Numerical studies in high Reynolds number aerodynamics

Two numerical approaches are presented for the computation of viscous compressible flows at high Reynolds' numbers. In the first approach, named global approach, the whole flow field, which includes viscous and inviscid regions, is determined as the solution of a single set of equations, which may be the full Navier-Stokes equations, or some approximate form of these equations. The second approach, named coupling approach, consists in solving two different sets of equations in their respective domains simultaneously; one of the two sets governs an inviscid flow whose boundary conditions are provided by the viscous effects, determined by the other set.The discussion of the global approach is centred on two particular features of the finite-difference method used: a discretization technique, directly in the physical plane with arbitrary meshes: and a mesh adaptation technique, which combines a coordinate transformation to fit the mesh system to particular lines in the flow, and a technique of dichotomy for mesh refinement. Numerical results are presented for an axisymmetric compression corner and a shock-boundary layer interaction on a flat plate, both in supersonic regime, and for a transonic nozzle flow.For the coupling approach, emphasis is given firstly to the improvement resulting from an interacting analysis where the viscous and inviscid computations are matched, and not only patched. It is shown that the parabolic problems associated with simple viscous theories are always replaced by elliptic problems, even for supersonic flows, and that “supercritical interactions” or “critical points”, as defined by Crocco-Lees, are removed. Secondly, a new coupling method, fully automatized and capable of solving directly a well-posed problem for supersonic flow, is illustrated by examples involving shock wave-boundary layer interactions and reverse flow bubbles; they concern flows over symmetrical transonic airfoils and supersonic compression ramps.     

Numerical investigation of supersonic flow for axisymmetric cones

An existing curvilinear finite-volume code with robust shock-capturing scheme was modified to allow for simulations of supersonic flow for axisymmetric cone geometries. It is shown how for an axisymmetric coordinate system the convective and viscous flux derivatives in the circumferential direction reduce to a y-momentum equation source term. The advantage of this approach over an axisymmetric code is that the governing equations and the discretization do not need to be changed. This paper provides a detailed derivation of the axisymmetric source terms from the full Navier-Stokes equations. Results are shown for a sharp and a blunt cone for approach flow Mach numbers of M = 3.5 and M = 7.99.     

Strive for accuracy—improvement of predictions

Finite-difference approximations to the governing equations of fluid motinos are discussed for viscous three-dimensional flows for Reynolds numbers of several hundred, for three-dimensional incompressible and compressible boundary layers and for inviscid near-sonic supersonic flows. It is shown how the boundary conditions of the problems chosen influence the solutions. If certain conditions to be discussed in the following are not met, neither convergence nor uniqueness of the solution can be guaranteed. Comparison of the predictions described herein with experimental data of the recent literature asserts the validity of the solutions given. The examples chosen include: Viscous flows through biological vessels with bends and bifurcations, formation of Taylor-Görtler vortices in spherical gaps, three-dimensional viscous displacement effect on wings in transonic flow and front and embedded shocks in flow fields with Mach numbers slightly above one.     

Solution of viscous transonic flow over wings

Knowledge of the viscous flow about wings is very important in 3-D wing design. In transonic flow about a typical supercritical wing, the viscous effect results in a sizable reduction of the lift-to-drag ratio. The Reynolds number dependence of the flow is not clearly defined, and no known similitude exists that can be used to scale the experimental data for a particular design. Recent advances in computer technology and numerical technique have relieved the difficulty of obtaining a theoretical solution somewhat, but the lack of a proper reliable method of treating the turbulence in a time-averaged Navier-Stokes solution remains the major stumbling block.For this paper, a “zonal” approach has been used for a viscid-inviscid interaction analysis to yield an iterative solution for the viscous flow about wings in the transonic flow regime. The chord Reynolds number considered was of the order of 10⁶ and above so that the flow was predominantly turbulent. The inviscid flow field was obtained by solving the 3-D potential flow equation. A parabolic coordinate mapping was used in the computation, in conjunction with a finite volume formulation. A new approximate factorization scheme has been developed for the iterative solution of the inviscid flow. A special far field asymptotic boundary condition that improves the accuracy and convergence of the method was derived. For the 3-D boundary layer calculation, the integral method of Myring-Smith-Stock was extensively modified to make it suitable for the interaction calculation. The effect of wing thickness was taken into account and the 3-D viscous wake was computed. The interaction calculation was formulated with a set of coupling conditions that includes the source flux distribution due to the surface boundary layer on the wing, the flux jump distribution due to the viscous wake, and the effect of the viscous wake curvature. The transpiration boundary conditions have been used for the inviscid flow in the coupled calculation. In addition, a method was devised so that the results of an analysis of the trailing edge strong interaction solution for a 2-D viscous airfoil could be adapted for the normal pressure correction near the trailing edge. The theory has been applied to supercritical wing geometries of practical interest. The converged viscous flow results compare favorably with experimental pressure data.     

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.     

On the application of boundary conditions to time dependent computations for quasi one-dimensional fluid flows

A study is made of the influence of boundary and initial conditions on time-dependent finite-difference solutions of quasi-one-dimensional duct flows. Several questions are addressed: (1) Under what conditions will a time-dependent solution converge to a steady-state supersonic flow, (2) Under what conditions will it converge to subsonic flow and (3) What conditions are necessary to insure a particular unique solution for subsonic flows. The results provide an orientation, or way of thinking, about the role of such conditions in time-dependent solutions of steady-state flows. The results also show that supersonic solutions are readily obtained by holding only pressure and temperature fixed at the duct inlet, and allowing velocity to float. However, subsonic solutions require pressure, temperature and velocity to be fixed at both the duct inlet and exit. If no conditions are held fixed at the exit, the results always converge to the supersonic solution, even if the fixed inlet mass flow is less than critical. In such a case, the program appears to generate additional mass flow between the inlet and throat, sufficient to choke the flow. These results also have some impact on two- and three-dimensional time-dependent solutions where subsonic flow is present on some or all portions of the flow boundaries.     

Shock fitting in conical supersonic full potential flows

The full potential equation is solved implicitly for supersonic conical flows with the bow shock fitted as an external boundary. Existing potential flow computational procedures capture the embedded crossflow shock within the context of transonic relaxation schemes. In this study, the crossflow shock is fitted for the first time in potential flow as an internal boundary. Hence, all shocks are implicitly fit making the computation fully conservative. For thin elliptic cones, the outer segment of the crossflow shock was found to be oblique to the incoming or supersonic crossflow. This behavior was not found for circular cones. The shock fitted solutions are compared to both Euler and potential captured solutions. The full potential shock fitted results are in favorable agreement with conservative captured solutions.     

An investigation of transonic flow over axisymmetric rigid body

The aim of this study is to investigate transonic flow over the axisymmetric rigid body of revolutions using matched asymptotic expansions of high Reynolds number flow. For this purpose the triple-deck model is employed. It allows to study the flow separation near a junction line where a circular cylinder is connected to a divergent conical body. It is found that in the axisymmetric transonic flow the interaction region is governed by the viscous-inviscid interaction process, where the axisymmetric Karman-Guderley equation in the inviscid part of the flow should be coupled with Prandtl’s boundary layer equations for the viscous sublayer. The coupled governing equations of the interaction region is solved using a semi-direct numerical method considering proper boundary conditions. Numerical results imply that incipience of separation may appear over the axisymmetric rigid body subject to body shape and transonic axisymmetric nature makes the flow much less prone to separation as compared to the two-dimensional flow.     

Identification of large coherent structures in supersonic axisymmetric wakes

Direct numerical simulation data of supersonic axisymmetric wakes are analysed for the existence of large coherent structures. Wakes at Ma=2.46 are considered with results being presented for cases at Reynolds numbers Re_D=30,000 and 100,000. Criteria for identification of coherent structures in free-shear flows found in the literature are compiled and discussed, and the role of compressibility is addressed. In particular, the ability and reliability of visualisation techniques intended for incompressible shear-flows to educe meaningful structures in supersonic wakes is scrutinised. It is shown that some of these methods retain their usefulness for identification of vortical structures as long as the swirling rate is larger than the local compression and expansion rates in the flow field. As a measure for the validity of this condition in a given flow the ‘vortex compressibility parameter’ is proposed which is derived here. Best ‘visibility’ of coherent structures is achieved by employing visualisation techniques and proper orthogonal decomposition in combination with the introduction of artificial perturbations (forcing of the wake). The existence of both helical and longitudinal structures in the shear layer and of hairpin-like structures in the developing wake is demonstrated. In addition, elongated tubes of streamwise vorticity are observed to emanate from the region of recirculating flow.     

Comparison of different integration schemes based on the concept of characteristics as applied to the ablated blunt body problem

Three different versions of a finite difference code based on the concept of characteristics are discussed and their results are compared in a practical application (inviscid flow past an ablated blunt body in supersonic flow).     

Modified kinetic flux vector splitting (m-KFVS) method for compressible flows

To resolve many flow features accurately, like accurate capture of suction peak in subsonic flows and crisp shocks in flows with discontinuities, to minimise the loss in stagnation pressure in isentropic flows or even flow separation in viscous flows require an accurate and low dissipative numerical scheme. The first order kinetic flux vector splitting (KFVS) method has been found to be very robust but suffers from the problem of having much more numerical diffusion than required, resulting in inaccurate computation of the above flow features. However, numerical dissipation can be reduced by refining the grid or by using higher order kinetic schemes. In flows with strong shock waves, the higher order schemes require limiters, which reduce the local order of accuracy to first order, resulting in degradation of flow features in many cases. Further, these schemes require more points in the stencil and hence consume more computational time and memory. In this paper, we present a low dissipative modified KFVS (m-KFVS) method which leads to improved splitting of inviscid fluxes. The m-KFVS method captures the above flow features more accurately compared to first order KFVS and the results are comparable to second order accurate KFVS method, by still using the first order stencil.     

A viscous inverse method for aerodynamic design

A numerical technique to solve two-dimensional inverse problems that arise in aerodynamic design is presented. The approach, which is well-established for inviscid, rotational flows, is here extended to the viscous case. Two-dimensional and axisymmetric configurations are here considered. The solution of the inverse problem is given as the steady state of an ideal transient during which the flowfield assesses itself to the boundary conditions by changing the boundary contour. Comparisons with theoretical and experimental results are used to validate the numerical procedure.     

Study of subsonic�Csupersonic gas flow through micro/nanoscale nozzles using unstructured DSMC solver

We use an extended direct simulation Monte Carlo (DSMC) method, applicable to unstructured meshes, to numerically simulate a wide range of rarefaction regimes from subsonic to supersonic flows through micro/nanoscale converging–diverging nozzles. Our unstructured DSMC method considers a uniform distribution of particles, employs proper subcell geometry, and follows an appropriate particle tracking algorithm. Using the unstructured DSMC, we study the effects of back pressure, gas/surface interactions (diffuse/specular reflections), and Knudsen number on the flow field in micro/nanoscale nozzles. If we apply the back pressure at the nozzle outlet, a boundary layer separation occurs before the outlet and a region with reverse flow appears inside the boundary layer. Meanwhile, the core region of inviscid flow experiences multiple shock-expansion waves. In order to accurately simulate the outflow, we extend a buffer zone at the nozzle outlet. We show that a high viscous force creation in the wall boundary layer prevents any supersonic flow formation in the divergent part of the nozzle if the Knudsen number exceeds a moderate magnitude. We also show that the wall boundary layer prevents forming any normal shock in the divergent part. In reality, Mach cores would appear at the nozzle center followed by bow shocks and expansion region. We compare the current DSMC results with the solution of the Navier–Stokes equations subject to the velocity slip and temperature jump boundary conditions. We use OpenFOAM as a compressible flow solver to treat the Navier–Stokes equations.     

Compressible laminar flow around a wall-mounted cubic obstacle

The fully-compressible, viscous and non-stationary Navier-Stokes equations are solved for the subsonic flow over a block placed on the floor of a channel. The Reynolds number is varied from 50 to 250. The Mach number is varied between 0.1 and 0.6. In all cases studied the flow field proves to be steady. Several distinct flow features are identified: a horseshoe vortex system, inward bending flow at the side walls of the obstacle, a horizontal vortex at the downstream upper-half of the obstacle and a downstream wake containing two counter-rotating vortices. The shape and size of these flow features are mainly dominated by the Reynolds number. For higher Reynolds numbers, both the horseshoe vortex and the wake region extend over a significantly larger area. The correlation of the position of the separation and attachment point with the Reynolds number has been calculated. Increasing the Mach number (at a fixed Reynolds number of 150) shows its influence in the reduced size, due to compression, of both the wake region and the horseshoe vortex.     

Shock wave reflection from the axis of symmetry in a nonuniform flow with the formation of a circulatory flow zone

Two supersonic axisymmetric flows, in which the incidence of a shock wave on the axis of symmetry leads to the formation of return (circulatory) flow zones, are numerically modeled. The former of these flows is of the wake type with a ram pressure deficit in the vicinity of the axis of symmetry. The numerical results obtained are in qualitative and quantitative agreement with the available experimental data. The latter flow is generated by a hypersonic spherically-symmetric source placed on the axis of a cylindrical channel. An analysis of the obtained solutions, together with their comparison with the data in the literature, suggests that the formation of the above-mentioned circulatory flow zones is physical, rather than computational, in nature. There is also reason to believe that the phenomenon under consideration is hydrodynamic, rather than dissipative, in nature.     

An axis treatment for flow equations in cylindrical coordinates based on parity conditions

A novel axis treatment using parity conditions is presented for flow equations in cylindrical coordinates that are represented in azimuthal Fourier modes. The correct parity states of scalars and the velocity vector are derived such that symmetry conditions for each Fourier mode of the respective variable can be determined. These symmetries are then used to construct finite-difference and filter stencils at and near the axis, and an interpolation scheme for the computation of terms premultiplied by 1/r. A grid convergence study demonstrates that the axis treatment retains the formal accuracy of the spatial discretization scheme employed. Two further test cases are considered for evaluation of the axis treatment, the propagation of an acoustic pulse and direct numerical simulation of a fully turbulent supersonic axisymmetric wake. The results demonstrate the applicability of the axis treatment for non-axisymmetric flows.     

A review and comparative study of upwind biased schemes for compressible flow computation. Part III: Multidimensional extension on unstructured grids

Summary The edge based Galerkin finite element formulation is used as the basic building block for the construction of multidimensional generalizations, on unstructured grids, of several higher order upwind biased procedures originally designed for the solution of the 1D compressible Euler system of equations. The use of a central type discretization for the viscous flux terms enables the simulation of multidimensional flows governed by the laminar compressible Navier Stokes equations. Numerical issues related to the development and implementation of multidimensional solution algorithms are considered. A number of inviscid and viscous flow simulations, in different flow regimes, are analyzed to enable the reader to assess the performance of the surveyed formulations.     

Complex-grid spectral algorithms for inviscid linear instability of boundary-layer flows

We present a suite of algorithms designed to obtain accurate numerical solutions of the generalised eigenvalue problem governing inviscid linear instability of boundary-layer type of flow in both the incompressible and compressible regimes on planar and axisymmetric curved geometries. The large gradient problems which occur in the governing equations at critical layers are treated by diverting the integration path into the complex plane, making use of complex mappings. The need for expansion of the basic flow profiles in truncated Taylor series is circumvented by solving the boundary-layer equations directly on the same (complex) grid used for the instability calculations. Iterative and direct solution algorithms are employed and the performance of the resulting algorithms using nonlinear radiation or homogeneous Dirichlet far-field boundary conditions is examined. The dependence of the solution on the parameters of the complex mappings is discussed. Results of incompressible and supersonic flow examples are presented; their excellent agreement with established works demonstrates the accuracy and robustness of the new methods presented. Means of improving the efficiency of the proposed spectral algorithms are suggested.     

Consistent strongly implicit iterative procedures for two-dimensional unsteady and three-dimensional space-marching flow calculations

First and second-order accurate time consistent versions of the coupled strongly implicit procedure (CSIP) have been developed and investigated for diffusion, potential flow and reduced Navier-Stokes (RNS) equations. Typical examples for the flow over an airfoil and a flat plate at incidence are presented. The method is also applicable to space marching for 3-D flows. Primitive variable forms of the (RNS) and the boundary region equations are considered for low speed flow near the trailing edge of a finite-span plate and for supersonic flow over a cone at incidence, respectively. The composite velocity formulation is considered for flow over a cylinder of rectangular cross-section.