Two-dimensional Boundary Layer Research Articles

Linear stability analysis of compressible boundary layer over an insulated wall using compound matrix method: Existence of multiple unstable modes for Mach number beyond 3

The linear spatial stability of a parallel two-dimensional (2D) compressible boundary layer on an adiabatic plate is investigated by considering both 2D and three-dimensional (3D) disturbances. The compound matrix method is employed here, for the first time, for compressible flows, which, unlike other conventional techniques, can efficiently eliminate the stiffness of the equations governing the spectral amplitudes. The method is first validated with published results in the literature corresponding to spatial and temporal instability of flows ranging from low subsonic to high supersonic Mach numbers (M), which shows a good match depending upon the proper choice of free-stream temperature and the wall dispersion relation. Subsequently, flow compressibility effects and the spanwise variation of disturbances are also investigated for M ranging from low subsonic to high supersonic cases (from M = 0.1 to 6). Mack (AGARD Report No. 709, 1984) reported the existence of two unstable modes for M > 3 from viscous calculations (the so-called “second mode”) that subsequently fuse to create only one unstable zone when M increases. Our calculations show a series of higher-order unstable modes for M > 3 in addition to the findings of Ma and Zhong [“Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions,” J. Fluid Mech. 488, 31–78 (2003)], where such higher-order modes for supersonic boundary layers are all noted to be spatially stable. The number and the frequency extent of the corresponding unstable zones increase with an increase in M beyond 3 while propagating downstream at a higher speed than those corresponding to incompressible, subsonic, and low supersonic (M < 2) cases.

Similarity solutions of a class of unsteady laminar boundary layer

The study of laminar unsteady boundary layer flows is essential for understanding the transition from laminar to turbulent flow, as well as the origins of turbulence. However, finding solutions to this phenomenon poses significant challenges. In this study, we introduce a novel method that employs a similarity transformation to convert the two-dimensional unsteady laminar boundary layer equations into a single partial differential equation with constant coefficients. By applying this transformation, we successfully derive similarity solutions for flat plate boundary layer flow, expressed in terms of Kummer functions. For convergent boundary layer flow, we derive an approximate analytical solution that includes both shock wave and soliton wave solutions. The superposition of these solutions provides evidence for the existence of solitons or soliton-like coherent structures (SCS) within boundary layers. Additionally, this paper explores two- and three-dimensional laminar flows, as well as three-dimensional turbulent flow equations, revealing that they all incorporate third-order derivatives with respect to spatial coordinates. This finding suggests that all viscous fluid motions have the potential to exhibit solitons/like coherent structures (SCS).

Analyzing heat transfer behavior in two-dimensional Darcy–Forchheimer porous medium using magnetized nanoparticles

This study investigates the two-dimensional boundary layer flow and heat transfer of an incompressible fluid over a plate embedded in a Darcy–Forchheimer porous medium of magnetite ternary hybrid ferrofluid in the presence of suction or injection. It uniquely includes the effects of thermal radiation and viscous dissipation in the energy equation, offering a comprehensive analysis of the phenomena involved. The research focuses on three nanoparticle (NP) combinations of ferrofluid particles ( Fe 3 O 4 ) and metal oxide NPs ( CuO / TiO 2 ) dispersed in an aquatic base fluid with different volume fractions. The governing nonlinear partial differential equations are transformed into ordinary differential equations via similarity transformations and solved using the Runge–Kutta-based shooting technique. The study evaluates key physical quantities, such as the skin friction coefficient and Nusselt number, and examines the effects of dimensionless parameters, including the absorbent Reynolds number (Re), expansion ratio (α), Peclet number (Pe), Eckert number (Ec), NP volume fractions ( φ 1 , φ 2 , and φ 3 ) , porous medium parameter (r), and Darcy-porous medium parameter (F), on velocity and temperature profiles. The results indicate that increasing the porous medium parameters r and F leads to opposite shear stress effects in the dual-porosity medium. Higher Peclet number (Pe) and Eckert number (Ec) values enhance heat transfer rates within the thermal boundary layers. Moreover, elevated Reynolds number (Re) and expansion ratio (α) values result in opposing radial velocity profiles within the momentum boundary layers. This comprehensive analysis provides valuable insights into the complex interactions governing ternary ferrofluid flow and thermal performance in dual-porosity media, contributing to advancements in fluid system applications.

Darcy-Forchheimer MHD micropolar water based hybrid nanofluid flow, heat and mass transfer features past on stretching/shrinking surface with slip and radiation effects

Hybrid nanofluids (HNF) play a vital role in enhancing the heat transfer characteristics of all types of traditional fluids, both in industrial and experimental applications. In this regard, the laminar two-dimensional (2D) boundary layer magnetohydrodynamic (MHD) Darcy-Forchheimer flow, heat transfer, and mass transfers characteristics of Cu–MoS2/micropolar water-based hybrid nanofluid have been considered over a stretching/shrinking surface. The thermal radiation and partial slip effects are considered in a porous medium. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using appropriate similarity transformations. The equations are numerically solved using the shooting method in Maple software, and dual solutions are obtained for different ranges of the applied parameters. The effects of the different physical parameters and nanoparticle volume fractions on velocity, microrotation, temperature, and concentration profiles along with skin friction, couple stress, Nusselt and Sherwood numbers are examined. The main findings of this study show that the velocity profiles decrease with an increase in the suction, the Darcy-Forchheimer number, velocity slip, magnetic and micromaterial parameters, while oppositely, it increases with an increase in nanoparticle volume fractions. Moreover, an increase in the magnetic field, nanoparticle volume fractions, and thermal radiation increases the temperature profiles, while an increase in the Prandtl number, suction, and thermal slip parameters decreases it. An increase in nanoparticle volume fractions decreases skin friction, the couple stress coefficient, and the Nusselt number, but increases the Sherwood number with the variation of suction.

Heat transfer analysis of mixed convection boundary layer blood base nanofluids with the influence of viscous dissipation

In this research study, we delve into the analysis of two-dimensional boundary layer incompressible viscous flow of blood based nanofluids, incorporating a couple stress model. This investigation encompasses considerations of heat transmission and viscous dissipation effects. Specifically, focus on a particular type of nanoparticle, Magnesium oxide (MgO), dispersed in blood, the base fluid. By employing suitable similarity transformations, transform a set of partial differential equations (PDEs) into nonlinear ordinary differential equations (ODEs). To leverage the Homotopy Analysis Approach (HAM) to solve this system of equations. This study discerns the impacts of various factors on temperature and velocity distributions. Visualization through graphs depicting nanoparticle volume concentration, magnetic field strength, couple stress parameter, Eckert number, and Prandtl number aids in presenting our findings. These insights illuminate the intricate interplay among different parameters and offer insights for enhancing heat transfer in blood-based nanofluids. Additionally, scrutinize the thermal performance of the blood-based nanofluid, assessing the Nusselt number and local skin friction coefficient. Such analysis holds significance for ensuring patient safety and treatment efficacy, particularly in medical procedures involving blood circulation, where mixed convection plays a pivotal role. Understanding these dynamics is crucial for the design and optimization of various medical devices and therapies, ensuring optimal temperature maintenance during blood circulation-related operations.

Formation and local stability of a two-dimensional Prandtl boundary layer system in fluid dynamics

By the linearized method, Oleǐnik-Samokhin had constructed solutions to distinguish between two cases of the behavior of the fluid at the initial stage of its motion past the surface in Oleinǐk and Samokhin (1999). In this paper, we consider the development of the boundary layer about a body that gradually starts to move in a resting fluid. In the analytical frame, the formation of the layer shows that, when the layer starts to move, the local solutions of the Prandtl boundary layer system is positive. By the Crocco transformation, the Prandtl boundary layer system reduces a degenerate parabolic equation with a nonlinear boundary value condition. Then by the reciprocal transformation, we can obtain a divergence type parabolic equation. We quote a new kind of BV entropy solution matching up with this divergence type parabolic equation when t≥t0, where t0 is a small enough positive constant. By imposing some close connections between the velocity of the outflow and the geometric characteristic of the spatial domain, using Kružkov’s bi-variables method, the local stability of BV entropy solutions is proved. Thus, by the Crocco inverse transformation, in the analytical frame, we obtain the existence and the uniqueness of the global solution of the Prandtl boundary layer system for a regular spatial domain.

Numerical study of binary mixture and thermophoretic analysis near a solar radiative heat transfer

The main focus of the current article is to investigate the two-dimensional boundary layer flow of non-Newtonian fluid induced by a linear stretching surface in the streamwise direction. The flow features of non-Newtonian fluid (i.e. shear-thinning and thickening nature of the fluids) are explored with Carreau fluid model. In addition, thermophoresis, Brownian motion, thermal radiation (with Rosseland approximation), binary mixture and activation enthalpy are also incorporated in fluid model for better analysis of thermal and solutal properties. The mathematical modelling of under consider physical problem yields nonlinear partial differential system. The governing mathematical system is first simplified with scaling transformations for the high Reynolds number and then transformed into non-dimensional ordinary differential system by using the similarity variables. Since the governing mathematical system comprised on nonlinear boundary value problem, thus shooting method is utilized for the numerical solution of the fluid flow governing equations. The obtained results followed the qualitative manners of the previously reported data. The computed results for important physical quantities (i.e. velocity, temperature, and concentration) are presented through graphs and then interpreted physically. It is observed that the boundary layer thickness increased for shear thickening fluids (n>1) while it is reduced for shear thinning fluids (n<1). Also, the temperature of shear thinning fluid is higher than shear thickening fluid. In addition, heat generation and thermal radiations enhance the thermal boundary layer. Thermophoretic viscosity effect reduces the concentration profile significantly in the flow domain.

Significance of quadratic density variation on the heat transport phenomena in Careau dusty fluid subject to Lorentz force via stretching surface

The two-dimensional boundary layer of steady mixed convective magnetohydrodynamic Carreau dusty fluid flow across a stretched sheet was investigated in this study. The primary focus of this study is to examine the impact of Carreau nanofluid on dusty flow dynamics in the presence of the Lorentz force. The partial differential equations governing the fluid flow, temperature, and concentration fields for both the fluid and dust phases are transformed into ordinary differential equations using an appropriate set of similarity transformations. Numerical outcomes are acquired employing the Runge–Kutta technique combined with the shooting method, implemented on the MATLAB platform. The numerical outcomes are compared to previous research and confirmed to be in very good accord. Finally, the impacts of relevant factors of physical and technical importance on flow and thermal transport characteristics are visually and tabulated. The results for the linear density and the quadratic density profile, it is noted that the linear density decreases compared to the quadratic density for velocity, but the opposite behavior is observed for temperature profile. The advancement of dusty nanofluids has resulted in significant enhancements in the heat transfer mechanism, which finds application in manufacturing, industry, and nanotechnology research. Research on dusty flows benefits on is beneficial in air pollution studies, dust entrainment in a cloud formed during a nuclear, vehicle emissions of smoke and other pollutants, crude oil purification, petroleum production, combustion, industrial effluent emissions, capillary blood flow, paint sparing, and, more.

Numerical heat and mass transfer in magnetized Williamson liquid motion over a sensor wall engulfed in a micro-cantilever system: An application to thin-film fluidic cells

Central focus of this numerical analysis is to investigate the influence of magnetic body force and thermal radiation on the time-dependent electrically conducting two-dimensional viscous incompressible boundary layer flow of non-Newtonian Williamson fluid over a microcantilever sensor surface suspended in a squeezed regime numerically. A novel heat source/sink and Soret effects are included in the respective governing equations. The constructed time-dependent nonlinear coupled partial differential equations are solved by using a bobust RK-4 technique. Increasing magnetic number increases the velocity distribution and decreases the temperature and concentration profiles in the flow regime.Enhancing Soret number increases the mass diffusion profile. Increasing magnetic number suppress the skin-friction coefficient. Heat transport rate booted with rising radiation number and Sherwood number decreased with enlarged Schmidt number in the channel regime. Finally, the numerical accuracy of the present similarity solutions and used numerical method is validated with existing results and observed a remarkable agreement.

Frictional heating effects on entropy generation with nonlinear Rosseland radiation: Implementation of generalized differential quadrature method

This article examines heat transfer and entropy generation analysis for boundary layer flow, considering the impacts of frictional heating and nonlinear Rosseland thermal radiation. The equations governing momentum and energy in a two-dimensional boundary layer are transformed into self-similar equations via similarity operations. The generalized differential quadrature method (GDQM) is then used to numerically solve the no-dimensional governing equations. Entropy generation and Bejan numbers are computed using the acquired solutions and are then plotted against the emerging flow parameters. An in-depth analysis of temperature distribution, entropy generation, velocity profile, and Bejan number. A high level of agreement was seen when the results from the literature were replicated using GDQM to verify the accuracy of the numerical code. It is found that a self-similar solution exists in the presence of viscous dissipation. The primary contributor to entropy generation at the interface is fluid friction, in contrast to heat transfer. Further, the findings indicate that entropy generation positively correlates with the Prandtl number, heating parameter, radiation parameter, and Eckert number. Notably, the results show that a decrease in operational temperature causes a decrease in entropy production.

New momentum integral equation applicable to boundary layer flows under arbitrary pressure gradients

By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\beta _\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

A two-dimensional boundary layer model for combustion chamber simulation

A two-dimensional boundary layer model for combustion chamber simulation

Control of linear instabilities in two-dimensional airfoil boundary layers

The present study deals with the control of linear, modal and non-modal growth of perturbations in a trapezoidal wing boundary layer. The asymmetric NACA 2515 airfoil is chosen for the analysis. The boundary layer velocity profiles, at different locations on the airfoil, are obtained using the Falkner–Skan equations. The non-modal linear stability theory is used to obtain the amplification of the initial perturbations, for which the state-space approach is adopted. The control is brought about using a Linear Quadratic Regulator (LQR) controller by the means of addition of fluid momentum in the wall-normal direction from the wing surface. Maximum transient energy growth is calculated across a range of wavenumber pairs, and control is applied to the pair with the highest energy growth. In this numerical analysis, we show that the transient growth is higher towards the trailing edge because of the adverse pressure gradient of the flow. Upon control actuation, the reduction in the transient energy growth of perturbations is achieved up to a maximum of 54.6%.

PARAMETRIC STUDY OF SUCTION OR BLOWINGEFFECTS ON TURBULENT FLOW OVER A FLATPLATE

The two-dimensional, incompressible, and turbulent boundary layer flow over a flat plate with suction or blowing from a spanwise slot is examined numerically. The mathematical modeling involves the derivation of the governing partial differential equations of the problems. These are the continuity, the momentum, the energy and the (K-ε) turbulence model. Besides, the perfect gas law is also used. A numerical solution of the governing equations is approximated by using a finite volume method, with staggered grid and modified SIMPLE algorithm. A computer program in FORTRAN 90 is built to perform the numerical solution.The developed computational algorithm is tested for the flow over a flat plate (4m) long with uniform suction or blowing velocity ratios of (V/U∞ =± 0.0185, ± 0.0463 and ±0.0925 m/s) are imposed on the slot for Reynolds number of (1.36 x 107 ), based on the plate length. The position of the slot change in the range of (X/L=1/4, 1/2 and 3/4) from leading edge and also, change width of slot in the value equal (0.12, 0.2 and 0.28m).The plate temperature is (70 °C), with the free stream velocity and temperature are (8.6m/s) and (25 °C) respectively. In addition, the effects of pitch angles on the flow field are investigated in the range of (30о   150о).The numerical results show that, for a uniform blowing, location of slot equal (X/L=1/4) from leading edge, a significant reduction of skin friction coefficient, wall shear stress and boundary layer thickness [displacement and momentum] to occur. While, an increase in boundary layer shape factor. Reynolds stress (uv) is more decreased than [(uu) and (vv)], mean velocity profiles in wall coordinates and dimensionless distance (U+, y+) decreases. When slot location is moved downstream to locations (X/L=1/2 or 3/4) a similar behavior can be said and most effective slot is obtained as (slot at X/L= 3m) from leading edge. While width of slot equal (0.28m) is better than values equal (0.12m and 0.2m). An opposite observations for the case of suction. The numerical results are compared with available numerical results and experimental data and a satisfactory results are obtained.

Separation of the two-dimensional steady MHD boundary layer

In this paper, we study the separation of the two-dimensional steady MHD boundary layer. When the outer flow satisfies appropriate conditions, we find that there exists a separation point on the boundary. Moreover, the singularity of the solution and the separation point occur simultaneously. Our result shows that the magnetic field has a certain positive effect on preventing the occurrence of separation. This gives a rigorous mathematical verification to previous study (Buckmaster, 1969) on the MHD boundary layer separation.

The effect of slip on the development of flow separation due to a bump in a channel based on a two-fluid turbulence model

In the paper, two-fluid turbulence model simulations were used to calculate the flow of a statistically two-dimensional turbulent boundary layer over a hump. In this work, the finite element method is used for the numerical implementation of the turbulence equations. Galerkin least squares (GLS) stabilization was used to stabilize the discretized equations. The results of the two fluid models were compared with experimental measurements by Matai, R., Durbin P. A. and the SST turbulence model which exists in the Comsol Multiphysics software package. The implementation of the Comsol Multiphysics software package showed the stability and good accuracy of the two-fluid turbulence model than the SST turbulence model.

The lie group analysis method for heat transfer in steady boundary layer flow field of nanofluid

In order to reveal the mechanism of the abnormal movement (Brownian motion enhances thermal scattering) of nanoparticles on the fluid enhanced heat transfer, the two-phase model was used to study the abnormal convection and diffusion of viscous nanofluids in the flat boundary layer of porous medium. Firstly, for the two-dimensional steady boundary layer stagnation point flow of incompressible Newtonian-nanofluids, the nonlinear governing equations of the flow field and temperature field of nanofluids are established from the Oberbeck-Boussinesq approximate equations. Secondly, the modern Lie group analysis method is introduced, we give the Lie symmetry determining equation of the flow field partial differential equations and the characteristics of the solutions. Further, using the relationship between the Lie symmetries and the conserved quantities, the conservation vector form of the flow field and the group invariant solution are derived in detail, and the reduced order model of the nanofluid flat boundary layer is obtained. Finally, the correctness of the analytical results obtained by the Lie group method was verified for different values of the flow parameter Prandtl. Research has shown that the Lie group method can be used to analytically solve the velocity and temperature distribution functions of abnormal motion of nanoparticles. The fluid temperature increases with the increase of the volume fraction parameter of nanoparticles, but decreases with the increase of the Prandtl value of the base fluid, and decreases with the increase of the plate stretching speed. The Lie group analysis method in this paper provides reference value for numerical simulation solutions of various heat and mass transfer in nanofluids.

NUMERICAL STUDY OF CARREAU FLUID FLOW ALONG AN EXPONENTIAL CURVED STRETCHING SURFACE

A two-dimensional incompressible boundary layer Carneau fluid flow with heat-transfer analysis over a curved stretching surface is analyzed. The energy equation with the inclusion of thermal radiation and viscous dissipation effects is considered. The governing partial differential equations which govern such flow phenomena are transformed into suitable form of ordinary differential equations for integration by using stream function formulation. The developed non-linear problem has been solved by computational approach based on shooting technique using sixth-order Runge-Kutta method and Matlab built-in function bvp4c program. The effects of non-dimensional controlling parameters on temperature and velocity profile are analyzed with the aid of tables and figures. The surface drag force and Nusselt numbers are studied for the different values of the governing parameters. It is predicted that velocity of the fluid and boundary layer thickness is decreased when radius of curvature parameter &amp;delta; is increased. Furthermore, the temperature profile dwindles for the growing values of &amp;delta;. Other important information is that for shear-thinning fluid the velocity profile shows its increasing nature, whereas for shear-thickening fluid the opposite trend has been observed. For increasing values of curvature parameter &amp;delta; from 2 to 1000, the temperature distribution and velocity profile is decreased. The radiative heat flux is included to enhance the temperature of the system, so, for the increasing values of radiation parameter &lt;i&gt;R&lt;sub&gt;d&lt;/sub&gt;&lt;/i&gt; from 0.2-0.5 the temperature distribution is increased. Further, as the Biot number and Eckert number are increased from 0.2-2 and 0.1-1, respectively, the temperature distribution is increased.

Investigation of non-ideal effects in compressible boundary layers of dense vapors through direct numerical simulations

In this work, we present an investigation about the sources of dissipation in adiabatic boundary layers of non-ideal compressible fluid flows. Direct numerical simulations (DNS) of transitional, zero-pressure gradient boundary layer flows are performed for two fluids characterized by different complexity of the fluid molecules, namely, “air” and siloxane MM. Different sets of thermodynamic free-stream boundary conditions are selected to evaluate the influence of the fluid state on both the frictional loss and the dissipation mechanisms. The thermophysical properties of siloxane MM are calculated with a state-of-the-art equation of state. Results show that the dissipation due to both time-mean strain field, irreversible heat transfer, and turbulent dissipation differs significantly depending on both the molecular complexity of the fluid and its thermodynamic state. The dissipation coefficient calculated from the DNS results is then compared against the one obtained using a reduced-order model (ROM), which solves the two-dimensional boundary layer flow equations for an arbitrary fluid [M. Pini and C. De Servi, “Entropy generation in laminar boundary layers of non-ideal fluid flows,” in 2nd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion and Power (Springer, 2020), pp. 104–117]. Results from both the DNS and the ROM show that low values of the overall dissipation are observed in the case of fluids made of simple molecules, e.g., air, and if the fluid is at a thermodynamic state in the proximity of that of the vapor–liquid critical point.

Extending Closed-Loop Control of Two-Dimensional Supersonic Boundary-Layer Instabilities Beyond a Single Operating Point

This paper investigates two control strategies to maintain low perturbation amplitudes of second Mack mode instabilities in a two-dimensional boundary layer with freestream velocity variations of ±5% around M∞=4.5: the multimodel synthesis and the gain scheduling method, both based on the structured mixed H2/H∞ synthesis to handle simultaneously, in an optimal way, the problems of nominal performance, stability, and performance robustness. The multimodel synthesis is based on a single linear-time invariant (LTI) controller optimized over multiple operating points, whereas the gain scheduling method results in a control law based on the interpolation of multiple LTI controllers designed at the various operating points. Both methods are compared with simple LTI controllers based solely on the nominal operating point in two sensor–actuator architectures: in feedforward (respectively, feedback) the estimation sensor is located upstream (respectively, downstream) of the actuator. In the feedforward case, significant performance loss is observed in off-design conditions, regardless of the method used. In the feedback case, multimodel synthesis enhances slightly performance robustness (but improvement is weak because of the wide range of freestream velocity variation), while the gain scheduling approach succeeds in strongly reducing the perturbation amplitude over the entire range of variation.