Integral Boundary Layer Equations Research Articles

Airfoil Design Framework for Optimized Boundary-Layer Integral Parameters

An airfoil design framework is introduced in which boundary-layer integral parameters serve as the driving design mechanism. The method consists of generating a parameterized pressure distribution capable of producing the desired boundary-layer characteristics for inverse design use. Additionally, by deduction from the Squire–Young theory, the method allows for the determination of the pressure distribution that results in the minimum theoretical drag. To assess this design framework, several airfoils were developed based on the mission requirements of the RQ-4B Global Hawk aircraft. Numerical results obtained using a viscous-inviscid solver of the integral boundary layer and Euler equations showed that the optimized airfoils achieved profile drag reductions of 9.06 and 6.00%, respectively, for α=0° and L/Dmax design points. A validation experimental campaign was also performed using the optimized CA5427-72 airfoil. The acquired data produced the expected pressure distribution characteristics and aerodynamic performance improvements, typifying the efficacy of the design framework.

Convection boundary layers with transpiration: Non-similar solutions and scaling

We present a new formulation and novel scaling relations for natural convection boundary layers on horizontal surfaces with weak transpiration velocities Vi, for intermediate Schmidt numbers (or Prandtl numbers Pr) 1<Sc<20. Using characteristic scales derived from integral boundary layer equations, we define a new similarity variable η=(y/x)(Rex/Grx)1/4/Sc1/5 that has Vi within it, where Rex and Grx are the local Reynolds and Grashof numbers based on Vi and the concentration difference ΔC, respectively. Using η, and the dimensionless transpiration velocity ξ=(Rex/Grx1/5)5/4 as a non-similarity variable, we obtain novel, reduced, non-similar boundary layer equations. Numerical solution of these reduced boundary layer equations using a local non-similarity approach gives us the profiles of horizontal velocity and concentration, boundary layer thicknesses, wall shear stress, and mass flux as functions of Sc, for various ξ in the range ξ<1. Using normalizing functions, we then obtain scaling relations for these parameters; we observe extended similarity where the non-similarity parameter ξ itself appears in the obtained similarity scaling relations.

Calculation and Selection of Airfoil for Flapping-Wing Aircraft Based on Integral Boundary Layer Equations

The flight of a migratory bird-like flapping-wing aircraft is characterized by a low Reynolds number and unsteadiness. The selection of airfoil profiles is critical to designing an efficient flapping-wing aircraft. To choose the suitable airfoil for various wing sections, it is necessary to calculate the aerodynamic forces of the unsteady two-dimensional airfoil with a Reynolds number in the range of 105. While accurate, calculating this by solving the Navier–Stokes equations is impractical for early design stages due to its high consumption of computing resources and time. The computational demands for extending it to 3D aerodynamic calculations are even more prohibitive. In this paper, a relatively simple method is proposed. The two-dimensional unsteady panel method is utilized to derive the inviscid flow field, the unsteady integral boundary layer method is utilized to solve the boundary layer viscous flow, and the eN transition model is adopted to predict the position of the transition. These models are coupled with the semi-inverse interaction method to solve the aerodynamics of the unsteady low-Reynolds-number two-dimensional airfoil. The unsteady aerodynamics of the symmetric and cambered airfoils at different wing sections are calculated respectively by the proposed method. Mechanism analysis of the calculation results is conducted, and a symmetrical airfoil or a slightly cambered airfoil is recommended for the wing tip, a moderately cambered airfoil is suggested for the outer-wing section, and a highly cambered airfoil is suggested for the inner-wing section.

Effect of shear on local boundary layers in turbulent convection

In Rayleigh Bénard convection, for a range of Prandtl numbers $4.69 \leqslant Pr \leqslant 5.88$ and Rayleigh numbers $5.52\times 10^5 \leqslant Ra \leqslant 1.21\times 10^9$ , we study the effect of shear by the inherent large-scale flow (LSF) on the local boundary layers on the hot plate. The velocity distribution in a horizontal plane within the boundary layers at each $Ra$ , at any instant, is (A) unimodal with a peak at approximately the natural convection boundary layer velocities $V_{bl}$ ; (B) bimodal with the first peak between $V_{bl}$ and $V_{L}$ , the shear velocities created by the LSF close to the plate; or (C) unimodal with the peak at approximately $V_{L}$ . Type A distributions occur more at lower $Ra$ , while type C occur more at higher $Ra$ , with type B occurring more at intermediate $Ra$ . We show that the second peak of the bimodal type B distributions, and the peak of the unimodal type C distributions, scale as $V_{L}$ scales with $Ra$ . We then show that the areas of such regions that have velocities of the order of $V_{L}$ increase exponentially with increase in $Ra$ and then saturate. The velocities in the remaining regions, which contribute to the first peak of the bimodal type B distributions and the single peak of type A distributions, are also affected by the shear. We show that the Reynolds number based on these velocities scale as $Re_{bs}$ , the Reynolds number based on the boundary layer velocities forced externally by the shear due to the LSF, which we obtained as a perturbation solution of the scaling relations derived from integral boundary layer equations. For $Pr=1$ and aspect ratio $\varGamma =1$ , $Re_{bs} \sim Ra^{0.375}$ for small shear, similar to the observed flux scaling in a possible ultimate regime. The velocity at the edge of the natural convection boundary layers was found to increase with $Ra$ as $Ra^{0.35}$ ; since $V_{bl}\sim Ra^{1/3}$ , this suggests a possible shear domination of the boundary layers at high $Ra$ . The effect of shear, however, decreases with increase in $Pr$ and with increase in $\varGamma$ , and becomes negligible for $Pr\geqslant 100$ at $\varGamma =1$ or for $\varGamma \geqslant 20$ at $Pr=1$ , causing $Re_{bs}\sim Ra_w^{1/3}$ .

The characteristics of laminar film condensation of pure vapor flow in a vertical tube

In this study, we investigate the characteristics of laminar film condensation of pure vapor flow in a vertical tube, taking into account (i) the vapor flow rate reduction owing to condensation, (ii) the finiteness of the vapor boundary-layer thickness, (iii) the curvature of the vapor layer, and (iv) the pressure variation. An approximate integral two-phase boundary-layer equation model is derived, and a condition that pressure recovery occurs is determined from the asymptotic analysis around the tube inlet. Furthermore, the present model is further simplified to obtain an analytical solution if the degree of subcooling is sufficiently large. The asymptotic solution for sufficiently large degrees of subcooling gives a good approximation to the numerical solution. The tube length of complete condensation is estimated. Comparison with vertical plane condensation demonstrates that the film thickness and the Nusselt number increased and decreased, respectively, by the vapor flow reduction for large degrees of subcooling.

Pseudo-Two-Dimensional Modeling and Validation of a Hybrid Rocket Combustion Chamber

A pseudo-two-dimensional (pseudo-2-D) model for the cylindrical combustion chamber of a hybrid rocket is developed in this paper with the purpose of its integration in a system design tool used for engine predesign phases. A compromise of simplicity and accuracy is thus required. Integral transient boundary-layer equations are included and coupled with the mass and energy balances at the fuel surface, where fuel pyrolysis is described by an Arrhenius law. Studies assessing the model in the nonreactive case provide satisfactory results regarding the description of the boundary layer. Concerning the hybrid case, the computed steady-state fuel regression rates are found to be in the low region of the reference values coming from semiempirical laws for several oxidizer/fuel couples and oxidizer mass fluxes, producing errors up to 50% for certain sources. The effect of mass flux and motor size is correctly simulated. Eventually, the whole engine operation is assessed through the comparison of the computed results with three experimental tests realized at our laboratory and using an 87.5% hydrogen-peroxide/high-density-polyethylene couple. Corresponding results lead to regression rate errors varying in a wide range (from 10 to 40%). Such differences, although high, remain acceptable for our engine predesign phase application.

Scaling in concentration-driven convection boundary layers with transpiration

Abstract

Comparison between temporal and spatial direct numerical simulations for bypass transition flows

ABSTRACT Temporally developing direct numerical simulations (T-DNS) are performed for free-stream turbulence induced bypass transition flow over a flat-plate under zero-pressure gradient, and results are validated using experimental data and spatially developing DNS (S-DNS) results. The temporal simulations predict the growth of near-wall Klebanoff modes in the pre-transition regime and their subsequent breakdown due to the sinuous-like instability, where the increase in free-stream turbulence intensity and length scale enhanced the instability. A formulation for the domain translation velocity is developed using the momentum integral boundary layer equation, to estimate the translation of the domain along the plate. The formulation was found to be robust for a range of free-stream turbulence intensities, and both the boundary layer growth and free-stream decay compared well with spatial simulations, confirming that the same translation velocity can be used for the entire domain. The optimal domain size for a temporal simulation is dictated by the length scale of the turbulent spots and errors due to streamwise periodicity, and is estimated to be to , where is the boundary layer thickness at the transition onset. The T-DNS prediction of the integral boundary layer parameters, mean flow, second and higher-order statistics, stress budgets, and free-stream decay were comparable to those of S-DNS, even though they were obtained on ten times smaller streamwise domain and numerical grid. However, T-DNS are limited for bypass transtion flows for which leading-edge effects are unimportant, and may have limitations for the accurate predictions of the near-wall Klebanoff modes that extend up to breakdown. Nonetheless, temporal approach is identified to be a viable inexpensive alternative to the spatial approach for canonical test cases for transition flow physics analysis.

Viiflow—A New Inverse Viscous-Inviscid Interaction Method

Viscous–inviscid interaction is the fastest method in computational fluid dynamics available for airfoil design and analysis that is able to capture viscous effects on the flow. Despite advances in (Reynolds-averaged) Navier–Stokes equations solvers and models, it is arguably the first choice for preliminary and even detailed design. A new method of coupling the boundary-layer equations to the inviscid flow is presented. The boundary-layer equations are solved with given displacement thickness, and a Newton method couples the viscous and inviscid regimes. This leads to a significant decrease in computational effort as compared to a joint solution of the boundary-layer equations with the coupling problem. The software Viiflow uses this mechanism, and examples that couple the integral boundary-layer equations with a panel method are presented: assessing accuracy using wind-tunnel measurements, using the software for a fluid–structure interaction problem, and a runtime performance comparison demonstrating a significant improvement as compared to the popular viscous–inviscid interaction airfoil analysis software XFOIL for higher numbers of panels.

Preliminary Design Method for Dense-Gas Supersonic Axial Turbine Stages

A fast preliminary design methodology for supersonic organic Rankine cycle (ORC) stator and rotor axial turbine blades with low degree of reaction is presented. First, the stator and rotor blade mean-line profiles are designed by using the two-dimensional (2D) method of characteristics (MOC), extended to gases governed by general equations of state (EOS). We focus more specifically on working fluids with medium to high molecular complexity, operating at thermodynamic conditions such that the fundamental derivative of gas dynamics Γ is lower than one in a significant portion of the flow field. For rotor blades, MOC is combined with a free-vortex method to achieve a smooth deflection of the supersonic incoming flow. A numerical approach is developed for solving the unique incidence problem in the case of gases governed by general EOS. Both stator and rotor blade geometries designed according to the inviscid MOC model are subsequently corrected to account for the development of viscous boundary layers by solving the compressible integral boundary layer equations extended to dense gases. The resulting blade designs are assessed by means of computational fluid dynamics (CFD) simulations based on a high-order finite volume solver equipped with advanced thermodynamic and transport-property models. Properly accounting for dense gas and viscous effects at an early design stage is found to improve the expected performance of ORC turbine rows significantly and delivers valuable baseline profiles for any further optimization.

An extension of a strong viscous–inviscid coupling method for modeling the effects of vortex generators

The aim of this work is to provide insights into the advantages and the limitations of the extension of a strong viscous–inviscid interactive code for modeling the effects of vortex generators, with focus on the calibration and validation for wind turbine airfoils. The proposed methodology relies on the approach proposed by other authors in the past and introduces an alternative formulation for the lag dissipation within the integral boundary-layer equations whose effects on the numerical prediction are evaluated. Besides the verification of the aeronautical test cases, particular attention is devoted to the validation of the results for several airfoils commonly used in wind turbine design. A blind test for airfoils from an industrial test case is conducted as well. Results show that the maximum lift coefficient can be well predicted by the present implementation, whereas the angle of maximum lift is a little higher than in wind tunnel measurements. Apart from that, the drag coefficients of airfoils with vortex generators cannot be predicted by the current implementation, but it is expected that the maximum lift and the stall angle of attack are more important in the design phase, since they determine power, loads, and noise.

Perfectly Matched Layer Boundary Integral Equation Method for Wave Scattering in a Layered Medium

For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces, and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Green's function of the background medium must evaluate the expensive Sommefeld integrals. Alternative BIE methods based on the free-space Green's function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the Green's function of the PML-transformed free space to define the boundary integral operators. The method is efficient, since the Green's function of the PML-transformed free space is easy to evaluate and the PMLs are very effective in truncating the unbounded interfaces. Numerical examples are presented to validate our method and demonstrate its accuracy.

Measurement and modelling of the turbulent boundary layer near the attachment line of a swept wing

ABSTRACTThis work is motivated by the need for low-order aerodynamic models to predict accurately the effect on profile drag of controlling attachment line transition. Head's entrainment method(1), a rapid integral boundary layer technique used for design studies on swept wings, suffers from the governing swept-tapered turbulent integral boundary layer equations being ill-posed in the vicinity of the attachment line. This singularity has been treated using crude extrapolations of the attachment-line similarity solution for over half a century, but this approach is unlikely to deliver accurate predictions of the effect of changes in the attachment line flow on profile drag. An experimental study has been carried out to explore the nature of the turbulent flow in the vicinity of a highly swept swept attachment line and has revealed a quite complex, non-monotonic development of the momentum thickness in this region. It has also revealed lower levels of twist in the boundary layer velocity profiles than anticipated from the highly curved character of the inviscid flow streamlines. These observations have prompted an alternative approach to the modelling of the flow in this region which not only successfully eliminates the lack of robustness in the swept-tapered equations but which also matches the experimental results to within ±5%.

Robust viscous-inviscid interaction scheme for application on unstructured meshes

A coupled viscous-inviscid interaction scheme combining the continuity equation for potential flow with the three-dimensional integral boundary layer equations is presented. The inviscid problem is discretized by a finite-element approach whereas an upwind-biased finite-volume scheme is employed for the boundary layer equations. The discretization is applicable to unstructured tetrahedral-triangular meshes and results in a sparse system of non-linear equations which is solved by a Newton-type method. The mathematical reasons for the singularities commonly associated with the integral boundary layer equations in separated flow regions are analyzed and the connection between the mathematical singularities and the numerical ill-conditioning is discussed. It is shown that, by a suitable choice of closure relations, it is possible to obtain a boundary layer model free from numerical ill-conditioning in separated flow regions. The accuracy of the coupled viscous-inviscid model is investigated in a number of test cases including transitional and mildly separated flow over two different natural laminar flow airfoils and three-dimensional flow over a swept wing. It is concluded that the coupled method is able to provide reasonably accurate predictions of viscous and inviscid flow field quantities for the investigated cases.

Conjugated heat transfer from a uniformly heated plate and a plate fin with uniform base heat flux

The relationship between maximum temperature and heating rate is determined for an internally heated flat plate, which is cooled on both sides by either forced or natural convection. Convection heat transfer is calculated using the energy integral boundary layer equations instead of solving governing partial differential equations numerically. The results are presented in simple analytical correlations.Solutions to a heated plate were used in the study of a plate fin with a uniform heat flux at the base in cases of forced and natural convection. New simple analytical equations for the maximum fin temperature were derived using the result on the heated plate. Solutions to a fin with uniform base heat flux include also those to a uniform fin base temperature as limiting cases. The validity of the simple new solutions to an internally heated plate and a fin with a uniform heat flux at the base were verified by comparing results to numerically obtained conjugated heat transfer solutions, analytical solutions, and to some experimental data for a heated plate.

Adaptive space‐time boundary element methods for the wave equation

AbstractFor the solution of the wave equation a space‐time energetic boundary integral formulation is used. The resulting single layer boundary integral equation is discretised by a conforming ansatz space on the lateral boundary. To derive an adaptive scheme an a posteriori error estimator based on the representation formula is used. Finally, numerical examples for a one‐dimensional spatial domain are presented. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite-volume scheme for the solution of integral boundary layer equations

An unstructured-mesh finite-volume formulation for the solution of systems of steady conservation laws on embedded surfaces is presented. The formulation is invariant to the choice of local tangential coordinate systems and is stabilized by a novel up-winding scheme applicable also to mixed-hyperbolic systems. The formulation results in a system of non-linear equations which is solved by a quasi-Newton method. While the finite volume scheme is applicable to a range of conservation laws, it is here implemented for the solution of the integral boundary layer equations, as a first step in developing a fully coupled viscous-inviscid interaction method. For validation purposes, integral boundary layer quantities computed using a minimal set of three-dimensional turbulent integral boundary layer equations are compared to experimental data and an established computer code for two-dimensional problems. The validation shows that the proposed formulation is stable, yields a well-conditioned global Jacobian, is conservative on curved surfaces and invariant to rotation as well as convergent with regard to mesh refinement.

Towards an Analytical Model for Film Cooling Prediction using Integral Turbulent Boundary layer

The objective of this work is to develop deep theoretical methods that are based on the solution of the integral boundary layer equations for investigating film cooling in liquid rocket engine. The integral model assumes that heat is transferred from hot free stream gas to the liquid film both by convection and radiation. The mass is transferred to the free srteam gas by the well-known blowing process. Downstream of the liquid film, the gas effectiveness is obtained by solving boundary layer integral equations. It incorporates a differential model for calorimeter mixing between liquid vapors in the boundary layer with the free stream gas entrained in the boundary layer. Comparisons with existing theoretical and experimental results indicate the film coating trends were well predicted by the present integral model proposed by us.

Analysis on the heat transfer characteristics of a micro-channel type porous-sheets Stirling regenerator

To avoid the high flow frictional loss associated with conventional wire mesh Stirling regenerators, a micro-channel type stacked porous-sheets Stirling regenerator is investigated. An analytical solution is derived for the transient heat transfer characteristics of the fully developed reciprocating laminar flow under prescribed wall temperature profiles. The complex Nusselt number (Nu) is expressed as a function of kinetic Reynolds number (Reω) and Prandtl number (Pr). At low Reω of less than 10, the real part of Nu has an almost constant value of 6.0, approximately equal to the known real-valued Nu for the fully developed unidirectional laminar flow under constant wall heat flux, while the imaginary part is negligible, thus “scaling effect” can be utilized to enhance heat transfer. At higher Reω, both the real and imaginary parts of Nu increase with the increase of Reω and Pr, and the phase shift between the temperature difference and the heat flux gradually increases and approaches 45°. Approximate analytical solutions are also deduced for the entrance region from the integral boundary layer equations in both cases of “Thermally developing flow” and “Simultaneously developing flow”. The heat transfer is enhanced in the entrance region and the local Nu in the flow direction approaches the corresponding values of fully developed flow. The analytical results are confirmed by dynamic mesh CFD results, and the obtained Nu∼ Reω data and patterns generally agree with available analytical and experimental data from published literatures. Application of the analytical results to the design and optimization of Stirling regenerator are also shown.

Three-dimensional viscous-inviscid coupling method for wind turbine computations

In this paper, a computational model for predicting the aerodynamic behavior of wind turbine wakes and blades subjected to unsteady motions and viscous effects is presented. The model is based on a three-dimensional panel method using a surface distribution of quadrilateral sources and doublets, which is coupled to a viscous boundary layer solver. Unlike Navier-Stokes codes that need to solve the entire flow domain, the panel method solves the flow around a complex geometry by distributing singularity elements on the body surface, obtaining a faster solution and making this type of codes suitable for the design of wind turbines. A free-wake model has been employed to simulate the wake behind a wind turbine by using vortex filaments that carry the vorticity shed by the trailing edge of the blades. Viscous and rotational effects inside the boundary layer are taken into account via the transpiration velocity concept, applied using strip theory with the cross sectional angle of attack as coupling parameter. The transpiration velocity is obtained from the solution of the integral boundary layer equations with extension for rotational effects. It is found that viscosity plays a very important role in the predictions of blade aerodynamics and wake dynamics, especially at high angles of attack just before and after boundary layer separation takes place. The present code is validated in detail against the well-known MEXICO experiment and a set of non-rotating cases. Copyright © 2014 John Wiley & Sons, Ltd.