An inverse integral 3D compressible boundary layer method and coupling with transonic inviscid solution

Summary A new velocity profile, which has a simple expression and agrees well with experimental data in a wide range, is proposed in the present paper. Based on this profile, the governing equations of the 3D compressible inverse boundary layer method are deduced. The steady transonic viscous flow around a 3D wing can be calculated as follows: the inviscid flow is calculated by using nonisentropic full potential equation; the viscous flow is calculated by using present boundary layer method; the viscous and inviscid solutions are coupled by using semi-inverse method. Numerical results agree well with the experimental data and required computer resources are less, so that it has broad prospects in the engineering application.     

A simple interaction law for viscous–inviscid interaction

The viscous–inviscid interaction (VII) philosophy for modelling aerodynamic boundary layers is discussed. ‘Traditionally’ the shear-layer equations are solved with pressure prescribed by the inviscid flow, but then the solution breaks down in a singularity related to flow separation. In the quasi-simultaneous coupling approach this singularity is overcome by making use of an interaction law. A novel mathematical analysis is presented of the essential properties of such interaction laws, which is based on classical theory for non-negative matrices. The performance of a highly simplified interaction law is demonstrated for separated airfoil flow beyond maximum lift.     

Fully non-linear two-layer flow over arbitrary topography

Steady, two-dimensional, two-layer flow over an arbitrary topography is considered. The fluid in each layer is assumed to be inviscid and incompressible and flows irrotationally. The interfacial surface is found using a boundary integral formulation, and the resulting integrodifferential equations are solved iteratively using Newton's method. A linear theory is presented for a given topography and the non-linear theory is compared against this to show how the non-linearity affects the problem.     

Computations of high‐speed compressible flows with adaptive cell‐centered finite element method

Abstract

An upwind cell‐centered finite element formulation is combined with an adaptive meshing technique to solve Navier‐Stokes equations for high‐speed inviscid and viscous compressible flows. The finite element formulation and the computational procedure are described. An adaptive meshing technique is applied to increase the analysis solution accuracy, as well as to minimize the computational time and the computer memory requirement. The efficiency of the combined method is evaluated by the examples of Mach 2.6 inviscid flow in a channel with compression and expansion ramps, Mach 6.47 inviscid and viscous flows past a cylinder, and Mach 4 viscous flow over a flat plate.     

Boundary elements and optimization for linearized compressible flows around immersed airfoils

For two-dimensional inviscid compressible flows the stream function may be used as the field variable. Although the relevant equation is nonlinear, it can be linearized for flows around slender bodies, such as airfoils. In multiply connected flow domains the boundary stream function values are not known a priori. In the present paper, an optimization approach is adopted to find these unknown values, as well as the entire solution field. In the proposed method of solution, the adjoint variable method of optimization is used to find the sensitivity coefficients of the objective function, which is constructed by using the Kutta condition. The boundary element method is used to discretize the flow and adjoint equations at each iteration of the optimization procedure. Numerical solutions are provided for two example problems for flows in a channel with one and two airfoils.     

A conservative embedded boundary method for an inviscid compressible flow coupled with a fragmenting structure

We present an embedded boundary method for the interaction between an inviscid compressible flow and a fragmenting structure. The fluid is discretized using a finite volume method combining Lax–Friedrichs fluxes near the opening fractures, where the density and pressure can be very low, with high‐order monotonicity‐preserving fluxes elsewhere. The fragmenting structure is discretized using a discrete element method based on particles, and fragmentation results from breaking the links between particles. The fluid‐solid coupling is achieved by an embedded boundary method using a cut‐cell finite volume method that ensures exact conservation of mass, momentum, and energy in the fluid. A time explicit approach is used for the computation of the energy and momentum transfer between the solid and the fluid. The embedded boundary method ensures that the exchange of fluid and solid momentum and energy is balanced. Numerical results are presented for two‐dimensional and three‐dimensional fragmenting structures interacting with shocked flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Mechanism of instability of Falkner-Skan flows

Summary. It is shown that the mechanism of linear instability of boundary-layer flows driven by favorable and adverse pressure gradients can be understood as a kinematic resonant interaction between inviscid and viscous partial modes. This kind of interaction has been proposed by Baines, Majumdar and Mitsudera [4] for the Blasius boundary layer. Here, this proposal has been examined for more general flows and quantitative confirmation has been obtained. Piecewise linear approximations of Falkner-Skan velocity profiles are taken as the mean flows. To understand the mechanism, it proves sufficient to examine eigensolutions of the viscous part obtained by enforcing no-slip. This leads to the prediction of the parameters for maximum growth in the space of Reynolds number and wave number. In the case of adverse pressure gradient the inviscid flow itself is unstable due to the presence of an inflexion point. We show that the instability mechanism stated above has a role in the flows of this kind, too.     

Dynamic analysis of open membrane structures interacting with air

Vibrations of open membrane structures including interaction with air are presented in the paper. Free and forced linear harmonic vibration problems are considered in the analysis. It is assumed that the air is compressible and inviscid. The aerodynamic pressure associated with structure deformations is described by boundary integral equation. The finite element method for the structure and the boundary element method for the air are used. To discretize the surface of the structure, triangular curvilinear 6-node elements are applied. The effects of the air compressibility and the aerodynamic radiation damping are investigated. The considerable decrease of the lowest natural frequencies in consequence of considering the effect of the surrounding air is observed. Numerical examples are given.     

Unstructured mesh methods for the solution of the unsteady compressible flow equations with moving boundary components

A review of a procedure for the simulation of time-dependent, inviscid and turbulent viscous, compressible flows involving geometries that change in time is presented. The adopted discretization technique employs unstructured meshes and both explicit and implicit time-stepping schemes. A dual time-stepping procedure and an ALE formulation enable flows involving moving boundary components to be included. Techniques that have been developed to maintain the validity of the unstructured mesh and to allow for the capture of moving flow features are also reviewed. Using the in-house developed techniques, some examples are included to demonstrate the use of the approach for the simulation of a number of flows of practical industrial interest.     

Helmholtz decomposition revisited: Vorticity generation and trailing edge condition

The use of the Helmholtz decomposition for exterior incompressible viscous flows is examined, with special emphasis on the issue of the boundary conditions for the vorticity. The problem is addressed by using the decomposition for the infinite space; that is, by using a representation for the velocity that is valid for both the fluid region and the region inside the boundary surface. The motion of the boundary is described as the limiting case of a sequence of impulsive accelerations. It is shown that at each instant of velocity discontinuity, vorticity is generated by the boundary condition on the normal component of the velocity, for both inviscid and viscous flows. In viscous flows, the vorticity is then diffused into the surroundings: this yields that the no-slip conditions are thus automatically satisfied (since the presence of a vortex layer on the surface is required to obtain a velocity slip at the boundary). This result is then used to show that in order for the solution to the Euler equations to be the limit of the solution to the Navier-Stokes equations, a trailing-edge condition (that the vortices be shed as soon as they are formed) must be satisfied. The use of the results for a computational scheme is also discussed. Finally, Lighthill's transpiration velocity is interpreted in terms of Helmholtz decomposition, and extended to unsteady compressible flows.     

An updated interactive boundary layer method for high Reynolds number flows

The quasi‐simultaneous interactive boundary layer (IBL) method is improved with the iterative correction of an inviscid operator. The updated interactive boundary layer method (UIBL) presented in this work, uses the Hess–Smith panel method (HSPM) as an inviscid operator to update the outer flow calculation and the inviscid velocity in the interaction law (IL). The discretization of the Hilbert integral (HI) from the original method is modified to reduce the error introduced by the calculation of the HI in a restricted domain. The method is tested on a flat plate with a small indentation for two‐dimensional, steady, incompressible and laminar flow. The UIBL method is capable to predict the flow separation and reattachment with good accuracy. The accuracy of the results is competitive with the numerical solution of the Navier–Stokes equations (NSE). Copyright © 2005 John Wiley & Sons, Ltd.     

Micropolar fluid jet impingement on a curved surface

This paper deals with an analysis of a normally impinging micropolar fluid jet on a curved surface. The flow near the stagnation point in the impingement region is divided into inviscid and viscous flow regions. The inviscid flow solution is governed by Euler's equations of motion expressed in curvilinear coordinate system. The viscous flow solution is governed by the zeroth and the first order boundary layer equations. These boundary layer equations are solved by assuming power series expansions for both velocity and microrotation fields which give rise to two systems of ordinary coupled differential equations. The effects of surface curvature and material parameters on boundary layer characteristics have been studied and presented graphically. The gradients of zeroth order velocity and microrotation at the wall decrease and the zeroth order displacement and momentum thicknesses increase with decrease in the value of surface curvature. The reduction in curvature results in the reduction in the gradients of first order velocity and microrotation at the wall as well as first order displacement and momentum thicknesses.     

The shape oscillations of a two-dimensional drop including viscous effects

The implementation of a boundary integral method for potential flow is presented for the case of a two-dimensional drop freely oscillating in a vacuum. Calculations using a standard boundary integral formulation and a double-layer potential boundary integral formulation are compared for the case of an inviscid drop with a clean interface. Additional comparisons are made using a least-squares spectral transform method for interpolating and differentiating versus more common methods using cubic splines and central differences. The boundary integral method for potential (i.e. inviscid) flow is extended in two viscous examples to approximate (i) the weak viscous effects in the bulk fluid far from the clean interface, or (ii) the surface viscous effects in an inviscid drop arising from an interface that is highly contaminated with an insoluble surfactant.The addition of an incompressibility constraint, implemented in a least-squares sense, to the standard boundary integral formulation is shown to significantly improve its ability to preserve the conserved quantities of volume and total energy. Nevertheless, the double-layer potential boundary integral formulation, despite its more complicated form, is found to be computationally more efficient than the standard formulation. The use of the least-squares spectral transform method is shown to be more accurate, and in certain conditions more efficient, than using cubic splines and sixth-order central differences for time-evolution of this system. Simulations approximating the damping effects of clean viscous drop are found to be consistent with small deformation theory while the calculations incorporating the damping effects of surface dilatational viscosity are shown to dissipate the energy of the oscillations at a rate that is neither exponential nor algebraic.     

Simulation of micro flows with moving boundaries using high-order upwind FV method on unstructured grids

 Numerical methods are presented for the simulation of steady and unsteady micro gas flows with moving boundaries found in micro scale fluidic devices. Both steady and unsteady flows are calculated by using an implicit real-time discretization and a dual-time stepping scheme implemented in a high-order upwind finite-volume unstructured-grid Navier–Stokes solver. For moving boundary problems, a new dynamic mesh method has been developed which is shown to be robust in handling large mesh deformation. Micro-scale flows studied with the methods developed include flow in micro channels, unsteady flow around a micro cylinder in oscillation and transport processes in micro pumps. The simulation is based on the continuum fluid model (the compressible Navier–Stokes equations) with slip boundary conditions implemented in the context of unstructured grids as the micro flows studied are all in the slip flow regime. Results are presented to validate the methods and demonstrate their applications to the analysis and design of micro fluidic devices. The implicit dual-time stepping scheme is found to be robust and efficient in dealing with both steady and unsteady micro flows. The unstructured-grid solver proves to be very flexible in dealing with complex geometries such as micro pumps. This is the first known report on the use of finite-volume unstructured grid solver for studying micro flows based on the slip boundary condition with moving boundaries.     

Flow feature aligned grid adaptation

A flow feature aligned grid adaptation method is proposed for the solution of Euler and Navier–Stokes equations for compressible flows, motivated by the desire for an efficient grid system for an accurate and robust solution method to best resolve flow features of interest. The method includes extraction of the flow features; generation of the embedded flow feature aligned structured blocks combined with unstructured grid generation for the rest of the flowfield; and adaptation of the hybrid grid for high flow feature resolution. The feature alignment makes it possible to maintain the high resolution property for both shock waves and shear layers of the approximate Riemann solvers and the higher order reconstruction schemes based on one‐dimensional derivation and dimensional splitting. High grid efficiency is obtained with highly anisotropic directional grid corresponding to the feature directions. The computational procedure is described in details in the paper and its application to flow solutions involving shock waves, boundary layers, wakes and shock boundary layer interaction are demonstrated. Its accuracy, efficiency and robustness are discussed in comparison with an anisotropic unstructured grid adaptations for the shock boundary layer interaction case. Copyright © 2006 John Wiley & Sons, Ltd.     

Interfacial waves and hydraulic falls: Some applications to atmospheric flows in the lee of mountains

Two dimensional flow of a layer of constant density fluid over arbitrary topography, beneath a compressible, isothermal and stationary fluid is considered. Both downstream wave and critical flow solutions are obtained using a boundary integral formulation which is solved numerically by Newton's method. The resulting solutions are compared against waves produced behind similar obstacles in which the compressible upper layer is absent (single layer flow) and against the predictions of a linearised theory. The limiting waves predicted by the full non-linear equations are contrasted with those predicted by the forced Korteweg-de Vries theory. In particular, it is shown that at some parameter values a multiplicity of solutions exists in the full nonlinear theory.     

Analysis of interactions between shock waves and laminar boundary layers with heat transfer

Summary This paper deals with interactions between shock waves and laminar boundary layers on flat plates with heat transfer. In order to describe this phenomenon the boundary layer is divided into inner viscous layer and outer inviscid layer after Gadd. The boundary layer approximations are assumed to remain valid in the inner layer and the momentum integral equation for the layer is utilized instead of the Pohlhausen's wall condition. In the outer inviscid layer the motion is described by Euler's equation in terms of the isobar coordinates and the deflection angle is determined to match with that at the outer edge of the inner layer. The present theory predicts that self-induced separation does occur for highly cooled wall and yields results in good agreement with the measurement of Lewis et al.With 6 Figures     

Numerical analysis of two-dimensional compressible turbulent flows in a turbine stage

A numerical analysis based on the compressible Reynolds-averaged Navier-Stokes equation has been developed for the analysis of two-dimensional compressible turbulent flows in a turbine stage (nozzle and bucket). In the present flow analysis, governing equations are solved by the use of a time dependent explicit method and a two-equation model of turbulence is employed to estimate turbulence effects. To calculate nozzle and bucket flow fields simultaneously, a steady interaction between these flows is assumed. For spatial discretization of the governing equations, a control volume method combined with a body-fitted curvilinear coordinate system is developed to calculate the flows in arbitrarily shaped cascades. In order to assure the effectiveness of the present method, computations are carried out for a two-dimensional section at a blade midspan in a turbine stage. The method gives satisfactory results about boundary layers on blade surfaces, nozzle wake profiles and pitchwise averaged turbine design parameters at each blade exit.     

Detailed study of complex flow fields of aerodynamical configurations by using numerical methods

The mathematical physics of fluid flow in a compressible medium, leads to nonlinear partial differential equations or their equivalent integral versions. For the solution of these equations one has generally to resort to numerical methods using mostly finite difference or finite volume schemes, which are well established now. These field methods are very suitable for studying the physical features of complex flows. The present paper gives at first a short sketch of the numerical procedure and thereafter goes into the detailed analysis of the flow fields of delta wings, double-delta wings, delta shaped wing-canard combinations and space vehicles. Further examples include long span wings and wing-bodies at supercritical onflows, flows around propellers and rotors and finally some unsteady flows. The examples cited are selected topics from the extensive studies undertaken in the department of numerical aerodynamics of thedlr in Braunschweig in the course of the last few years.