and suction from a control slot) is found by solving iteratively the nonlinear Navier-Stokes and the adjoint equations in a forward/backward loop using direct numerical simulation. The optimization is performed in a finite time horizon. Large values of optimization horizon result in the instability of the adjoint equations. The control slot is located exactly in the region of transition. The results show that the control is able to significantly reduce the objective function, which is defined as the spatial and temporal integral of the quadratic deviation from the Blasius profile plus a term that quantifies the control cost. The physical mechanism with which the actuation interacts with the flow field is investigated and analysed in relation to the objective function employed. Examination of the joint probability density function shows that the control velocity is correlated with the streamwise velocity in the near wall region but this correlation is reduced as time elapses. The spanwise averaged velocity is distorted by the control action, resulting in a significant reduction of the skin friction coefficient. Results are presented with and without zero-net mass flow constraint of the actuation velocity. The skin friction coefficient drops below the laminar value if there is no mass constraint; it remains however larger than laminar when this constraint is imposed. Results are also compared with uniform blowing using the same time-average velocity obtained from the nonlinear optimal algorithm. [ABSTRACT FROM AUTHOR]

We present a model for ice formation in a thin, viscous liquid film driven by a Blasius boundary layer after heating is switched off along part of the flat plate. The flow is assumed to initially be in the Nelson et al. (J. Fluid Mech., vol. 284, 1995, pp. 159-169) steady-state configuration with a constant flux of liquid supplied at the tip of the plate, so that the film thickness grows like x1/4 in distance along the plate. Plate cooling is applied downstream of a point, Lx0, an O.L/-distance from the tip of the plate, where L is much larger than the film thickness. The cooling is assumed to be slow enough that the flow is quasi-steady. We present a thorough asymptotic derivation of the governing equations from the incompressible Navier-Stokes equations in each fluid and the corresponding Stefan problem for ice growth. The problem breaks down into two temporal regimes corresponding to the relative size of the temperature difference across the ice, which are analysed in detail asymptotically and numerically. In each regime, two distinct spatial regions arise, an outer region of the length scale of the plate, and an inner region close to x0 in which the film and air are driven over the growing ice layer. Moreover, in the early time regime, there is an additional intermediate region in which the air-water interface propagates a slope discontinuity downstream due to the sudden onset of the ice at the switch-off point. For each regime, we present ice profiles and growth rates, and show that for large times, the film is predicted to rupture in the outer region when the slope discontinuity becomes sufficiently enhanced. [ABSTRACT FROM AUTHOR]

A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between.Ageneral algorithm is provided to construct such approximants. To demonstrate the versatility of the method, approximants are constructed for three non-linear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili (FP) monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and FP monopole. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of a solution to a generalized Blasius equation with asymptotic boundary conditions is proved. A new numerical approximation method is proposed.

Este trabalho reporta aos resultados de um experimento sobre perda de carga e fator de atrito em tubos de polietileno de pequenos diâmetros. Utilizaram-se cinco tubos com os seguintes diâmetros internos: 10,0 mm, 12,9 mm, 16,1 mm, 17,4 mm e 19,7 mm. Oexperimento foi conduzido para números de Reynolds, no intervalo de 6000 a 72000, obtidos pela variação da vazão nos tubos, a uma temperatura média da água de 20oC. Os resultados foram analisados e, de acordo com as condições experimentais, o fator de atrito fda equação de Darcy-Weisbach pode ser estimado com c = 0,300 e m = 0,25. A equação de Blasius (c = 0,316 e m = 0,25) superestimou os valores do fator de atrito para todos os tubos analisados, porém esse fato não constitui limitação para sua utilização em projetos demicroirrigação. As análises mostraram que as duas equações proporcionam estimativas do fator de atrito com pequeno desvio-médio (5,1%).On this paper, the results of an experimental study on the hydraulic friction loss for small-diameter polyethylene pipes are reported. The experiment was carried out using a range of Reynolds number between 6000 to 72000, obtained by varying discharge at 20oC water temperature, with internal pipe diameters of 10.0 mm, 12.9 mm, 16.1 mm, 17.4 mm and 19.7 mm. According to the analysis results and experimental conditions, the friction factor (f) of theDarcy-Weisbach equation can be estimated with c = 0.300 and m = 0.25. The Blasius equation (c = 0.316 and m = 0.25) gives an overestimate of friction loss, although this fact is non-restrictive for micro-irrigation system designs. The analysis shows that both the Blasiusand the adjusted equation parameters allow for accurate friction factor estimates, characterized by low mean error (5.1%).