Integral Boundary Layer Research Articles

Nonlinear Separated Inviscid-Viscous Analysis of Oscillating Cascade Aerodynamics Using an Inverse Integral Method

A mathematical model is developed to analyze the unsteady flow through an harmonically oscillating cascade of airfoils, including separated flow. The model incorporates an inverse integral boundary layer solution with the time-marching Euler analysis NPHASE. An embedded composite grid formulation is incorporated, specifically a deforming C-grid embedded in a Cartesian H-grid, thereby simplifying grid generation. To reduce computational requirements, Fourier series unsteady periodic boundary conditions are implemented. The integral turbulent boundary layer model is closed with steady correlations adopted in a quasi-steady manner. To couple the inviscid and viscous solutions, the viscous effect is modeled in the unsteady Euler solution in a quasi-steady manner by a transpiration boundary condition. An isolated airfoil is used to compare the steady interaction model with experimental data. Then a flat plate cascade is used to verify the unsteady flow solver with linear theory predictions. An experimental unsteady aerodynamics data set of a loaded cascade with separated meanflow executing torsional oscillations compared favorably with the analysis. The code is then utilized to study the effect of flow separation on the unsteady aerodynamics.

Simple Numerical Method to Compute Viscous Lift Loss of Wings

It is well known that the lift of a wing is less than the value computed on the basis of potential  ow because of the presence of a boundary layer on the surface. A method of computing this lift loss is described based on numerically solving the integral equations of the three-dimensional boundary-layer  ow in a streamline coordinate system. When applied to several swept, tapered, and twisted wings, the method closely predicts the lift curve measured in a wind tunnel. Similar computations with a twodimensional (strip) model overestimate wing viscous lift loss by about 10%. The numerical effort required to solve the integral boundary-layer equation set is much less than that required to solve either the threedimensional boundary-layer equations or the Navier – Stokes equations, with little loss in accuracy. It could easily be used, therefore, for preliminary estimates of climb and cruise wing performance. A simple procedure is presented for identifying cross ow separation.

Discharge Characteristics of Low-Density Gas Flow through Parallel Plate Channel with Narrow Gap.

Low-density choked flow through a parallel plate channel with a narrow gap is studied experimentally for a wide range of Reynolds numbers. In the region from continuum flow to slip flow, the choked flow is analyzed by one-dimensional flow approximation based on the integrated boundary layer equations, taking into account the slip at the channel walls. The calculated discharge coefficients and pressure distributions agree well with the experimental results for a wide range of Reynolds numbers, including the slip flow region. A nondimensional correlation equation of the discharge coefficient is proposed as a function of Rehs/l for Rehs/l<2×102. In the region from slip flow to molecular flow, the experimental results agree well with those of previous work. A nondimensional correlation equation of Cdl/hs is proposed as a function of Re for 7<Re<5×101.

Simultaneous Viscous-Inviscid Coupling via Transpiration

In viscous-inviscid coupling analysis, the direct coupling technique and the inverse coupling technique are commonly adopted. However, stability and convergence of the algorithms derived are usually very unsatisfactory. Here, by using the transpiration technique to simulate the effect of the displacement thickness, a new simultaneous coupling method is derived. The integral boundary layer equations and the full potential equation are chosen to be the viscous-inviscid coupled system. After discretization, the Newton-Raphson technique is proposed to solve the coupled nonlinear system. Several numerical results a re used to demonstrate the accuracy and efficiency of the proposed method.

Two-dimensional viscous flow past flexible sail sections close to ideal incidence

AbstractA model of the viscous flow past a flexible sail section operating near the ideal incidence is described. Viscous effects are calculated via weak viscous-inviscid interaction of a panel method and an integral boundary layer method, and a new model for the leading edge separation bubble is introduced. The flexible section is allowed to deform in response to the pressure and shear stresses acting on it. Results are presented for the effect of incidence, excess length and Reynolds number on the development of the boundary layers on each side of the section and the consequences for the lift and drag of the section are described. The numerical model is compared with experimental results, giving in general good agreement and shedding light on the physics of the viscous flow past flexible membranes.

AN IMPROVED VARIANT OF INTEGRAL BOUNDARY-LAYER THEORY II. MASS TRANSFER

An improved variant of the integral boundary-layer theory was presented in Part I of this paper for flow around axisymmetric bodies with rigid or moving surfaces in the intermediate range of Reynolds numbers. It is applied here to predict the mass transfer coefficients in the continuous phase from bubbles and drops, over a wide range of Reynolds numbers and surface mobility. Good agreement is obtained with known limiting cases for mass transfer from spheres.

Experimental investigation of rarefied gas flow through rectangular slits and nozzles

The flow of a rarefied gas through rectangular configuratons of different geometries has been experimentally studied to determine their discharge coefficient characteristics. The configurations used are a set of sharp-edged slit orifices, a smooth converging nozzle and a tube. The range of the Reynolds number based on the throat conditions varied from 0.01 to 100. The equivalent Knudsen number range based on the upstream conditions and inlet diameter varied from 0.0521 to 2.521. The results for the smooth nozzle are compared with calculations using a numerical method with one-dimensional stream tube approximation based on integrated boundary layer equations. The slit and the tube results are compared with the experimental results of Sreekanth and Davis [1988].

ANALYSIS/DESIGN CALCULATIONS OF TRANSONIC FLOWS

SUMMARY Analysis/design calculations of transonic flow are discussed and several improvements are made. The nonisentropic potential method is used to calculate the inviscid transonic flow analysis problem instead of the traditional potential method. An inverse integral 3D boundary layer method is used to calculate the boundary layer in the viscous transonic flow analysis problem. The viscous/inviscid interaction calculations are carried out by a semi-inverse coupling scheme. In design problem calculations, an improved residual-correction method is used. Three individual methods are combined in a global algorithm and computing code. The improvements speed up the convergence, increase applicability and computational efficiency. Some numerical results are given to illustrate that the present method provides an effective engineering tool of high accuracy and efficiency in three dimensional transonic analysis and design situations.

Discharge Characteristics of Low-Density Gas Flow through Parallel Plate Channel with Narrow Gap.

Low-density choked flow through a parallel plate channel with a narrow gap is studied experimentally for a wide range of Reynolds Numbers. In the region from continuum flow to slip flow, the choked flow is analysed by one-dimensional flow approximation, based on the integrated boundary layer equations, taking into account the slip at the channel walls. The calculated discharge coefficients and pressure distributions agree well with the experimental results for a wide range of Reynolds Numbers, including the slip flow region. A nondimensional correlation equation of the discharge coefficient is proposed as a function of Rehs/l for Rehs/l <2×102. In the region from slip flow to molecular flow, the experimental results agree well with those of previous work. A nondimensional correlation equation of Cdl/hs is proposed as a function of Re for 7<Re<5×101.

Shock wave/boundary layer interaction with heat addition by nonequilibrium phase transition

This investigation deals with the interaction of a turbulent boundary layer and a normal shock wave caused by supercritical heat addition in homogeneously condensing flows. The method used is a combination of analytical and numerical procedures, which is based on the perturbation method of Bohning & Zierep for the shock/boundary layer interaction in adiabatic flow. The homogeneous condensation process is described by the classical nucleation theory of Volmer together with the molecular droplet growth law of Hertz and Knudsen. The cooling rate, the nucleation rate, the surface-averaged droplet radius and the condensate mass fraction are analyzed along several streamlines in the boundary layer. The strength of the shock wave caused by supercritical heat addition in the outer flow field weakens and ultimately vanishes near the boundary edge by the influence of viscous effects. Comparison is made of the integral boundary layer variables and the friction coefficient in adiabatic flow with those in diabatic flow.

Effect of pressure gradient on turbulent spots

Turbulent spots were generated artificially to simulate natural spots in the transitional boundary layer. The spots at the final stage under zero and adverse pressure gradients were measured using conventional hotwire anemometry technique. Ensemble averages of the velocity traces were calculated and velocity profiles through the spots and the surrounding region were plotted. The spots are represented by the ensemble average of the velocity perturbations. The integral boundary layer properties are derived and compared through the spots. The effect of the pressure gradient on the turbulent spots is analysed with the structure of the spots and the ‘calmed region’ following the spots.

Inviscid-Viscous Coupled Solution for Unsteady Flows Through Vibrating Blades: Part 1—Description of the Method

An efficient coupled approach between inviscid Euler and integral boundary layer solutions has been developed for quasi-3-D unsteady flows induced by vibrating blades. For unsteady laminar and turbulent boundary layers, steady correlations are adopted in a quasi-steady way to close the integral boundary layer model. This quasi-steady adoption of the correlations is assessed by numerical test results using a direct solution of the unsteady momentum integral equation. To conduct the coupling between the inviscid and viscous solutions for strongly interactive flows, the unsteady Euler and integral boundary layer equations are simultaneously time-marched using a multistep Runge–Kutta scheme, and the boundary layer displacement effect is accounted for by a first order transpiration model. This time-resolved coupling method converges at conditions with considerable boundary layer separation.

Model of two-dimensional vortex motion with an entrainment mechanism

A model for describing the vertically averaged vortex motions of an incompressible viscous fluid with an arbitrary vertical structure of the bottom Ekman boundary layer is proposed. An approach similar to that adopted in [1] is used: the second moments of the deviations from the average velocities required in order to close the vorticity equation are calculated by means of the Ekman solution for gradient flows, which makes it possible to take the integral bottom boundary layer effect into account. As a result, these terms lead to a specific form of nonlinear friction with a coefficient that depends on the vorticity of the average flow. In the particular case of a constant vertical turbulent transfer coefficient the inaccuracies of the model described in [1] can be eliminated. The generalized vorticity equation obtained has solutions of the vorticity spot type with a uniform internal vorticity distribution, which can be effectively investigated by means of appropriate algorithms [2]. The mechanism of entrainment at the vorticity front is illustrated with reference to the example of the evolution of vorticity spots. An exact solution of the problem of the evolution of an elliptic vortex (generalized Kirchhoff vortex), which in the case of fairly strong anticyclonic vorticity degenerates first into a line segment (vortex sheet) and then into a point vortex, is constructed. Equations describing the dynamics of an elliptic vorticity spot in an external field with a linear dependence of the velocity on the horizontal coordinates and generalizing the classical Chaplygin-Kida model [3, 4] are constructed.

Numerical modelling of cavities on axisymmetric bodies at zero and non‐zero angle of attack

AbstractThe flow about submerged, fully cavitating axisymmetric bodies at both zero and non‐zero angle of attack is considered in this paper. A cavity closure model that relates the point of detachment, the angle that the separating streamline makes with the body and the cavity length is described. The direct boundary element method is used to solve the potential flow problem and to determine the cavity shape. A momentum integral boundary layer solver is included in the formulation so that shear stresses can be incorporated into the drag calculations. The numerical predictions based on the proposed closure model are compared with water tunnel measurements and photographs.

A model of effective segregation coefficient, accounting for convection in the solute layer at the growth interface

A model of the effective segregation coefficient k eff is obtained by using integral boundary layer analysis. The model provides a simple relation between k eff and the actual physical thickness of the solute layer δ D, the convective velocity in the solute boundary layer V D, the equilibrium segregation coefficient k, the growth rate R and the characteristic lenght of the growth interface L. A good agreement is obtained between the model and the data in the literature (e.g. the data used to fit the BPS model). The model provides a criterion for the maximum convective velocity, allowing diffusion-controlled segregation. The criterion indicates that diffusion-controlled segregation of solvent-solute system with k ≪ 0.1 may not be feasible even in microgravity.

Wake effects on the prediction of transonic viscous flows around airfoils

A semi-inverse viscous/inviscid coupling technique for the calculations of compressible airfoil flows is presented. The Euler equations in integral form are solved in the inviscid part and the integral compressible boundary-layer equations in inverse form are solved in the viscous part. Both displacement and curvature effects have been included in the wake computation. Interaction is realized using the equivalent sources method. This technique has been applied to the calculation of the aerodynamic characteristics of airfoils both in subsonic and transonic regimes. It has been shown that the contributions of body displacement, wake displacement, and wake curvature effects are particularly important in the evaluation of the aerodynamic coefficients, especially drag. Different ways of computing the equivalent sources fluxes in the wake and the influence of the fixed-wake dividing streamline shape have been investigated and are discussed.

Compressible turbulent boundary layers with heat addition by homogeneous condensation

Homogeneous condensation in transonic viscous flow is computed using a combination of analytical and numerical methods. The procedure is based on the perturbance method for the shock/boundary-layer interaction of Bohning and Zierep for adiabatic flow. In this paper, the model was extended for applications to boundarylayer flows with internal nonequilibrium heat addition. For the simulation of the homogeneous condensation process we use the classical nucleation theory of Volmer and the molecular Hertz-Knudsen droplet growth law. One momentum equation of the Navier-Stokes equations is solved by a time-stepping procedure combined with a mixing length turbulence model. Objective in view is the analysis of the macroscopic phenomena of nonequilibrium heat addition from homogeneous condensation of water vapor in transonic flows with high velocity and temperature gradients normal to the main flow direction. In detail, the cooling rate, the nucleation rate, the droplet radius, and the condensate mass fraction in the boundary layer are discussed. Approaching the wall surface and in comparison to the inviscid diabatic flow just outside of the boundary layer the nucleation process and the deposition of droplets, i.e. the condensate fraction contours, are shifted downstream, continuously or stepwise, depending on the pressure gradients at the boundary-layer edge. From the comparison of the integral boundary layer variables we conclude that the friction coefficients change only moderately, whereas the displacement and momentum thickness are found to be more sensitive to the homogeneous condensation onset.

Numerical predictions of transonic viscous flows around aerofoils through an Euler/boundary layer interaction method

Abstract A semi-inverse viscous/inviscid coupling technique for the calculation of compressible flows is presented. Euler equations in integral form are solved for the inviscid part and integral compressible boundary-layer equations in direct and inverse form are solved for the viscous one. Interaction is realised using the equivalent sources approach.The technique has been applied to the calculation of the aerodynamic characteristics of aerofoils both in cruise and take-off or landing configurations. To treat the complex geometries involved in the latter case, a multi-block approach, based on structured ‘H’ grid topology, has been developed and validated for different configurations.

Local and integral turbulent boundary layer characteristics on a convex surface in diffusor flow

A semiempirical method is proposed for computing the local and integral characteristics of the turbulent boundary layer on a convex surface under conditions of a positive longitudinal pressure gradient. By using the method proposed, graphical dependences are obtained that characterize the influence of curvature on the form-parameter, the thickness of the momentum loss, the thickness of the viscous sublayer, and the relative flow coefficient.

Boundary layer development in supersonic MHD generators

A numerical model of the turbulent boundary layers in the gas dynamic channel of a supersonic MHD generator is constructed. This model describes the development and structure of the layers in the nozzle, on the electrode and insulating walls of the duct, in the two-dimensional approximation. The characteristics of the boundary layers in various generator operating regimes are investigated numerically. The integral boundary layer thicknesses characterizing the nonuniformity of the gas dynamic and electrodynamic quantities are calculated. The limits of applicability of the integral calculation method are determined for typical MHD generator operating conditions.