Application of the method of integral relations to boundary layer flows over blunt bodies

Calculations of boundary layer flows past blunt bodies at angles of incidence are presented. Using the method of integral relations together with the method of lines, the full three-dimensional boundary layer equations are reduced to a system of first order ordinary differential equations. The streamwise shear stress function θ and the cross-flow velocity component V are represented as suitable functions of the streamwise velocity component U. The role of the zone of dependence is automatically satisfied by the choice of differencing in the method of lines. Solutions correct to the second order are obtained in the positive shear region for flow over an ellipsoid at 30° incidence. The results are compared with corresponding finite difference solutions.     

Mathematical simulation of turbulent separated transonic flows around the bodies of revolution

The properties of the turbulent separated flows around boat-tailed objects, especially in transonic regimes, are very complex and have still not been fully understood. With a variation of the free-stream Mach number, the flow structure, the size and location of separation areas, internal supersonic regions, and the position and intensity of internal shocks vary significantly. These flow properties determine the complexity of the numerical modeling problem and high demands on the algorithms used. The paper presents a comparison of the numerical results obtained on the basis of different mathematical models with the experimental data. The investigations into fundamental properties of transonic flow transformation are presented as well.     

Simulations of viscous transonic flows over lifting airfoils and wings

In this paper, the hierarchical formulation for steady viscous transonic flow simulations of Refs. [Hafez M, Wahba E. Numerical Simulations of Transonic Aerodynamic Flows. AIAA paper 03-3564, 2003] and [Hafez M, Wahba E. Viscous/Inviscid Interaction Procedures for Compressible Aerodynamic Flow Simulations (in press)] is reviewed and a simplified version for the calculation of the vortical velocity components is presented. The results are compared to available solutions of standard Navier-Stokes equations for laminar flows.     

Some questions concerning the initial fields in finite difference computation of two-dimensional steady transonic flows

To study the influence of initial fields on computation results, the author, starting from a two-dimensional steady transonic small-disturbance equation, has computed the shock wave strengths and lift coefficients for the NACA0012 airfoil by the mixed finite difference relaxation iteration method. The computed results show that if (1) incident flows are supercritical, (2) a nonconservative difference scheme is used to capture shock waves, and (3) the computations are iterated with the initial fields at the lower entropy values (zero values or computed results of flow fields with smaller angles of attack or Mach numbers), then correct converged solutions can be obtained. Otherwise, erroneous converged solutions might be obtained. If incident flows are subcritical and shock waves do not appear in the flow field, then the computed results do not depend on the choice of the initial fields. This means that the converged solution is unique.     

SUPG finite element computation of inviscid supersonic flows with YZβ shock-Capturing

Stabilization and shock-capturing parameters introduced recently for the Streamline-Upwind/Petrov-Galerkin (SUPG) formulation of compressible flows based on conservation variables are assessed in test computations with inviscid supersonic flows and different types of finite element meshes. The new shock-capturing parameters, categorized as “YZβ Shock-Capturing” in this paper, are compared to earlier parameters derived based on the entropy variables. In addition to being much simpler, the new shock-capturing parameters yield better shock quality in the test computations, with more substantial improvements seen for triangular elements.     

Entropy corrections to supersonic conical nonlinear potential flows

Entropy corrections are applied to full potential supersonic conical flows that have the how shock fit as a boundary. The entropy corrections require the implementation of the Rankine-Hugoniot shock relations instead of the isentropic shock conditions. In addition, the pressure must be corrected to account for the bow shock induced entropy variation. For high Mach number and/or large deflection angles, the correction to the potential pressures can be of the same order of magnitude as the Euler pressures. Considering the simplistic nature of the corrections, remarkably accurate results are achieved for circular and elliptic cones. Additional corrections account for embedded crossflow shocks.     

Phenomena peculiar to underexpanded flows in supersonic micronozzles

In the present study, the characteristics of supersonic flows in micronozzles are experimentally and computationally investigated for Reynolds numbers ranging from 618 to 5560. In the experiments, the flows are created in a rectangular contoured nozzle whose heights at its throat and exit are 286 and 500 μm, respectively. The number-density distribution along the nozzle centerline is measured using the laser-induced fluorescence technique under an underexpanded condition for each Reynolds number. The experimental results reveal that the underexpanded flow expands along the streamwise direction in a range where the cross-sectional area of the nozzle is constant although the flow in such a range has been believed to be compressed owing to friction. The results also reveal that the unexpected range where the flow expands extends with a decrease in Reynolds number. In the computations, the Navier–Stokes equations are solved numerically. The computational results agree very well with the experimental results; i.e., the computational code used in the present study is validated by the experiments. By using the computational results, the reason for the appearance of the phenomena peculiar to supersonic micronozzle flows is discussed. As a result, it is found that information about the back pressure under which the flow is underexpanded can reach into the inside of a micronozzle. Such a property induces the unexpected phenomena observed in the experiments.     

A computerized analysis of supersonic nonuniform flows over sharp and spherically blunted cones at angle of attack

A system of computer programs has been developed to predict supersonic inviscid and viscous nonuniform flow fields over sharp and spherically blunted cones at angle of attack. For blunt cones the flow fields considered were axisymmetric wake flows positioned such that the flow in the subsonic nose region remained axisymmetric. For sharp cones, both axisymmetric wake flows and two-dimensional shear flows were considered. The programs used in solving inviscid flow fields incorporate a modified inverse method for solving subsonic flow regions and modified axiymmetric and three-dimensional method of characteristics procedures for solving the supersonic flow regions. Body properties predicted by the inviscid solutions were used as edge data for solution of the corresponding laminar boundary-layers over the bodies. The viscous flow solutions were obtained using axisymmetric and full three-dimensional boundary-layer programs. Typical results from inviscid calculations have shown the development of strong adverse pressure gradients over both sharp and blunt cones in wake flows. In addition a thin entropy layer was found near the surface of both bodies; however, the normal pressure gradient was found to be negligible for the nonuniform flows considered. For the sharp cone in shear flow, property variation along the body was found to be almost linear. In all cases the aerodynamic coefficients were found to be significantly affected by the free-stream nonuniformity. Typical viscous flow field results have shown that relative to uniform flow values the skin friction and heat transfer increase along the windward streamline of both blunt and sharp cones in the nonuniform flows considered. Decreasing the width of the wake in wake flow increases the heat transfer and skin friction.     

Application of the method of integral relations to unsteady transonic flows with shocks

A solution technique for unsteady transonic small perturbations is presented which is based on semi-discrete finite element concepts constructed in time with linear interpolation. A shock searching finite-difference scheme is used in the assembled element equations. The mixed method (finite element in time, finite difference in space) is tested versus a known solution.     

Streamwise computation of duct flows

A computational method is presented to calculate momentum and energy transport in two-dimensional viscous compressible duct flows. The flow in the duct is partitioned into finite streams. The difference equations are then obtained by applying momentum and energy conservation principles directly to the individual streams. The method is applicable to laminar and turbulent flows.     

Behaviour of an immersed boundary method in unsteady flows over sharp-edged bodies

The behaviour of the immersed boundary method proposed by Goldstein et al. [Goldstein D, Handler R, Sirovich L. Modelling a no-slip boundary condition with an external force field. J Comput Phys 1993;105:354-66] as a second-order damped control system is investigated. The natural frequency and the damping coefficient are introduced as driving parameters of the method. The comparison between the velocity response at forced points in the startup flow over a square cylinder with the theoretical response of a second-order damped oscillator is performed. The role of each parameter appears clearly. At the beginning of the startup flow, the response time depends directly on the natural frequency, and this parameter determines the level of residual velocities achieved in an unsteady flow. The damping coefficient drives the oscillation of the velocity response at the beginning of the startup flow, but has negligible influence during the establishment and in the unsteady flow. At forced points facing no unsteady perturbation from the flow, the zero-velocity set point is reached asymptotically, as usual in second-order damped-systems. Through the simulation of the flow over a blunt flat plat at Re=1000, it is observed that the initial thickness of the mixing layer due to the separation at the edge may vary during the simulation because the sharpness of the edge increases as the residual velocities decrease. This insight gained on the behaviour of the response allows a time-step optimisation, which, completed with comparisons to reference literature results, confirms the feedback forcing method a competitive tool for accessing near-wall unsteady flow over sharp-edged bodies.     

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.     

The numerical computation of turbulent flows

The paper reviews the problem of making numerical predictions of turbulent flow. It advocates that computational economy, range of applicability and physical realism are best served at present by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy $\text{k}$ and its dissipation rate ?, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour. The width of applicability of the model is demonstrated by reference to numerical computations of nine substantially different kinds of turbulent flow.     

Streamwise computation of three-dimensional parabolic flows

A new approach to calculate three-dimensional parabolic flows is presented. The flow field is computed by calculating velocity along a set of streamlines. The dependent variables commonly used in the computation of three-dimensional flows are the three velocity components. In contrast, the dependent variables in the present approach are the streamwise velocity and the two coordinates, in the cross-stream plane, of the chosen streamlines. The streamwise velocity is calculated from the finite difference equations obtained by applying Euler's momentum theorem to streamtubes constructed around the chosen streamlines; the streamline coordinates are calculated from the conservation of mass. Results of the calculations, based on the present approach, are compared with the experimental data for flow through rectangular ducts; the agreement is satisfactory.     

A non-mapped complex variable method for the computation of unseparated airfoil flows

A new complex variable method is developed for the analysis of incompressible potential flow over lifting airfoils. The method is built around the numerical solution of a principal-value Cauchy integral equation. Unlike previous complex variable methods, no mappings are required, and the numerical solution is obtained entirely in the physical ($\text{Z}$) plane. Zero normal velocity over the airfoil is satisfied analytically at every point on the surface, not merely at discrete locations. No restrictions are placed on the airfoil contour, and the solution method producers both the potential distribution around the airfoil and the circulation.     

Analysis of unsteady supersonic two-phase flows by the particle-in-cell method

A procedure for analyzing the unsteady flow of gas-particle suspensions, based on the Particle-In-Cell (PIC) method, is presented. Interactions between the phases are incorporated in the Lagrangian portion of the PIC calculation, while discrete ‘mass points’ are used to represent each of the phases of the suspension during the material transport. Test calculations presented to demonstrate the properties of the PIC method as applied to suspension flows include studies of an impulsively accelerated solid boundary moving with constant velocity into a suspension initially at rest, and of suspension flows in shock tubes. Shock tube calculations are also used to demonstrate the need for a fine mesh in order to resolve details of the relaxation zone behind the shock, and to investigate instabilities arising from too large a time increment.     

Parallel finite-element computation of 3D flows

The authors describe their work on the massively parallel finite-element computation of compressible and incompressible flows with the CM-200 and CM-5 Connection Machines. Their computations are based on implicit methods, and their parallel implementations are based on the assumption that the mesh is unstructured. Computations for flow problems involving moving boundaries and interfaces are achieved by using the deformable-spatial-domain/stabilized-space-time method. Using special mesh update schemes, the frequency of remeshing is minimized to reduce the projection errors involved and also to make parallelizing the computations easier. This method and its implementation on massively parallel supercomputers provide a capability for solving a large class of practical problems involving free surfaces, two-liquid interfaces, and fluid-structure interactions     

Operator splitting methods for the computation of reacting flows

If the modeling of the chemistry of a reacting flow system is to consider more than a simple, one-step, global reaction the attainment of a numerical solution of the relevant governing equations is generally jeopardised by the problem of stiffness. In the present work a general approach to overcoming the stiffness hazard is presented. It makes use of operator-splitting techniques, the novel feature being that splitting of the chemical source terms themselves is permitted, with the aid of a new chemical splitting parameter. For the case of elliptic flow with a unimolecular reversible chemical reaction a convergence analysis provides an insight into the conditions under which a converged solution will be obtained. The existence of an optimum value of the chemical splitting parameter indicates the possibility of accelerating convergence. Finally, calculated results for some test problems, involving linear and nonlinear chemical kinetic models, illustrate the viability and usefulness of methods based on the new approach.     

A fine grid finite element computation of two-dimensional high Reynolds number flows

A velocity—pressure integrated, mixed interpolation, Galerkin finite element computation of the Navier-Stokes equations using fine grids, is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions: and the pressure was interpolated using linear shape functions defined on a triangular element, which is contained inside the quadratic element for velocity variables. Comprehensive computational results for a cavity flow for Reynolds number of 400 through 10,000 and a laminar backward-facing step flow for Reynolds number of 100 through 900 are presented in this paper. Many high Reynolds number flows involve convection dominated motion as well as diffusion dominated motion (such as the fluid motion inside the subtle pressure driven recirculation zones where the local Reynolds number may become vanishingly small) in the flow domain. The computational results for both of the fluid motions compared favorably with the high accuracy finite difference computational results and/or experimental data available.     

Numerical computation of transient coaxial entry tube flows

A numerical program was developed to compute transient laminar flows in two dimensions including multicomponent mixing and chemical reaction. The program can compute both incompressible flows and compressible flows at all speeds, and it is applied to describe transient and steady state solutions for low subsonic, coaxial entry, tube flows. Single component, non-reacting flows comprise most of the solutions, but one steady state solution is presented for trace concentration constituents engaging in a second order reaction. Numerical stability was obtained by adding at each calculation point a correction for numerical diffusion errors caused by truncation of the Taylor series used to finite difference the conservation equations. Transient computations were made for fluids initially at rest, then subjected to step velocity inputs that were uniform across each region of the entry plane and were held constant throughout the computation period. For center tube to annulus velocity ratios of 0.5 and 2.0, the bulk fluid in the tube initially moved in plug flow, but strong radial flows developed near the injection plane which moved the fluid into the high shear region between the jets and away from the tube wall. The entrance flows penetrated the bulk flow until steady state was attained. A computation with only the center jet flowing developed a recirculation vortex in the annulus that propagated downstream. The calculation of steady state reacting flows showed formation of a third specie in the mixing zone. Both short and long tube solutions are presented.